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._.
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TRANSLATIONS OF MATHEMATICAL MONOGRAPHS Volume 26
GEOMETRIC THEORY OF FUNCTIONS OF A COMPLEX VARIABLE
by G. M. GOLUZIN
• AMERICAN MATHEMATICAL SOCIETY Providence, Rhode Island 02904 1969 r
~\fpr~it\l
nf lA/aC'hinnf-nn
I :I... .,.",.,.••
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-"'=:':::-'!!!i!!!~!i"o-
672 rEOMETPMqECKAH TEOPMH ~YHKUM~ KOMOflEKCHOrO OEPEMEHHOrO
TABLE OF CONTENTS
r. M. rOflY311H
Page
S/O,%'~'A : note on the auth or """"" r,' EPreface to the second edition I ~ ,.i:~~:: ... . !S;l/ Preface (to the £lrst edit1on) M3AaHRe BTOpOe nOA PeAaK~HeH
...,...
1
,ii.,_
. 3
,j
(.),f/~
~
,...{.., "~
. 4
Introductory geometric considerations ................................................:
B. M. CMHpHOBa
"
Chapter I. The convergence of sequences of analytic and harmonic functions
§l.
113AaTeJlbCTBO "HayKa" rJlaBH(llI PeAaKW1H ~H3"'KO-MaTeMaTHqeCKOli JhiTepaTypbI
•
Translated from the Russian by Scripta Technica
11 .14
§3. The convergence of sequences of harmonic functions
19
§l.
~~a~:i:~~~~.~.~..~~.~.~.~.~~~~~~ ..~~~.~.i.~.~~.~~.~.~~.~.~~ ..~~~~~~.~~.~
23 23
Univalent conformal mapping
§2. Riemann's theorem
25
§3. The correspondence of boundaries under conformal mapping
31
§4. Distortion theorems
47
§5. Convergence theorems on the conformal mapping of a sequence of domains'
54
§6.
621'.
Modular and automorphic functions
'§7. Normal families of analytic functions. Applications
Standard Book Number 821-81576-0 Library of Congress Card Number 70-82894
67
Chapter III. Realization of conformal mapping of simply connected . domains
§ 1.
Copyright (2) 1969 by the American Mathematical Society
I
Printed in the United States of America All Rights Reserved No portion of this book may be reproduced without the written permisswn of the publisher
§2. Parametric representation of univalent functions
76 89
§3. Variation of univalent functions
99
Chapter IV. Extremal questions and inequalities holding in classes of univalent functions
J
110 "
§2. Sharpening of the distortion theorems
'/(
76
Conformal mapping of domains bounded by rectilinear and circular polygons
§1. Rotation theorems
/;; " ... ,.,;.4,
The convergence of sequences of analytic functions
hapter II.
eu
11
§2. The condensation principle
*
MOCKBa 1966
5
iii
.110 118
--------~~._-
g;;g;;---.--.-:==----'-"---~-..~-=::=:::::-=---=-==-===-----~~~~ .......-=----=-----'-=._-----~..,.;.
IV
v
§3. Extrema and majorizations of the type of the distortion theorems
128
§4. Application of the method of variations to other extremal problems §5. Limits of convexity and starlikeness
140 165
§6. Covering of segments and areas
170
§7. Lemmas on the mean modulus. Bounds for the coefficients
182
§8. The relative growth of coefficients of univalent functions §9. Sharp bounds on the coefficients Chapter V. Univalent conformal mapping of multiply connected domains
@
§2. The hyperbolic metric principle
336
§3. Lindelof's principle
339
190
351
196
§6. The hyperconvergence of power series
356
205
§7. A nonanalytic generalization of the Schwarz lemma. A theorem on covering of disks
360
§8. Majorization of subordinate analytic functions
368
210
§3. Univalent mapping of a multiply connected domain onto a helical domain
216
§4. Some relationships involving the mapping functions
222
VI. Mapping of multiply connected domains onto a disk
329
341
§2. Univalent mapping of a multiply connected domain onto a plane with parallel rectilinear cuts
Chapter
-11.AninvariantformOftheSChwarz lemma
, . Harmonic measure. The simplest applications
205
**
329
§5. On the number of asymptotic values of entire functions of finite order
§l. Univalent conformal mapping of a doubly connected domain onto an annulus
§5. Convergence theorems for univalent mapping of a sequence of domains §6. Univalent mapping of multiply connected domains onto circular domains. The continuity method §7. Proof of Brouwer's theorem
Chapter VIII. Majorization principles and their applications
'i,;{ChaDter
IX. Boundary value problems for analytic functions defined on a disk
:.
§1. §2. §3.
228
§4. §5.
234 244
Chapter
X. Boundary questions for functions that are analytic inside a rectifiable contour
254
417
§1. The correspondence of boundaries under conformal mapping
.417
§ 1. Conformal mapping of a multiply connected domain onto a disk
254
§2. Correspondence of boundaries under a mapping of a multiply connected domain onto a disk
262
§3. On the limiting values of Cauchy's integral
430
§3. Dirichlet's problem and Green's function
266
§4. CaUChy's formula
435
§4. Application to a univalent mapping of multiply connected domains
275
§5. Classes of functions. Cauchy's formula
438
277
§6. On the extrema of mean moduli
441
§7. Approximation in mean and the theory of orthogonal polynomials
448
§5. Mapping of an n-connected domain onto an n-sheeted disk §6. Some identities connecting a univalent conformal mapping and the Dirichlet problem Chapter VII. The measure properties of closed sets in the plane
§2. Privalov's uniqueness theorem
428
283
jCbapter XI. Some supplementary information
454
293
.,
§1. Gluing theorems
454
§2. Conformal mapping of simply connected Riemann surfaces
461
§l. The transfinite diameter and Ceby~ev's constant
293
§2. Bounds for the transfinite diameter
300
§3. An extremum for bounded functions in multiply connected domains
467
309 314
§4. The three-disk theorem
;
476
§5. Transformation of analytic functions by means of polynomials
.480
321
§6. On p-valent functions
§3. The capacity of a closed bounded set.. §4 .. Harmonic measure of closed bounded sets
·.. ·
§5. An application to meromorphic functions of bounded form
··
·
,
487
vi
§7. Some remarks on the Caratheodory-Fejer problem and on an analogous problem
§8. Some inequalities for bounded functions :: §9. A method of variations in the. theory of analytic functions
497 514 526 A NOTE ON THE AUTHOR
The scientific works of Gennadil Mihallovi~ Goluzin
545
Bibliography
549
:
Supplement. Methods and results of the geometric theory of functions Introduction
563 563
§l.
Basic methods of the geometric theory of functions of a complex variable §2. Univalent functions in a disk and in an annulus
§3. Functions that are analytic in multiply connected domains
565 577 629
Bibliography for the Supplement
651
Subject Index
673
G. M. GOLUZIN (1906-1952)
'1:4,': The author of the present book, Gennadil Mihanovi~ Goluzin, was born in '\" ';\1' 1906 in the town of Tor~ok. In 1924, he entered the Mathematics Division of ';\~;Leningrad State University. From then until his death, he never interrupted his '\~"',:~ '!~>Cloooection with the University. In 1~29 he defended his graduation thesis, which , ,!~t;iJ,Vas then published in MatematiC'eskil Sbornik. After his graduation from the Uni'ursity, Goluzin began teaching; in 1936 he brilliantly defended his doctoral dissertation and in 1938 he was appointed professor. From 1939 on, he held a chair on the theory of functions of a complex variable at Leningrad State University, "Where, with great energy and devotion, he taught young scientists and also supervised their scientific research. Over a period of years he conducted the basic ,\j~ourse
as well as a number of special courses, and also led a seminar on the
geometric theory of functions of a complex variable. From the founding o( the Leningrad division of the Mathematical Institute of tbe Academy of Sciences of the USSR in 1940, Goluzin simultaneously worked also at that institute, where he carried on extensive scientific activity. The outstanding scientific services of Goluzin were recognized by his being awarded the first prize of the University in 1946 and the Government prize in 1947. The present book is another result of Goluzin's scientific and pedagogical labors. Goluzin died January 17, 1952, about the time of the printing of the first edition, after a long illness. In one of his earlier works, Goluzin used Loewner's parametric representation for the class S of functions [(z) = z + C2z2 + .•• that are regular and unialent in the disk Iz I < 1, to obtain the precise form of the so-called rotation eorem, that is, a sharp bound for 1 arg
f' (z) I
in the class S. For many years,
,atbematicians had been unable to solve this difficult problem. In subsequent works, 'Goluzin continued to develop systematically the method f parametric representations and devised a general method of investigating a ery broad class of extremal problems. 1
--!===.";'--=~~~
2
A NOTE ON THE AUTHOR
3
Goluzin developed his own variant of the method of interior variations, and with it obtained a number of profound results. His variational method became one of the basic contemporary methods for investigating univalent functions. Goluzin obtained extremely powerful results in the problem of estimating the PREFACE TO THE SECOND EDITION
order of growth of the coefficients of univalent functions. In particular, he found a solution, in a certain sense definitive, of the problem of the relative growth of the coefficients of functions in the class 5: If
lenl
the author's death. In the last decade, an extensive literature has appeared on
~ A lna, where A 1 is a constant, a~ 0, and n assumes values equal
to the terms of some arithmetic progression, then
len I s A 2n a for
all n, where
A 2 is a constant depending only on A land a. For the coefficients of functions in this same class, he found the inequality
len I < %en,
The first edition of Goluzin's monograph was published in 1952, shortly after
which was the first advanc e in the problem of the coefficients since
themes closely related to the content of this monograph, and many of these results were obtained in the works of Goluzin's pupils. A survey of this literature is given in a special supplement. The text of the book has undergone only slight modifications, which we shall ;'::j)oint out. In the beginning of the book, the well-known theorems of Runge and Walsh on the approximation of regular functions with polynomials have been omitted
the well-known result of Littlewood (1925). To the pen of G. M. Goluzin belong a number of works in other divisions of
since these are easy to find in the literature on the subject. The theorem on the
the theory of functions of a complex variable and also in mathematical physics.
construction of a mapping function by means of orthogonal polynomials has also
We mention his investigations on conformal mapping of multiply connected domains
been omitted. Additional material has been included in Chapters IV and XI. Speci-
onto domains of the canonical type and his work dealing with the solution of the
fically, a few additional remarks have been inserted at the end of § I of Chapter IV.
Dirichlet problem for Laplace's equation in the case of domains bounded by circles
In § 2 of that chapter, some modification has been made in the arrangement of the
or spheres.
exposition: relations (3 )-(5) from a 1951 article of Goluzin have been inserted
Even a simple listing of the numerous and valuable works of Goluzin would require an excessive amount of space. The reader can get an idea of them by
and Theorem 4 from a 1951 article of Goluzin, have been added. The material at the end of §3 of Chapter IV has been extended on the basis of other works of
reading the present book. Gennadir Mihallovi~ Goluzin was an extremely unassuming man, who IIBde a vivid impression of personal charm and outstanding ability. June 7, 1966
before the formulation of Theorem 1, and a corollary to Theorem 3 of Lebedev,
Academician V. I. Smirnov
Goluzin [1946f, 1947, 1951d] and put in a separate section (§4). This has necessitated a renumbering of the remaining sections of the chapter. Theorems 3 and 4 of Chapter V, § 3 have been moved to §6 of that chapter (where they are numbered Theorems 6 and 7) and their proofs are omitted. Chapt~r IX has been extended by the addition of the four sections 6-9, in which the articles by Goluzin in [1940; 1946a, 1950, 1952) are incorporated almost without change. All the additions not included in Goluzin's works are indicated by the angle brackets
<).
Three bibliographic lists have been added to the book. One of these corresponds to references made in the main text, another to the supplement. In addition, a complete list is given of Goluzin's works. 1965
V. I. S.
----------
~.
----=-
~
_~~=
----l£2iZt&i
'~7Em=
~
4
5
PREFACE
INTRODUCTORY GEOMETRIC CONSIDERATIONS
(to the first edition)
The geometric theory of functions of a complex variable makes a study of
This book is based- primarily on lectures given by the author at Leningrad State University in the course entitled "the geometric theory of functions of a
jlnalytic functions defined by some geometric property or other and a study of "'variOUS geometric properties of certain classes of analytic functions. Therefore,
complex variable" and, in part, in the course entitled "supplementary topics in
it naturally rests on a number of general geometric concepts that are encountered
the theory of functions of a complex variable." It includes an exposition of the
in present-day mathematics. Here we propose to make some brief remarks about
following topics: the theory of a univalent conformal mapping of simply connected
(these concepts, in order of their occurrence, that are associated with the complex
and multiply connected domains, conformal mapping of multiply connected domains
"plane and that we shall need in our subsequent exposition.
onto a disk, applications of conformal mapping to the study of interior and boundary properties of analytic functions, and general questions of a geometric na ture dealing with analytic functions. In addition to various general problems in the geometric theory of functions, many particular problems under study at the present time are also considered. However, the book does not pretend to expound all questions related to the geometric theory of functions or to treat all topics with the same degree of completeness. This would have been an impossible task for the author. Therefore, it is only natural that certain questions in the theory of functions
Sets of points in a plane. We shall usually denote sets of points in the plane by capital letters. We shall denote by lower-case letters individual points in the plane and we shall note the complex numbers corresponding to them by the same letters. If a point a belongs
The book is intended for a reader who has already mastered the fuodamentals of the theory of functions of a complex varia ble as taught in most university Courses. G. M. Goluzin
a set
E, we indicate this fact by writing
a €
E. If
every point of a set E belongs to a set F, we write E C F and we say that E is a subset of F or that E is contained in F. Every point in the plane has a collection of neighborhoods. A neighborhood
have remained untouched, whereas the treatment of others has not been sufficiently complete. In many such cases, the author has given appropriate bibliographic references.
to
of a given point a is the set of all interior points of some disk with center at a (or sometimes any set of points containing such a set). A neighborhood is said to be sufficiently small if the radius of the disk is sufficiently small. The complex plane is supplemented with an ideal point "", called the point at infinity, which is thought of as lying outside every circle. A neighborhood of the point at infinity is the set of all points lying outside some circle. Such a neighborhood
is considered sufficiently small if the radius of the circle referred to is sufficiently great. We note that the distinctive nature of the point "" disappears if We
map the complex plane onto the complex sphere by means of a stereographic
projection. A set of points is said to be bounded if it lies entirely inside some circle. A point a in the complex plane is called a cluster point or a point of accumulation of a set if every neighborhood of a includes points of that set other than a. A cluster point of a set mayor may not belong to that set. A point in a set that is not a cluster point of that set is called an isolated point of the set.
--------.-----
6
---------=-=--=---;----=--
-----------==-------_=--------co----~:_:-----
INTRODUCTORY GEOMETRIC CONSIDERAnONS
__
=::=--~..:::::_~-
==-----=-==-;;;;;~~...::,,=-
__
~=-_~_=_~~--~~
INTRODUCTORY GEOMETRIC CONSIDERATIONS
An infinite set of points always has at least one cluster point. If an infinite set is bounded, this assertion constitutes the well-known Bolzano-Weierstrass
t"'h E k' If a
--=--:::::===
;_=_----~~~~~~--___=_=__
7
set E is contained in a set F, then the set of all points belonging to F
·'that do not belong to E is called the difference of these sets and is denoted by
theorem, and in this case all the cluster points are finite. On the other hand, if an infinite set is unbounded, one of its cluster points has to be the point at infinity.
Set operations have the following properties: The union of a finite number of
For any cluster point a of a set, there exists a sequence of points belonging to the set that converg~s to a. A sequence of points may also converge to the
losed sets and the intersection of an arbitrary combination of closed sets are
~losed sets; the intersection of a finite number of open sets and the union of an
point at infinity. For a sequence to converge to a finite point, it is necessary and
bitrary combination of open sets are open sets.
sufficient that the distance between any two numbers in the sequence, from some
Domains and curves. One of the basic geometric concepts in the theory of
point on, be less than an arbitrary prenamed positive number.
ctions of a complex variable is the concept of a domain. A domain is defined
A set of points is said to be closed if each of its cluster points belongs to it.
s an open set any two points in which can be connected by a broken line con-
Corresponding to any set E is the closed set obtained by inclusion of all the
sisting entirely of points of that set (the property of connectedness). The bound-
cluster points of E. We denote this closed set by E and call it the closure of E.
'.ary points of a domain are those points in the complex plane that do not belong to
The distance between two disjoint sets (that is, without points in common)
"the domain but are cluster points of it. If a domain is other than the entire plane,
is defined as the greatest lower bound of distances between two points, one
it necessarily has boundary points. The set of all boundary points of a domain
belonging to one set and the other belonging to the other set. We know that if
is called its boundary. The boundary of a domain is a closed set. Those points
both sets are closed, this distance is posi tive and the greatest lower bound re-
in the plane that are neither interior nor boundary points of a domain are called
ferred to is actually attained by some pair of points (one in each set).
exterior points of the domain. 1) Every exterior point of a domain has a neighbor-
Another important property of closed sets is given by the well-known HeineBorel covering theorem:
·hood containing no points of the domain.
lf a closed bounded set E is covered by a set of disks such that every point of E lies inside one of these disks, then a finite number of these disks will also co ver the set E.
with this, a domain itself is sometimes called an open domain.
The union of a domain and its boundary is called a closed domain. In contrast A domain is said to be simply connected if its boundary consists either of a continuum or of a single point or if the domain itself is the entire complex plane. Otherwise, a domain is said to be multiply connected. Specifically, it is said to
A closed set consisting of more than one point is called a continuum if it
be
cannot be decomposed into two nonempty disjoint closed sets. We shall call a
doubly, triply", " n-connected if its boundary consists of 2, 3" •• , n dis-
'joint continua (possibly degenerate). All these regions are said to finitely COn-
set consisting of one point a degenerate continuum. A point is said to be an interior point of a set to which it belongs if some
",ected and the continua (including the degenerate ones) are called boundary I
.
"contmua.
neighborhood of it is contained in that set. A set consisting only of interior points is called an open set. Obviously, the complement of a closed set in the complex plane is an open set and vice versa. Let us recall the definitions of union, difference, and intersection of sets. Let E l' E 2 , ' " denote sets (either finitely or infinitely many). The set of points belonging to at least one of these sets is called the union of these sets and is denoted by E 1 U E 2 U ... or U kE k' The set of points belonging to all the sets E 1> E 2, •• " is called the intersection ofthese sets and is denoted by E 1 E 2 or
The role of domains in the study of closed and open sets is clear from tht; following theorem:
Every open set E in the complex plane is the union of finitely or countably . many domains. Another basic geometric concept in the theory of functions of a complex :variable is that of a curve.
n n· ..
1) These concepts are meaningful for any open set.
8
INTRODUCTORY GEOMETRIC CONSIDERA nONS
INTRODUCTORY GEOMETRIC CONSIDERATIONS
A continuous curve is a set of points in the plane the rectangular coordinates x, y of which can be written as continuous functions
x=r,o(t),
y=~(t)
of a real variable t in some finite interval set is a continuum.
as
t
~ero derivative z' (t) everywhere in [a, b] (a one-sided derivative at the two
'ndpoints). The requirement of smoothness is obviously equivalent to the requireot that the curve have everywhere a continuously turning tangent. A curve con-
(1)
S b.
isting of finitely many smooth curves is called a piecewise-smooth curve. The
It is easy to see that this
iefinition of a rectifiable curve will be given in Chapter X. Finally, the simplest e of continuous curve is an analytic curve. This is a curve defined by an
However, the concept of a continuous curve is too general for our purposes. iMi
ation of the form z = z (t) for a
There are continuous curves that do not at all correspond to our intuitive idea of a curvt: as a one-dimensional figure. In fact, it is possible to construct a continuous curve that the curve have no multiple points, it will possess a number of clear-cut properties. Such curves are called simple curves or Jordan curves.
+ i~ (t),
a ~ t~b,
where z (t) can be expanded in a
Let us now look at certain questions dealing with domains.
Thus, the continuous curve (1) or, more briefly, the curve
z (t) =
S t S b,
Z(t)=CO+Cl(t-tO)+ ... , 'ith c 1 ,f. 0, about each value to in [a, b].. A continuous curve consisting of a ite number of analytic curves is called a piecewise-analytic curve.
that passes through every point of a given square. On the other hand, if we require
Z'=z (t),
9
Sometimes we make cuts along various Jordan curves in a domain._ Let B enote a domain. To make a cut along a Jordan curve L contained in B means
(2)
is called a Jordan curve if, for any two distinct values t' and til in the interval [a, b) with t' < t", we have z(t'),f. z(t") and z (til) ,f. z(b). The points z(a) and z(b) mayor may not coincide. In the first case, the curve is called a closed, in the second case a nonclosed Jordan curve.
,.~o delete from ~ng
B all points of the curve L. We shall consider only cuts consist-
of finitely or infinitely many sets of anal ytic arcs. We shall consider the infi-
nite case under the assumption that the endpoints of these arcs cluster only at tbe boundary of the domain. A cut in a domain B is said to be transverse if it
We have the following important theorem (due to Jordan):
Connects two (not necessarily distinct) boundary points of the domain B, which
A closed Jordan curve C partitions the plane (including 00) into two simply
serve as its endpoints and if all its other points lie inside B. It turns out that
connected domains both of which have C as their boundary. One of these domains is bounded and is called the interior of C. The other contains "" and is called the exterior of C. The complement of a nonclosed Jordan curve C consists of a: single simply connected domain containing' "" and having C as its boundary. 1)
if a
transverse cut in a finitely connected domain connects boundary points on ,different boundary continua but does not partition the domain into two parts, it decreases the connectedness by 1; on the other hand, a transverse cut in a : simply connected domain always partitions that domain into two simply connected
From nonclosed Jordan curves, one can construct continuous curves that are
,.parts. (This is a characteristic property of simply connected domains.) Analo,;,JOUSly, if a cut is a closed Jordan curve lying entirely in a domain B, it is called
not of the Jordan type. On the other hand, even a Jordan curve is sometimes too general. Then, depending on our purpose, we introduce curves of more restricted types, for example, smooth, piecewise-smooth, and rectifiable curves.
." circular cut. A circular cut always partitions a domain B into two domains. In »he case of a simply connected domain B, one of the domains bounded by a circular cut lies entirely in B (another characteristic property of simply connected J~mains). Finally, a cut that is an open Jordan curve lying entirely in a domain except possibly for one of its endpoints does not partition B into two parts. ;,:i'i' We note that, in a finitely connected domain B, it is always possible to make a
A curve (2) is said to be smooth if the function z (t) has a continuous non-
,il
1) Whereas the proofs of the assertions made up to now are extremely simple and the reader can reproduce them himself, the proofs of Jordan's theorem and of a number of assertions that we shall make below present considerable difficulty. The interested reader should consult special texts. With regard to Jordan's theorem itself, we note that there are two new and comparatively simple proofs of it due to Yol'pen [1950] and Filippov [1950].
~71!;'
transverse cut connecting points of two given boundary continua (that do not . degenerate to "") and also a circular cut encircling a given bounded boundary continuum but no other boundary points. Moreover, if a closed bounded set E is
'
,~y .. .'>~,. . • ,:t..,,"..... .
.~
;:~;':.
10
INTRODUCTORY GEOMETRIC CONSIDERATIONS
contained in a domain B, then one can make a circular cut in B that separates E from the boundary of B, that is, a cut that encircles E but does not touch any part of the boundary of the domain B. These results can be achieved by means of cuts consisting of a finite number of rectilinear segments.
CHAPTER I
THE CONVERGENCE OF SEQUENCES OF ANALYTIC AND HARMONIC FUNCTIONS
§l.
The convergence of sequences of analytic functions
Many divisions of the theory of functions of a complex variable, and in smrticular that division known as the geometric theory of functions, employ in their proofs the basic properties of the convergence of sequences of analytic '::functions. Thanks to these properties, such proofs are fairly simple and elegant
,in comparison with the analogous proofs of real analysis. Let us give some definitions. Let Ifn (z)l, for n = 1, 2," " (;sequence of single-valued functions defined on a set
denote a
E of points in the z-plane.
This sequence is said to converge at a point Zo € E if the sequence of numbers In (zo) converges. A sequence of such functions fn (z) is said to converge on E ','1£ it converges at every point of E. In sucn a case, we may speak of the limit iJunction f(z) = limn->oofn (z) defined on E. The sequence Ifn (z)l is said to converge uniformly on E to a function f(z), which is finite on E, if, for every t > 0, there exists an N > 0 such that the inequality n > N implies Ifn(z) - f(z)\ <( 'for every z € E. On the other hand, if f(z) 00 on E, the sequence Un (z)l is ~:$aid, by d,efinition, to converge uniformly on E to 00 if, for every M> 0, there ~:exists an N> 0 such that the inequality n> N implies Ifn (z)\ > M for all z € E.
=
'It is easy to show that a necessary and sufficient condition for uniform conver-
(> 0, there exist an N> 0 such that, for all m, n> N and all z € E, Ifm(z) - fn (z)! < (.
,Jence of a sequence to a finite function is that, for every
If the functions fn (z) are defined on a domain B, we shall need, besides " the concept of uniform convergence of a sequence in the domain B, the concept of uniform convergence of a sequence in the interior of the domain B; this means 'uniform convergence of Ifn (z)! on every closed set E C B. Uniform convergence
11
- - - - - - - - - - - -............""'....= ....= ....--==='-iil_~=;;;;;;;=;;;-;o"iOiREiiiiiii..-==.==·ii'iifiii~ ..,.:,---~T~
12
I. SEQUENCES OF ANALYTIC AND HARMONIC FUNCTIONS
§l.
in the interior of B is a weaker requirement than uniform convergence in B.
A single-valued function f(z), defined and finite on a set E not including is said to be continuous on E if, for an arbitrary point z 0 €
°
B_
==~
-
-·-~E
azam
00
=
1, 2; .•• , .
!1I(z)-!o(z)=_1 \ fn(z')-/(z')d • 21ti ~
E and arbitrary
( > 0, there exists a 0 > such that the relations z € E and Iz - z 0 I < 0 imply If(z) - f(zo)!
13
TIlE CONVERGENCE OF SEQUENCES OF ANALYTIC FUNCTIONS
is regular in K. From (1) and (2), we have, for n z' -
__
Z
(3)
Z ,
l'
and similarly
(4) (k) (z) _ £(k) (z) =.!!... \ In (z') - I (z') dz'. n J0 211/ J (z' _ Z)Hl I f a sequence Ifn (z)! of functions that are continuous on a set E converges uni
l' formly to a finite function f(z) defined on E, this function f(z) is also continu
But the sequence Ifn(z)! converges uniformly on y' to [(z). Consequently, for ous on E. To see this, let z 0 € E. Then, for given ( > 0, there exists 2. number
given ( > 0, there exists an N> 0 such that, for n> Nand z' on y', we have n such that, for all z € E, we have If'I (z) - f(z)1 < d3. Furthermore, there exists
;',Ifn (z') - f(z')\ < (. In view of this, we obtain from (3) and (4) the result that, for a number 8:> 0 such that, for all z € E satisfying the inequality Iz - zol < 0,
n> N, we have If'l (z) - fn (zo)1 < d3 (because of the continuity of fn (z) on E). There
r (5) fore, for z € E and Iz - zol < 8, we have
If~) (z) - !~k) (z) (r' kl~F € , lin (z)--!o(z) I
t.
1<
I/(z) - I (zo) I ~ I/(z) - In (z)
I+ lin (z) -
In (zo) I 1/11 (zo) - !(zo) I
+
which means that f(z) is continuous at the point Zo €
where rand
< e,
E. It then follows that,
r'
ties means that j
are the radii of the disks K and K'. The first of these inequali
1['1 (z)l
converges in K to the function
f 0 (z),
which by hypo-
thesis, must coincide with fez). Consequently, fez) is regular inside K. But K
if rhe functions fn (z) are continuous in the domain B and if a sequence of them
is an arbitr~y disk contained in B. Therefore, fez) is regular in B if B does
convetges in the interior of B to a finite function f(z), then f(z) is continuous in B.
not include
In the case of analytic functions, we have the following fundamental theorem of Weierstrass.
00. Thus, the second of inequalities (5) with [o<"k)(z) replaced by [(k)(z) shows that the sequence If~k)(z)~ n = 1, 2,"·, converges uniformly in K to f(k) (z) because the right-hand member can, by a suitabfe choice of (, be made arbitrarily small for all z in K simultaneously. To prove the unifocm con vergence of If~k) (z)l in the interior of B, we note that every closed bounded set
Theorem 1. If a sequence of functions fn(z) that are regular in a domain B E C B can be covered by a finite number of disks contained, along with their converges uniformly in the interior of B to a finite function [(z), then fez) is boundaries, in B. Specifically, for every point c € E, there exists a closed disk regular in B and each sequence {f"k)(z)l of derivatives (for k = 1, 2,,") con with center at c and contained in B. The set of these disks (that is, the disk verges uni[ormly in the interior o[ B to [(k)(z). corresponding to all c € E) completely covers E. By the Heine-Borel theorem, Proof. Let K denote any closed disk contained in B and let K' denote a there exists a finite collection of these disks which also covers E. Let us denote closed disk of greater radius, also contained in B, that is concentric with K. these dis~s by K m for m = I, 2,·, . , mo. Then, by what we have proved, for any Let y' denote the boundary of K'. Then, from Cauchy's formula, we have, for l > 0 and each m = 1, • " , mo, there exists a number Nm > 0 such that for z € Km z € K, / and n> N m we have If~k) (z) - r
N, the inequality I[~k) (z) - [(k)(z)1 < ( T' l' holds for all points in the circles K m and hence for all z € E; that is, the se k=l, 2, ... quence I[~k) (z)! converges uniformly in E. If B does not include .,., the theo On the other hand, since f(z) is continuous in B, the function rem is proved. If the domain B does include 00, we still need to show that fez) !O(Z)=_l \ L
I z I ). R,
where R is sufficiently large. This can be shown in the same way by
14
Iz I > R
using Cauchy's formulas for
rains a convergent subsequence. Several proofs in analysis are made more compli
(that is, formulas (1) with the first formula
applying also to fn (00» and considering the corresponding function completes the proof of the theorem.
fo (z).
cated by the fact that this is not always the case. Not every sequence of functions
This
defined, continuous, and uniformly bounded on a given interval contains a conver gent subsequence, for instance the sequence Isin
As regards uniformly convergent sequences of regular functions, we shall
2, ...•
the answer to the above question is affirmative. As a preliminary to our discus
Theorem 2. Let If.. (z) I, where n = 1, 2"", denote a sequence of functions
sion, let us adopt some conventions on terminology. A family of functions
that are regular in a domain B. Suppose that Un (z)l converges uniformly in the
i
net>!, n = I,
For an extremely broad class of sequences of analytic functions, however,
prove another theorem, which has numerous applications.
interior of B to a regular function fez)
15
§2. THE CONDENSATION PRINCIPLE
I. SEQUENCES OF ANALYTIC AND HARMONIC FUNCTIONS
V(z)} defined on a subset E of the z-plane is said to be uniformly bounded on E
const. Suppose that each of the functions
fn (z) assumes a given value a at no more than p (p ~ 0) points of the domain B.
;,.if there exists a finite positive number M stich that the inequality If(z)
Then the function f(z) assumes the value a at no more than p points of B.
:every function in the family and every z €
done. Under these conditions, there exists an m> 0 such that If(z) all the neighborhoods
n.
al
> m on
Since the sequence lfn (z)l converges uniformly on the
circles Yk, there exists an n such that Ifn (z) - f(z)1 •. " p + 1. Since
<m
on each Yk, k
= 1,'"
E. If the functions f(z) are defined in
the interior of a domain is a weaker requirement dian uniform boundedoess in a domain.
ficiently small in each case that the circles do not touch each other and there const, this can be
for
!l! is said to be uniformly bounded in the interior of B. Uniform boundedness in
B, k = 1, 2"", P + 1 Around each of the points zk' k = 1,"', P + 1, let us draw a circle Yk C B suf that f(z) assumes the value a at p + 1 distinct points z k €
i
< M holds
a domain B and are uniformly bounded on every closed set E C B, then the family
Proof. Let us suppose to hegin with that B does not include "". Suppose
are no zero of the functions f (z) - a on them. Since f(z)
m=
"
Theorem 1 (the condens.ation principle). Let Ifn (z)l, n = 1, 2,' . "
denote
if a sequence of functions that are regular in a domain B. Suppose that the sequence is uniformly bounded in the interior of B. Then this sequence lfn (z)l contains a subsequence that converges uniformly in the interior of B to a regular function.
Proof. The proof consists of four pans.
fn (z) - a=(j(z) - a)+ifn (z) -fez)~
1°. Let us show first that the functions fn (z), n
=
I, 2, .. " are equicontinu
> 0,
we conclude on the basis of Rouche's theorem, that the function fn (z) - a does
ous on any given closed bounded set E C B. This means that, for every
indeed have zeros inside each circle Yk since the function f(z) - a does. Con
there exists a 0> 0 such that, for every pair of points z 1 and z 2 in E with
f
sequently fn (z) assumes the value a at no fewer than p + 1 points of the domain
IZI - z21 < 0,
B, which contradicts the hypothesis of the theorem.
us denote by 01 the (positive) distance between the set E and the boundary of
If the domain B includes
00
but is not the entire plane, we can choose a
suitable fractional linear function to map it onto a domain not including
00,
and
apply to the image functions the result just proved. Finally, the case in which the domain B is the entire z-pIane is excluded because, in such a case, the function f(z) must necessarily be constant. This completes the proof of the theorem.
§ 2.
the inequality Ifn(zl) - fn(z2)!
the domain B. Let E 1 denote the union of E and the set of all points whose tained in B. The sequence Un (z)l is uniformly bounded on it, let us say, by some number M. Let ZI and Z2 denote two points in E such that IZI - z21
<
0 1/4. Let K denote the closed disk of radius 0 d2 with center at z l ' This disk together with its boundary Y is contained in E l' In accordance with Cauchy's formula, for n
=
1, 2,' .. ,
f n (z 1)=~~ In (z') 21t1'
In many branches of mathematic,l1 analysis in which existence questions a given sequence of functions all defined on the same domain of definition con
holds for all n = 1, 2,···. Let
distance from E does not exceed od2. This set E 1 is closed and it is con
The condensation principle
are answered, it is important to have an affirmative answer to the question whether
Z -Z1
Cons equently,
T
dz'
,
f n (z ~)=~ ~ In (z') 21t1' Z
T
-zs
dz' •
17
§2. THE CONDENSATION PRINCIPLE
I. SEQUENCES OF AN AL YTIC AND HARMONIC FUNCTIONS
16
contained in B. Since the set of functions fn,n(z) is equicontinuous on K, it
IIn (Zl) - In (Zg) I= 12~l '~/n (Z') (z' _Z;I)(:~ Z2) dz'l
follows that, for given
f
T
&1
1/(') ZI- Z n Z (. _ ) ( '_
~ -2 max z' ET
that is,
2
Z
ZI
Z
&IMlzl-Z21 4M ) I ~ -2 ~ & = -.- 1Zl Z2 -.!. • -.!. Ul 2 4
4M
lin (Zl) - In (Zg) I.:;; -~ I Zl 1
Zg
I
Zg ,
> 0, there exists a 0 > 0 such that the inequality lin. n (Zl) - In. n (Zg)
\;'1
holds for arbitrary z l ' z 2 €
K such that Iz 1
1< ; -
(1)
z 21 < 0 and for all n. Let us 012. In each closed square con
cover the z-plane with a grid of squares of side
taining points' of K, let us choose arbitrarily a point with rational coordinates
1·
Ji
belonging to K. Let us denote these chosen points, of which there are finitely many, by rl' r2"", rp ' Since the sequence Ifn,n(z)} converges at these points,
Therefore, if 0 is the smaller of the two numbers 0 1/4, 01f!4M, then
for the same
Ifn(Z1)-fn(Z2)!
f
> 0 as in
(1) there exists an N
> 0 such that the inequalities m,
n> N imply lim. m(rk) - In. n(r k)
. 2°. Let us show that the sequence {fn (z)}, n = 1, 2, ... , contains a sub
1< ; .
(2)
sequence that converges at all points of B with rational coordinates. Since such·
for all k = 1, 2," . , p. Now, let z denote an arbitrary point in K. It lies in one
points constitute a countable set, they can be arranged in a sequence rl' r2, • ".
of the (closed) squares of the grid that contain points belonging to K. Its distance
From the bounded number sequence Ifn (r1) ~ n = 1, 2,' . " let us choose a con
from the point rk (in the same square) does not exceed
vergent subsequence and let us denote by Ifn,l(Z)I, n = 1, 2,""
from (1) we have
the corre
sponding sequence of functions. This sequence converges at the point z = rl' From the bounded number sequence Ifn,1 ('2) I, let us choose a convergent sub sequence and let us denote the corresponding sequence of fuOctions by Ifn,2(Z)~ where n = 1, 2, .. '. This sequence obviously converges not only at z = r2 but
11m. m(z)-Im.m(rk)
01..;2 < o.
1< ;,
Therefore,
I/n.n(z)-ln,n(rk)I<;. Adding the three inequalities (2) and (3), we obtain, for m, n > N,
(3)
I/m.m(z)-ln. n(z)l<e.
also at z = r1 (because it is a subsequence of the convergent sequence Ifn,l (z) D.
Since N is the same for all z €
Proceeding as before, we choose from the function sequence Ifn,2 (z)) a subse
verges uniformly on K to a finite function and by Theorem 1 of
quence {fn,3(Z)}, n = 1, 2," " that converges at the points z = r1> '2, '3' Proceeding in this way, we obtain sequences Ifn,k(Z)~ n = 1, 2,"', that con
is regular in B. If the domain B does not include tion principle.
verge at all the points rl' r2,'" , rk' The diagonal sequence Ifn, n (z)l, where n = 1, 2,"" is contained, except for a finite number of its initial terms, in each.
with a proof that the same sequence I fn,n(z)} converges uniformly outside some
of the sequences Ifn,k(Z)} and hence it converges at each point '1' r2," " is, at all points of the domain B with rational coordinates. 1)
that
3°. Let us show that the diagonal sequence {fn, n(z) 1 described above con
verges uniformly on an arbitrary closed bounded set E C B. By virtue of the
K, this means that the sequence Ifn,n(z)l con § 1 this function
R. Since the sequence Ifn,n(z)l converges uniformly on [zl = R to a f n, n ( 1/z) which are regular on the disk \z\ ~ 11R converges uniformly on Izi = 11R. Consequently, for given f > 0, there exists an N > 0 such that the inequalities m, n > N imply circle Izi =
finite function just as before, the sequence of the functions
J-tnt, m (; ) lIn n(z)1
converges
OIl
some count
sane points of which are included
this proves the condensa
4°. If the domain B does include "", the above proof must be supplemented
Heine-Borel theorem, we need only prove this for an arbitrary closed disk K
1) For what follows, it is essential 001y that the sequence able set of points that is everywhere dense in B, that is, a set in every neighborhood of any point z € B.
00,
zl
In. It (
~ I< e. )
11R. In accordance with the maximum modulus principle, this will also be true in the open disk Iz1 < 11R; that is, the sequence Ifn,n(1lz)l converges uniformly on the closed disk Izi ~ 1/R. Consequently, the sequence Ifn,n(z)l for
1
=
___
R~
--
-~-
-
.-=~---
~-
19
§I. THE CONVERGENCE OF SEQUENCES OF HARMONIC FUNCTIONS
18
I. SEQUENCES OF ANALYTIC AND HARMONIC FUNCTIONS
converges uniformly on the set
Izl 2: R
It should be noted that the condensation principle itself can be expressed in to a finite function. This completes the
a different way, in which it takes its final (orm:
proof of the condensation principle. By using the condensation principle, we can derive the following convergence
=
"uni{ormly bounded in the interior o[ B.
I, 2,···, denote a sequence o[ [unctions
interior o{ B and that it converges on a set o{ points z" € B, k
The sufficiency of the condition follows from what was proved above. Let us
= I, 2, . ",
'E
B.
Proof. Let us suppose that the sequence {fn(z)} does not converge at some
[ n. 2 (a)
-->
A' as n
--+
00.
I, 2,···. Thus we have [n.l(a)
The sequences {fn, I(Z)} and
I [n.2(Z) I
-->
maxz€ E l[n(z)l-" "" as n
--+ "".
I, 2, ••. , such that [n (z n)
--> 00
I[nk(z) - [(z)1
A and
I, 2," "
I r:..2 (z)l,
for n
§ I,
=
= I,
[i (z),
<1
for all z € E, and consequently
--> ""
as k
--> 00.
This
regular functions when we are dealing with the theory of normal families (§ 7,
2, ...), these
Chapter II). §3. The convergence of sequences of harmonic functions
which is regular in the domain B, must accordingly
The theorems proved above for analytic functions have their analogues for
vanish on any set of points that has a duster point in the interior of B. Therefore, it vanishes identically in B. But this contradicts the fact that this function is non Zero at the point a. Thus, the sequence
that converges uniformly in the interior of the domain B
of choosing convergent subsequences from various sequences of a given family
two s"equences converge to the same limit, with the result that [:.1 (z ,,) (:jzk) --+ 0 as n -> 00. Therefore, [!(zk) - [; (zk) = 0 for k = I, 2,·· '. The function [~(z) -
=
We note that we shall return later to questions dealing with the possibility
and hence fiCa) ~ n(a). But since the sequences If:.l(z)l and 1[~.2(z)l are (k
E, for n
But Un (z) I contains a subsequence
completes the proof of the necessity.
these functions are regu
Z"
--> 00.
However, this is incompatible with the fact that [nk(znk)
larin B. Since [:,l(a)-> A and [~.2(a)-A', we have n(a)=Aand[i(a)=A' subsequences of Un (z)}, it follows that, for any point
Then there must exist points zn € as n
zEB
that converge uniformly in the interior of B to finite limit functions
[~(z) and [i(z). In accordance with Theorem 1 of
IJJl such that
link (Z)! < I +max I/(z) \ =M
are uniformly
bounded in the interior of B. Therefore, in accordance with the condensation principle, these sequences have subsequences l{~,l(z)l and
not uniformly bounded on some closed set
there must exist a sequence Un (z)l of functions {n (z) €
to some regular function fez). In particular, from some k on we have
converge to distinct limits A and A'. Let us denote these subsequences by =
c B,
{fn k(z)l, k = I, 2, .. "
point a € B. Then, the number sequence I[n (a)} contains two subsequences that Un.I(Z)! and l[n.2(z)I, where n
mis
prove its necessity. If the family
that
has a cluster point in the interior o{ B. Then, the sequence I[n(z)! converges 0[
an infinite family
"o{ B to a regular [unction, I) it is necessary and sufficient that the {amily IJJl be
that are regular in a domain B. Suppose that {fn (z)! is uniformly bounded in the
uni[ormly in the interior
m denote
these [unctions to contain a subsequence that converges uniformly in the interior
test for sequences of regular functions. Theorem 2 (Vitali). ,Let {fn (zH, n
Let
o[ [unctions fez) that are regular in a domain B. For an arbitrary sequence o[
harmonic functions. We' recall that a harmonic function in a given domain B of the complex plane
lfn (z)l must converge everywhere in the
domain B. The uniformity of this convergence in the interior of B now follows
is a real single-valued function u (x, y) with continuous second derivatives that satisfies Laplace's equation ~u
2
shown implies that the sequence Un (z)l converges at all points of the domain B
+ a u/J y 2 = O. (We note that if B includes "", then u (x, y) must be bounded in a neighborhood of 00.) If u (x, y),
with rational coordinates. This completes the proof of the theorem.
or more simply u (z), is a harmonic function in the open disk
from 3° and 4° of the proof of the condensation principle since, what has been
One can easily see that uniform boundedness of the functions [n (z) in the interior of B is also a necessary condition for the validity of the assertion in Vitali's theorem. This is true because the limit function fez) is bounded on an arbitrary closed set E
cB
and hence the functions [n (z) are unifonnly bounded on E.
"~'" 'I
= a2 u/ax 2
Iz -
a\
< Rand
w ~" 1:3'
;r:;
1) This property is called the property of compactness of the family IJJt and the
principle stated is called the compactness principle.
20
I. SEQUENCES OF ANALYTIC AND HARMONIC FUNCTIONS
continuous in the closed disk
Iz - al
s R,
§l.
=2~ ~ II (z)at (z' ~~:2a) dO,
Then it contains twO subsequences that converge in B to distinct limit functions
z' =a+Re i9
where lR means "the real part of", is valid in the open disk Iz
-al < R.
(1)
u(z) and u'" (z) that are harmonic and that coincide in the domain B' . Conse quently, their difference in B' is zero. But then, the difference of their harmonic
This
conjugates v(z) and v* (z) must, by the Cauchy-Riemann conditions, be constant
formula reduces to a more familiar one if we note that, by setting z = a + re i ¢, we get
at (Z' + z - 2a) _
z'-z
-
R~ - r· R2-2Rrcos('!'-fl)+r B
,in B' . 1) This in turn implies that the corresponding analytic functions whose '-'real parts are u(z) and u* (z) coincide in B ' . But these analytic functions will 2
'
On the other hand, if u (z) is a harmonic function in the domain is continuous in Iz - al ~ R, then the analogous formula
(
then coincide throughout the entire domain B. Therefore, the supposedly distinct )
"',harmonic functions u(z) and u* (z) must coincide in B. This contradiction
Iz _ al > Rand
',proves the convergence of the sequence {un (z)l in B. That the convergence is ~niform in the interior of
, 2"
1l(Z)=-2~ ~ U(z')m(Z't~:2a)de is valid in the domain
21
Proof. Suppose that the sequence lUn (z)l does not converge in the domain B.
2"
ll(Z)
THE CONVERGENCE OF SEQUENCES OF HARMONIC FUNCTIONS
then Poisson's formula
B now follows from 3° and 4° of the proof of the con
densation principle for analytic functions. (3)
We conclude this chapter with another important theorem.
Iz - al > R.
Theorem 4 (Harnack). Let {un (z)l, n
=
1, 2"", denote a sequence of
Since harmonic functions can be regarded as functions of a complex variable z, the definitions of convergence given in § 1 can be applied to them. Further
fit functions that are harmonic in a domain B. Suppose that, for each z € B, the
{' sequence lUn (z)l is a monotonic increasing sequence (that is, un +l(Z) 2: un (z),
more, using the same line of reasoning as in §1 for the proof of the Weierstrass theorem, except that instead of Cauchy's formula we use formula (1) or (3), we can prove the following theorem.
" for n
1, 2, ... ). Then, the sequence {un (z)} converges uniformly in the inte
rior of B either to a harmonic function or to the function with constant value + 00. Proof. We may assume that all the functions un (z) are nonnegative since
otherwise we would consider not the functions un(z) but the functions un(z)
Theorem 1. Let {un (z)l, n = 1, 2,· • "
denote a sequence of functions that are harmonic in a domain B. If {un (z)J converges uniformly in the interior of B
to a finite function, then the limit function is also harmonic in B.
u 1 (z) for n = 1, 2, ....
ill
r~ular functions is no longer valid. For example, the sequence defined by U2k (z) = 0 and u2k -1 (z) = y (imaginary part of z) for k = 1, 2, .• " converges on the real axis but nowhere else. However, we do have Theorem 3. Let {un (z)} denote a sequence of functions that are harmonic and uniformly bounded in the interior of a domain B. If {un (z)} converges in Some subdomain B ' of B, it converges uniformly in the interior of B.
Iz - a\
always have
r' - r 9 ",:::::: at (Z' + z - 2a ) B
that are harmonic in a domain B. If {un (z)} is uniformly bounded in the interior of B, it contains a subsequence that converges uniformly in the interior of B.
In contrast, Vitali's theorem in the form in which it was given in §2 for
Iz - al S r contained in the domain B and a S r ' also contained in B. Since, in accordance with (2), we
Consider an arbitrary disk larger disk
Furthermore, by using the same line of reasoning as in §2, we can prove Theorem 2. Let lun(z)}, for n = 1, 2" • " denote a sequence of functions
=
-~ ~
I..
, whenever
Ii' - a\ :.: r'
and
z' _ z
Iz - a\ S r,
applied to the function Un (z)
2
~
r' - r·
Ir'
_ .. \2 ,
it follows on the basis of formula (1) as
(taking R = r ' ) that
r'-r
r'+r
lln(a)r'+r E;l1n(Z)~lln(a)r'r' Now, let us set u(z)
=
(4)
limn->ooun(z). Because of the monotonicity of the B. Inequality (4) shows that, if
sequence {un (z)l, this limit exists for z €
1) The conjugate functions v(z) and v*(z) may not be single-valued in B, but this fact is of no significance in the present cas e.
...
~~~~~~,=====
22
,,",-~;s=::~~~~z:~_.~.
'·!Z2
I. SEQUENCES OF ANALYTIC AND HARMONIC FUNCTIONS
u (a) is finite for a € B, then the functions un (z) are uniformly bounded in a sufficiently small neighborhood of the point a, and, if u (a) = +"", then IUn (z)} converges uniformly in that neighborhood to +00. Consider the set of all points in the domain B at which the function u (z) is finite and its complement, the set of points' at which u(z)
=
CHAPTER II
+"". From what has been said, both these sets are open.
But obviously, this can be true only when one of the two sets is empty. If the
\,{ 'I'.
function u(z) is everywhere finite in B, then the functions un (z) are uniformly
THE PRINCIPLES OF CONFORMAL MAPPING OF SIMPL Y CONN ECTED DOMAINS
bounded in the interior of B. By Theorem 3, the convergence of IUn (z)1 to U (z) is uniform in the interior of B, and by Theorem I, the function u (z) is harmonic in B. On the other hand, if u(z) = +"" in B, then Iun(z)} converges to +00 uni
§ 1.
formly in the interior of B. We note that if the domain B includes the point 00, then we must, as usual, use formula (3). This completes the proof of the theorem.
Univalent conformal mapping
One of the basic questions in the theory of functions of a complex variable is the srudy of analytic functions based on the nature of the mappings produced by these functions. Suppose that a function ( = f(z) is regular in a domain B except possibly at certain points at which it has poles (in which case the funcI
tion is said to be meromorphic in B). Then we, say that the function f maps the domain B in the z-plane univalently onto some domain B' in the (-plane if it sets up a one-to-one correspondence between points of the domains B and B' . The domain B' is then called the image of the domain B under the mapping (=
f(z), and B is called the preimage of B '. If ( the domain B onto B', the derivative
f'( z)
=
f(z) is a univalent mapping of
is nonzero at all finite points of the
domain B at which f(z) is regular.1) To see this, suppose that ['(a) = 0 at some point a € B. Then, the inverse function could be expanded in a neighbor hood of the point b = f(a) € B' in a series of fractional powers of ~ - b) and hence there would not be a one-to-one correspondence in neighborhoods of the points a € B and b € B '. The nonvanishing of the derivative means that we are dealing here with a conformal mapping. Therefore, a univalent mapping is also called a uriivalent conformal mapping (and, in the case of simply connected do mains, it is called simply a conformal mapping) of the domain B onto the domain
B': The function (
= f(z)
defining a univalent conformal mapping is called a
univalent function. Its inverse is a univalent function defined on the image.
1) However, nonvanishing of the derivative everywhere in the domain B does not en sure univalence of the mapping. For example, the derivative of the function ~ = fez) = l)n with n > 3 is nonzero throughout the disk < 1, but this function is not uni valent in that disk.
(z -
Izl
23
24
II. CONFORMAL MAPPING OF SIMPLY CONNECTED DOMAINS
25
§2. RIEMANN'S THEOREM
If a function fez) that is univalent in a domain B is not regular in B, it obvi ously has only one pole (and that a simple one) in B.
point (that is, when it is a punctured plane). Let us show that this is true. (In the second case, we may assume without loss of generality that the boundary point
The first question arising in the study of univalent conformal mappings is
is "",) If such a mapping is possible we have in both cases a mapping function
whether a given domain can be mapped univalendy into another domain. We men
that is regular on the entire plane except possibly at "" and, in addition, bounded.
tion certain necessary conditions for this. In the case of finitely connected do
, Such a function would have to be a constant, and a constant does not perform the
mains, a necessary condition is that the connectedness of the two domains Band
\
, desired mapping.
B I be of the same order: Let us show this. We may assume without loss of gen erality that the domains
B and B' both include"" and that the mapping in ques
However, we shall see below that, with the exception of these two cases, every simply connected domain can be mapped univalendy onto a disk. In fact,
tion maps "" into "" (we Can always arrange for this to be the case by means of
there are infinitely many ways of doing this; that is, there are infinitely many
suitable fractional-linear transformations). Let us enclose the boundary continua
distinct functions that carry out the desired mapping. This last is a consequence
of the domain B inside closed broken lines that have no multiple points, that lie outside each other, and that enclose each a single boundary continuum. The map ping then maps these broken lines inco closed piecewise-analytic Jordan curves lying inside B I and outside each other. If B I is of lower connectedness than B,
,of the fact that there exist infinitely many distinct fractional-linear functions that map a disk conformally into itself.
§2. Riemann's theorem
then at least one of these curves will not include boundary points of the domain
B '. Since the function inverse to the original function is regular inside such a curve and since the corresponding point describes a preimage C as it moves around that curve, the interior of the curve will correspond to the entire interior
The following theorem of Riemann answers the question posed at the end of
,
the preceding section and thus is a geometric principle of formation of analytic " functions. As a preliminary, we shall prove a simple lemma, which has numerous appli
of the preimage C, which, consequently, must be completely contained in B. But this contradicts the fact that there are boundary points of the domain B within C. This means that the domain B' cannot be of lower connectedness than B. By re versing the roles of Band B', we can show that B' cannot be of higher connect edness either. In the same way we can show that it is impossible to map a domain of infinite connectedness univalendy onto a domain of finite connectedness. Therefore, when we now pose the question of the possibility of univalent conformal mapping of various domains onto a given simply connected domain, we can confine ourselves to an examination of simply connected domains.
cations in the theory of conformal mappings.
Lemma 1 (Schwarz). Let fez) denote a function that is regular in the disk
Izi < R. Suppose that f(O) = 0 and If(z)1 ~ M for Izi < R. Then, If(z)1 ~ (MIR)lz! in the disk Izi < R; also, I!'(O)I ~ MIR. Equality holds in these two conditional inequalities (in the first at points z is a real constant. Proof. The function ¢(z)
=
*
0) if and only if fez)
=eia.(MIR) z,
f(z)/z is regular in the disk lzl
where a
if we define
connected domain, we need only know their mappings onto any of the standard do
1¢(z)1 on any closed disk Izi ~ r, where r < R, is attained on its boundary, we have 1¢(z)1 ~ Mlr for Izi ~ r. If we fix z and let r approach R, we see that ·I¢(z)! ~ MIR. This yietds the two ine
mains, for example, a disk, since the required mapping can then be obtained by
qualities asserted in the lemma.
In order to know how to map a simply connected domain onto another simply
first mapping one of the given domains onto the disk and then mapping the disk onto the other domain. The question arises whether an arbitrary simply connected domain can be mapped onco a disk. It turns out that there are two cases when this is impossible, namely when the domain is the entire plane and when it has a single boundary
¢(O)
=
!'(O). Since the maximum value of
Suppose now that Then,
¢(z)
I¢ (zo)\
=
MIR, where Zo is a point in the disk jzl
< R.
I¢(z)\ attains its maximum at the point zo' which is possible only if
= const = (MIR) eiQ.,
of the lemma.
that is, if fez)
=
eia.(MIR) z. This completes the proof
26
( =
plane that has more than one boundary point. Then, there exists a unique func tion (= f(z) that is regular in B and that maps the domain B univalentlyonto the disk 1'1 < 1 in suck a way that a given point Zo € B and a given direction at Zo are mapped into the point (= 0 and the direction of the positive real axis.
To prove this last, consider the family 3JI of all functions (
< 1.
f(z) that are
=
regular in the domain B, that map B univalendyonto domains contained in the disk
Proof. First of all, it is easy to map the domain B univalently onto a domain
1'1
1(1 < 1 in such a way that the mapping function f(z) is normalized by the conditions f(O) = 0, ['(0) > O.
z = 0, univalently onto the disk
Theorem 1 (Riemann).l) Let B denote a simply connected domain in the z
contained in the disk
27
§2. RIEMANN'S THEOREM
II. CONFORMAL MAPPING OF SIMPLY CONNECTED DOMAINS
1'1
< 1,
and that have the properties f(O)
=
Specifically, if B has exterior points, let c de
> O. < a:S
0 and ['(0)
nonempty since, for example, the functions f(z) = az, for 0
This set is I helong to it.
Let us pose the following extremal problem: Out of all the functions belong
note a finite exterior point. Then the function (* = 1/(z - c) maps the domain B
ing to the family :lJI, find the one such that ['(0) is greatest. Let us first prove
onto a simply connected domain B* including the exterior point
that this problem has a solution. We note that there exists a positive number p
0Cl.
Since B* is
a bounded domain, we can use a similarity transformation to map it into a domain contained io the disk
1(I < 1.
such that the disk 1 z!
On the other hand, if B has no exterior points,
disk, the functions fez) €
is contained in the domain B and hence that, in this
mare regular and satisfy the
conditions of the Schwarz
then, since it has more than one boundary point and is simply connected, its
lemma with R = p and M = 1. This yields ['(O):s l/p; that is, the numbers ['(0)
boundary consists of a continuum. Let a and b denote two finite boundary points.
are bounded above. Let a denote the least upper bound of all these numbers.
J(z - a)/(z - b). This function can be extended in the (simply connected) domain B along an arbitrary path; hence, it is single-valued in B. Since its inverse is a rational function and hence is single-valued on the entire z*-plane, the function f maps the domain B univalently onto a domain B*. The domain B* has the property that, if a point c € B*, then the point - c is an exterior point of B* since an arbitrary pair of points z*, - z* always corresponds to the same point z. Since B* has exterior points, it can be mapped, just as be fore, univalencly onto a domain contained in the disk 1'1 < 1. On the basis of this, it will be sufficient to prove Riemann's theorem for domains contained in the disk 1'1 < 1.
From the example given above, of functions in 9Jl, we have a
Consider the function z*
=
Furthermore, we can always assume that the given poiot in the domain B and the given direction at that point are the points z
=
0 and the direction of the posi
tive real axis since, in the opposite case, we could use a suitable fractional linear transformation to map the disk
1
zl
<1
into itself in such a way as to map
the domain B into a domain possessing this property.
Izl < 1
where n = 1, 2, " ' , denote a sequence of functions in
Let
msuch that
and including the point
lin (z)\,
[' (0) n
--+
a.
To this sequence we apply the condensation principle, which asserts that this sequence has a subsequence that converges inside B to a regular function fo(z) and that satisfies the conditions fo(O) = 0 and f~ (0) = a> O. By Theorem 2 of §1, Chapter 1 (with p
=
1), the function fo(z), since it is not identically constant,
assumes in the domain B an arbitrary value at no more than one point; that is,
fo(z) is univalent in B. Finally, since the inequality If(z)! Ifo(z)!
<1
<1
in B implies that
in B (for Ifo(z)1 to be equal to 1 in B is impossible by virtue of the
maximum principle), we conclude that fo(z) € :lJI. Let us now show that the extremal.function (= fo(z) maps the domain B onto
1"
the entire disk the disk
1'1
<1
< 1. If this
were not the case, there would exist a point a in
that would not belong to the image of B, which we denote by B o'
Consi~er the two-sheeted disk
1(I < 1
with unique branching point at a
(that is, the Riemann surface obtained by taking two copies of the disk
Thus, it will be sufficient to prove that it is possible to map an arbitrary simply connected domain, contained in the disk
,
2: 1.
1'1
< 1,
making.a cut in each of them along the portion of the radius extending from a to the circle
1(1
=
1 and then gluing them crosswise along the cuts). Let us map
this disk onto the one-sheeted disk
!tl < 1
in such a way that the point (= 0
and the direction of the positive real axis at that point is mapped into t = 0 and 1) This theorem was first stated by Riemann in his dissertation in 1851 though he did not prove it untillater. It was proved by Caratheodory [1912] and Koebe [1912] by the methods of the theory of functions. The proof that we give here was presented by Fejer and Riesz. See Rado [1922-1923a].
the direction of the positive real axis. To do this, we first make a fractional linear transformation of the disk
1(1 < 1
into itself in such a way that the branch
point a is mapped into the origin. Then, by means of the function
J[
we expand
the two-sheeted circle thus obtained into the one-sheeted disk. Finally, by means
of a new fractional-linear transformation of the disk
1'1 < I
The quantity R
into itself we estab
the point
lish the required correspondence of the coordinate origins. We denote the function that carries out this mapping by t
=
29
§2. RIEMANN'S THEOREM
II. CONFORMAL MAPPING OF SIMPLY CONNECTED DOMAINS
28
=
I/f~ (zo) is called the conformal radius of the domain B at
zOo
Second, the proof of Riemann's theorem is based on one of the many extremal
¢(,) and its inverse by' =t/J(t). Then, ¢(,)
can be extended in the domain Bo along an arbitrary path and hence it is single
,properties of functions that map a domain B univalently onto a disk. It
valued. The function t/J(t), being a rational and hence single-valued function by
that a number of extremal properties of such functions hold not only in the class
construction, satisfies the conditions of the Schwarz lemma with R = I and M = I, and it is not a linear fun~tion. Therefore, t lentlyonto a domain contained in the disk
=
¢(,) maps the domain Bo univa
1tl < 1.
Also, t/J '(0)
< I;
that is,
¢' (0) > 1. The composite function t = ¢(fo(z» = ¢(z), <11(0) = 0, 0, maps the domain B onto the same domain and hence it belongs to the family ~. But
=
f~ (0)
¢ '(0)
function'
=
> f~(O)
a, which contradicts the definition of a. Thus, the
=
fo(z) maps the domain B univalently onto the entire disk.
It remains for us to prove the uniqueness of the function mentioned in the theorem. Suppose that there are two such functions f/z) and f/z). Then, the function F(,)
=
f 1(f2- 1
(,»,
which is regular in the disk
1'1
must map this
disk univalently into itself. Consequendy, in accordance with the Schwarz lemma,
IF(')I S 1'1
in the disk
1'1
< I;
that is,
If1(z)1 S If2(z)!
in B. By reversing the
roles of the functions f 1(z) and f 2 (z), we can prove similarly that If1(z)[:: If2(z)! in B. Consequently, If1 (z)! '=
If/z)1
in B. Since the function f 1 (z)/f2 (z) is reg
ular in B and is of constant absolute value, it follows from the maximum modu lus principle that
f 1(z) '= f 2(z),
which proves the uniqueness. This concludes the
proof of the theorem.
We have a few remarks to make concerning Riemann's theorem.
,=
F (z) that is regular in B, that is normalized at a finite point Zo € B by the conditions F(zo) = 0 and F '(zo) = I, and that maps the domain B univalently onto the disk with center at 0 (an other form of Riemann's theorem). This is true because the function F(z) = fo(z)/f~(zo) is such a function, where fo(z), with fo(zo) = 0 and f~ (zo) > 0, is the function mentioned in Riemann's theorem and the radius of the disk onto which the function' = F (z) maps the domain B is R = I/f~ (zo)' If there existed a second function' = F1(z ), with F (zo) = 0 and F;(zo) = I, that maps the domain B onto a disk 1'1 < Rl' 1 then by Riemann's theorem, we would have F 1(z)/ R 1 = fo(z) and hence I/R 1 = f~(zo); that is, F (z) = fo(z)/f~(zo) = F(z). This completes the proof of the 1
,=
uniqueness of the mapping function' = F(z).
out
f univalent functions but also in the class of all regular functions that are nor
1
·zed in a suitable manner. We shall now present two extremal properties of this a) Minimization of the maximum modulus. Let B denote a simply con
ected domain that includes z = 0 and has more that one boundary point. Let m note the set of all functions F(z) with F (0) = 0 and F '(0) = I that are reg ar in B. The quantity M(F) = sup zt: B IF (z)1 is minimized by a unique function rhat maps the domain B univalently onto the disk Iz 1 < R. This minimum is equal the conformal radius R of the domain B at the point z = O. Proof. Let'
= fez), with f(O) = 0 and [' (0) > 0, denote a function that maps 1'1 < I and let z = ¢ (,) denote the in
e domain B univalently onto the disk
:verse function. To find the minimum of the quantity M(F), we need consider only the functions F (z) €
:ill for which M(F) is finite. Consider the corresponding
F(¢(,»/Jf(F). These functions are regular in the disk 1'1 < 1. It/J(')I < I in the disk 1'/ < 1. By the Schwarz lemma, we have IJ'(O)! S I; that is, M(F) ~ I/Qr'(O) = R. Here equality holds only for the function t/J(') '=" that is, for the function F(z) = cf(z), V:here c = 1/['(0).
<;iunctions t/J(') , Also, t/J(O)
=
=
0 and
This completes the proof of assertion a).
In the first place, we note, that, under the assumptions made regarding the domain B, there exists a unique function
turDS
From this extremal property of the mapping function we get certain properties
R of the domain B. In the first place, the conformal ,radius of a domain B does not decrease under a dilation since the family 31 is
"of the conformal radius
:'!hen decre,ased. In the second place, since M(z) is equal to the maximum of the ;'distances from the point z
=
0 to boundary points of the domain B, the conformal
radius R of the domain B does not exceed this maximum. On the other hand, if
,R were less than the minimum of these distances, the function f(z) would be regular in a disk Izl < R + f for some f> 0, F (0) = 0, and IF (z)1 < R in the 'disk 1 zl < R + •. In accordance with the Schwarz lemma, we would then have IF '(0)1 S R/(R + .) < I, which contradicts the fact that F' (0) = 1. This, together with what has been said above, shows that there are always boundary points of domain B on the circle
Izj
=
R.
30
the Riemann surface with which the domain B is put into one-to-one correspond
b) Lindelof s principle. I) Let 91(1;,) and 92(1;,) denote two functions that
are regular in the disk
II;,! < 1.
Suppose that 9 1 (0)
=
92(0). Suppose that 92(1;,)
is univalent. Suppose that 91(1;,) and 9/1;,) map the disk
II;,! < 1 onto
'fI; (0)
I :;::;;; 1
ence by the function
domains
Bland B 2 respectively, where B 1 is contained in B 2' Then I
and z
1)1 for which ~ (F) is finite. Using the same functions
9(1;,) as in a), we transform the integral (1) to the variable
ICI
function z = f 2 (9.(I;,» maps the disk 11;,1 < 1 onto a domain contained in the disk Izl < 1. Also, it maps the point I;, = 0 into the point z = O. Consequently =
=
lim ~ ~
=
p---..l I CI
IP' (q- (C»
q-' (C) I~ r dr d
'8
-1£ we set re'
(I;" where
1
!
00
P' ('fI (C» 'fI' (C) =
n anC , ao= 'fI' (0) = f' (0) ,
n=O
One immediate result of Linde1of' s principle is the fact that the conformal we have
radius of a domain increases when the domain is increased.
00
c) Minimization of area. Let B denote a simply connected domain that in
~(P)='lt'lim'
cludes the point z = 0 and has more than one boundary point. Let ~ denote the set of all functions F(z) with F(O~ = 0 and F '(0) = 1 that are regular in
I;, = fez) I;, = rei¢>:
~1(F)= ~ ~ IP'('fI(C»'fI'(c)I~rdrd'fl
with equality holding if and only if 9 1(1;,) = 9/(1;,), where lEI = 1. Proof. Let I;, = f 2(z) denote the function inverse to z = 9/1;,). Then, the
f/91 (1;,»
I;, = F (z).
Once again, in finding the minimum of the integral (1), we need only consider functions F (I;,) €
'fI~ (0) \'
19; (0) N; (0) I ~ 1 with eq uali ty holding if and only if Itl = 1, that is, if and only if 91(1;,) = 92«1;,).
31
§3. CORRESPONDENCE OF BOUNDARIES
II. CONFORMAL MAPPING OF SIMPLY CONNECTED DOMAINS
B. Then
the same function F as in a) is the unique function minimizing the quantity
~ (p) = ~ ~ I p' (z) I~ do
(1)
B
!
a
22nH
1
"I n"--1....1 P
~'lt'la0I~'
::.--
p---..l n=O
that is, ~ (F) ~ 17 lao 1 , with equality holding only if a 1 = a 2 = ... = 0, that is, only for the function F(z) = f(z)/['(O), This compleres the proof of assertionb). 2
The minimal property b) can then be used for an explicit construction of the map ping function.
where da is the element of area. 2) §3. The correspondence of boundaries under conformal mapping
Here, the integral (1) means the limit of a sequence of integrals over domains
B n , for n = 1, 2," " with the following properties: lim n B n = B; B n c B; B n C B n +1; every closed subset of B is contained in some B n (and hence in all sub sequent B ). It is easy to show that the value of the integral (1) is independent n
Since the boundary points of a domain do not belong to the domain, we were concerned in ,the preceding section with the establishment of a correspondence between interior points of a domain and its image. To study the correspondence
of the choice of sequence IB n 1 of domains with these properties. For example,
of boundary points, we need to take up the more difficult question of investigating
we can take as the domains B n the preimages of the disks I(, I < r n , where each r < 1, under a univalent mapping of the domain B onto the disk 11;,1 < 1 such n that z = 0 is mapped into I;, = O. In the case of a univalent function F (z), the
question. This investigation leads to a Dumber of remarkable results.
the behavior of a mapping function as we approach the boundary of the domain in We first establish a few auxiliary propositions.
integral (1) obviously gives the area of the image of the domain B under the map· ping
I;, = F (z). If F (z) is not univalent in B, the integral (1) gives the area of
Lemma 1 (Koebe). Let Iz~
longing to the disk on the circle
"izi 1) (A more general fonn of Lindelor s principle is gi ven in Chapter VIII, §3>. 2) Bieberbach [1914].
Izi
=
Izl < 1
1 and
Iz~ 1 denote two sequences of points be
that converge to distinct points a and b respectively
1. For every n, let An denote a Jordan arc lying in the disk
< 1 but outside some fixed neighborhood of the point z = O. Let fez) denote a function that is regular ~nd bounded in the disk I zl < 1 and that converges uni formly to 0 on the sequence
1,\n I.
Then fez) ==
o.
-------
====,,-,,-=-"'-~""-~~~~~~~~~~
-----------==-------
~
--------:;--~--- ~ -~·.-=;~~··~~.-:~£qE!c~~-=~~~1!'$,_~~~
II. CONFORMAL MAPPING OF SIMPLY CONNECTED DOMAINS
32
The uniform convergence of the function f(z) to 0 on the arcs An is to be understood in the sense that, for every inequality n
>N
f
> 0,
on An implies the inequality
Proof. Let us suppose that f(z)
i
there exists an
of the curve Ank with a, let us proceed along the curve Ank in either direction
N > 0 such that the
till we get to the first point at which this curve meets one of the other boundary
If(zl < f.
radii, which obviously belongs to the boundary of the sector adjacent to a. We denote by A~k the arc of the curve Ank from this point to the first point at which
O. We may as well assume that f(O} f, 0
because, in the opposite case, we could consider the function f(z}/ zP, where p
Ank touches a as we proceed along Ank in the opposite direction. Except for dleir endpoints, the arcs A~k' for k = 1, 2,00', lie entirely in the two sectors JJdjacent to a. Let 5 denote whichever of the two sectors contains an infinite I , lSet of arcs Ank • For these arcs we keep the same notation Ank , for k = 1, 2" 0••
denotes the order of the zero at z = 0 of the function f(z}, and one can easily verify that this function also satisfies all the conditions of the lemma and is non zero at z
=
1.'
O.
On
On A , we have n
f
n
-> 0 as n -> "". Let us partition the disk
A~k' we have If(z}\
< (nk'
We may assume that the sector 5 is symmetric
';i~bout the real axis since we can al ways arrange this by considering not only the 'i'function f(z) but the function f(eia. z) with suitable choice of real a, which also
If(z) I ~max If(z) I=$n' An where
33
§3. CORRESPONDENCE OF BOUNDARIES
Iz I < 1
{;,,"',jJ
\.'atisfies conditions analogous to the conditions of the lemma. Let A~k denote
into m equal sectors
that portion of the arc A~k extending from the endpoint of A~k lying above the
such that the points a and b do not lie at the ends of the boundary radii. For m
real axis to the first point at which A~k meets_the real axis. Let i~k denote the
sufficiently large, the points a and b will lie on the boundaries of distinct sec
arc symmetric to A~k about the real axis. On A~k we have
tors. If we now discard these sectors, the remaining portion of the disk will be partitioned into two groups of adjacent sectors, in each of which there will be at
I
as sum=-0 at If(z}!
f(z},
<M
If(z}1 < f nk •
If we
in B, we can now conclude that the function lj;(z) =
1lj;(z)1 < M(nk
~~rf(z)
as the boundary of two adjacent sectors (see Figure 1). One of these radii, which
.;the arc A~k U A~k since one of the factors whose product is the function lj;(z) is
0
wh~ch is regular in
Izi < 1
least two sectors. In each of these groups, let us choose one radius that serves
satisfies the inequality
on
;~ways no greater than M and the other is less than (nk' On the other hand, the
function 21ti
F(z)=Hz)~(1Jz) ... ~(1Jm-lz),
is regular in the disk arc
A:
k
U >:~k
Izi < 1.
1j=em ,
Then, on the closed Jordan curve consisting of the
and the arc obtained from it by means of a rotation about z
=
0
through angles br/m, 417/m, .. 0, 2(m - I} ,,/m, we have, on the basis of similar
considerations,
IF (z) I ~ $nkMzm-1,
o
The same inequality holds inside this curve and, in particular, at the point z = O.
But F(O) = If(0)1 2 m. Therefore, If(0)j2m ::SfnkM2m-l. Since f nk ->0 as k ->"",
it follows that f(O) = 0, which contradicts the assumption made at the beginning
of the proof. This completes the proof of the lemma.
Figure 1 we denote by a, intersects infinitely many of the curves intersected curves A ,for k nk
=
1\.
Lemma 2 (Lindelof). Suppose that the following holds: Let us index these
1, 2,0' '. Beginning with some point of intersection
1) The function f(z)
is regular in a domain B and If(z}! ::s M in B, 2) Zo is a finite point of the do main Band r is a positive number such that the eircZ e Iz - Zo I = r contains an arc
34
II. CONFORMAL MAPPING OF SIMPLY CONNECTED DOMAINS
3) For arbitrary ( > 0,
of length (217/m)r, where m is an integer, lying outside B. as we let z approach boundary points of
B lying in the disk
Iz - zol
through
points belonging to B, all the limiting values of If(z)! are less than or equal to L
Then
Proof. Obviously, we may take Zo = O. The common portion of the domain B and the domains obtained from it by a rotation about the origin through angles 217/m, 417/m, ... , 2(m - I}17/m is an open set including the points z
Izl = r.
=
0 but not
It consists of domains among which is a
domain B 1 including the points z = O. This last domain lies entirely in the disk
Iz I < r.
Iz -
a\ < p/2. Then, there exists a disk Iz - Zo I < r contained in the disk Iz - al < p such that there are exterior points of B on the circle Iz - Zo I = r and hence there exists an arc lying outside B. Lemma 2 with ( = 0 can be applied to the function f(z) - c. Consequently f(zo) = c. Since Zo is an arbitrary point in the disk Iz - al < p/2, we have J(z) == c in B, On the other hand, if the point a o is not a cluster point of the set of exterior points of the domain B, let us denote by b a finite boundary point
I/(zo) I ~ YeM m- l .
including points of the circle
35
§3. CORRESPONDENCE OF BOUNDARIES
Consider the function
distinct from a. Then, just as in the proof of Riemann's theorem, let us use the function z.. = y(z - a)j(z - b) to map the domain B uni val ently onto the domain B.. , which, in an arbitrary neighborhood of the point z.. = 0, will have exterior points. The function f(z) - c will now be regular as a function of z.. in B.. and it will bave the property mentioned in Lemma 3 at the point z = O. Therefore, using what we have already proved, we again obtain the result that f(z) == c in B. This com pletes the proof of the lemma.
F (z) =/ (Z)/(ljZ) .. '/(ljm-l z), which is regular in
2'"
(lj = e17'),
B l' If we let z approach the boundary of B 1 through points
Accessihle houndary points. Let us now examine the behavior of the function
(= f(z), which maps a given simply connected domain B univalently onto the disk \'1 < 1 close to the boundary of the domain B. Since such a domain can be
of B l' then, in accordance with condition 3) of the lemma, all the limiting values
mapped univalently by means of elementary functions onto a bounded domain,
of IF(z)! will be no greater than dIm-I. But then, we shall have IF(z)1 ~ (Mm-l
it will be sufficient in what follows to consider the case of a bounded domain B
at all points of the domain B l' To see this, let M1 denote the least upper bound
and it is for such a domain that we shall prove all the theorems.
of IF(z)! in B 1 • IfM 1 were greater than (M",-1 there would exist a sequence of points z k € B such that IF (z k)\ ~ MIas k ~ 0<1. Let denote a cluster
main B, the corresponding points ( in the disk
point of this sequence. It cannot be a boundary point of B 1 because then M1 ~
1(1 ==
1
Zo
(. M",-I. Therefore, IF(z~)1 = M1 ; that is, IF(z)1 attains a maximum at the point z = z~, which is impossible. Thus, !F(z)! ~(Mm-l in B 1 , In particular, since F(O) = [[(0)]"', we have If(o)\m ~ (M",-I, which completes the proof of the lemma.
First of all we note that, as points z € B approach the boundary of the do approach the circle
< 1'1 < 1.
corresponding point ( lies in the annulus 1 - (
Specifically, we may
take for;) the distance between the boundary of the domain B and the closed set constituting the preimage of the disk
As a corollary of this lemma, we have the following
if points of the disk
Lemma 3. If a function f(z) is regular and bounded in a simply connected
points z
domain B that has more than one boundary point and if there exists a neighbor
1(' < 1
1 uniformly in the sense that, for every (> 0, there exists a 8> 0 such that, if a point z € B lies at a distance less than 8 from the boundary of B, the
€B
I(I < 1
1'/ < 1 -
(. Analogously, we can show that
approach the circle
1(I
=
1, then the corresponding
approach the boundary of the domain B uniformly.
For the next step in our investigation, we introduce the concept of accessible
hood of a finite boundary point a such that the function f(z) approaches a con
boundary points of a domain. A boundary point c of a domain B is said to be an
stant c as z approaches boundary points lying in this neighborhood through
accessible boundary point if it can be joined with an arbitrary interior point of the
points belonging to B, then f(z) == c in B.
domain B by a continuous curve I: z
Proof. Let us first suppose that a is also a cluster point of the set of ex terior points of the domain B and let the disk
Iz
- a I < P denote the neighbor
hood referred to in the lemma. Let Zo denote an arbitrary point of the disk
=
z (t) for a ~ t ~ b that lies entirely in B
except for the endpoint c = z (b). Such a curve I can be replaced by a Jordan curve or even by a broken line with no multiple points that consists of a finite or infinite number of segments, in the latter case with endpoints clustering only about c. This is true because the arc z
=
z (t) for b -
(k-l
~ t ~ b-
(k
(where
(k
>0
for
~~ -~~~~~~~~-====~~"=-;o-=~~~=--=-~~~-===-~~:::-~=-_-:...--=:=-~~~~"~
36 k
=
~~~
"~~-~~~
~
_._-_._---'.--
.
~~_:~.~ _~-~~~~~~-~~~~=-=,-=-=".;';'---=~.~-
-'~""-~--'.==-""-=--"'''---
-
:
--= __ &+!ii_ ,...,.
II. CONFORMAL MAPPING OF SIMPLY CONNECTED DOMAINS
1, 2, .•• , wbere (k
> f k +1 , and wbere
hence lies at a positive distance
ok
f
k
->
0 as k
->
00
=-_=;;;==
--
.............
_
§3. CORRESPONDENCE OF BOUNDARIES
is a closed set and
from the boundary of tbe domain B. Tbere
fore, it can be partitioned into a finite number of arcs sufficiently small tbat the distance between any two points of anyone of tbese arcs is less tban
""""-!!i __ --=~_:::=.:.,"!~.~~
ok'
37
and l2 cannot be connected by a continuous curve lying both in B and in that neigbborhood. If two curves II and l2 that terminate at a single boundary point
d.
Then,
I
I
I
the broken line consisting of segments joining successive points of this partition will lie entirely in B. If we apply this for k = I, 2, ... , we see that the curve l can be replaced with a broken line consisting of an infinite number of segments. If
d
we discard any loops, we obtain a broken line without multiple points. This brok en line is a Jordan curve.
e
e
In connection with the concept of accessible boundary points, we bave an
c
important supplement to Jordan's theorem (d. introduction), presented by Schoen flies:
)
Figure 4
Figure 3
l
All points of a closed Jordan curve are accessible boundary points of the
domains defined by it. For continua of a more complicated type, there may not be
z 1 have common points in an arbitrary neighborhood of z l' these curves neces
an analogue of Jordan's theorem and its supplement. For example, the curve shown in Figure 2 and representing a spiral-shaped curve
sarily define the same accessible boundary points. Therefore, accessible bound ary points defined by a continuous curve and a broken line inscribed in that curve
without multiple points that approaches the circle C a
coincide, as shown above. This enables us to define accessible boundary points
symptotically partitions the plane into three simply con
by means of Jordan curves only. We have the following theorem
nected domains, and, for the two bounded domains, the circle C consists entirely of inaccessible boundary
1)
on the correspondence of accessible bound~
ary points:
points. Analogously for tbe domains sbown in Figures 3
Theorem 1. Given a univalent conformal mapping of a domain B in the z
and 4 and representing squares with rectilinear cuts that
plane onto the unit disk
cluster about the segment cd, all points of the segment
Figure 2
cd (with the exception of the point d in Figure 4) are inaccessible boundary points.
In these last two examples, the point e (see Figures 3 and 4) can be reached along two continuous curves lying in tbe domain, one on either side of the cut. To distinguish two boundary points at a point e, we identify the accessible bound ary point of the domain B not only by its position in tbe plane but also by the continuous curve in B that joins this point with an interior point. Let II and 12 define respectively accessible boundary points at z 1 and z 2' If z 1
i
z 2' tbese
\'1
< 1, to each accessible boundary point
main B we can assign a specific point 'Ion the circle
1'1 = I
zl
of the 00
such that, as z
approaches zl along an arbitrary curve II defining z l' the corresponding point , approaches
'1'
and also such that two distinct accessible boundary points of
the domain B correspond to two distinct points on
1'1
=
I and the set F of
= 1 corresponding to all accessible boundary points of the domain B is everywhere dense on "I = 1.
, points on
"I
Proof. We shall carry out tbe proof in several stages. 1°. Let us show that, as a point z approaches an accessible boundary point z 1 of the domain B by moving along a curve II: z = z (t), for a
S t S b, that de
accessible boundary points are distinct. On the other hand, if z 1 = z 2' we take
fines the accessible boundary point, the corresponding point' moves along a
the view tbat two curves II and l2 lying in B define at
Continuous curve -\:
zl
two distinct acces
sible points if and only if there exists a neigbborbood of the point z 1 in wbich II
1) See, for example, Caratheodory [1912].
,=
f(z(t)) in the disk
point on the circle "~I = 1.
1) Koebe [19151.
"I
and approaches a definite
~~~~:~:~:~;:,;;:~~=,~=.~~.'~~::~::;:"-~'~~'-::~::~- "~~-)i;~2,, ...,,:~~~~;:~;:": "~,._:;';;~.~. -":::-:;;~_~::~~~e~;~::~:~'~~:;:. __-~". _,=_-;-""....,,"__ ._~m ~ "_c'."~_.",,_
38
II. CONFORMAL MAPPING OF SIMPLY CONNECTED DOMAINS
Let us suppose the opposite. Then, there must exist on tbe circle
"I
=1
two distinct points' 1 and '2 such that there is on Al a sequence of arcs with endpoints that converge to
'1
and '2 and the inverse mapping function z = ¢ (,)
converges uniformly on these arcs to "I
< 1,
zl'
Since
it follows from Lemma 1, as applied to
¢(,) - z 1 == 0; that is, ¢( ,) == const in
¢(,) is bounded in the disk the function ¢(,) - zl' that
1'1 < 1,
and this is impossible. Conse
quently, as , moves along AI' it approaches a definite point on
1'1
2°. Let us show that, as z describes two curves II and l2 defining the
same accessible boundary point z 1 of the domain B as it approaches z l' the corresponding point , describes curves Al and A2 that terminate in a single point on the circle = 1.
'1
Let us suppose that the curves Al and A2 terminate at distinct points 1'1 = 1. In an arbitrary neighborhood of z l ' the curves II
and '2 on the circle
'1
.,.... ""~_~"",.,_,
39
§3. CORRESPONDENCE OF BOUNDARIES
in B, which is impossible. On the other hand., if the com
plex numbers zl and z2 are equal, II
U l2
is a closed Jordan curve which is
mapped into the boundary of the domain B l' In this case, if the domain B 1 has a boundary point distinct from points of the curve II the curve II
U l2"
U l2'
this point must lie inside
Consequently, the hypothesis of Lemma 3 is satisfied in a suf
ficiently small neighborhood of this point and we may again conclude that const in
= 1.
1'1
this yields fez) ==
_,,:c ~"""~:
B;
f(')
==
which again is impossible. This proves that the entire boundary of
B is the curve II U l2' But then, in an arbitrary neighborhood of the point zl' 1 the curves II and l2 can be joined by a continuous curve lying in the domain B 1 and hence in the domain B, and the curves II and l2 do not define distinct ac ,.cessible boundary points of the domain B. This proves that 4°. Finally, let us show chat the set
1'1
where dense on the circle
'1
f.
'2'
F referred to in the theorem is every
= 1. Let us suppose that the circle
"I
= 1 con
and l2 can be joined by a continuous curve and hence by a Jordan curve contained
tains an arc y no points of which correspond to accessible boundary points of the
both in the domain B and in this neighborhood. Therefore, as the neighborhood is
domain B. Let
constructed to a point, the endpoints of the corresponding Jordan curves in the
2, •.. , denote a sequence of points of the disk < 1 that converges to The corresponding points z n € B have at least one cluster point zo' which neces
'1
disk 1'1 < 1 approach and '2' On these curves, ¢(,) approaches formly. In accordance with Lemma 1, we again obtain ¢(,) == const in which is impossible. Consequently,
'1
= '2'
uni 1'1 < 1,
zl
'1
that z 1 and z 2 are defined by the Jordan curves II and l2' We may assume that they issue from a single point of the domain B and that they have no other points in common in B. Let \
and A2 denote the corresponding curves in the disk < 1. Let us suppose that they terminate at a single point'1 on I'I =- 1. The
UA2 partitions the disk into two domains, of one of which it con stitutes the entire boundary. Suppose that corresponding to this domain in B is a
closed curve Al domain
B l' The curve II U l2 is mapped into its boundary. If the complex num
bers z 1 and z 2 are distinct, there exists on the boundary of the domain B 1 a boundary point a of the domain B that is distinct from z 1 and z2 because, otherwise, the boundary of the domain B 1 would be the open Jordan curve II
denote an interior point of this arc and let {'n}' for n = 1,
'0'
\'1
sarily lies on the boundary of the domain B. Let {zn) denote a subsequence that converges to zo' Let us denote by lnk the straight line segment connecting
3°. Let us now show that two distinct points and '2 on the circle \,' = 1 correspond to two distinct accessible points zl and z2 of the domain B. Suppose
I"
'0
U l2
and thus would not be bounded, in contradiction with the assumption of bounded
znk with the closest point of the boundary of the domain B. This segment defines an accessible boundary point. From 1°, corresponding to this segment in the disk
1'1 < 1 1'1
is the curve A
nk
extending from the point'
to some point on the circle
= 1 not on the arc y. Also, ¢ (,) approaches Zo uniformly on A
On each of the curves A
nk
as k --.. 00. nk , we can choose an arc such that the sequences of end
points of these arcs converge to the point circle
'0
'rhen, we can apply Lemma 1 to the function
1'\ < 1,
¢(,) - zo, and we conclude that
which is impossible. Consequently, the arc y de
scribed above does not exist and the set =
in the one case and to a point on the
''I = 1 not on the arc y or to one of the endpoints of that arc in the other.
¢(,) == Zo in the disk
"I
nk
F is everywhere dense on the circle
1.
This completes the proof of Theorem 1.
Prime ends. To make a complete investigation of the correspondence of
1'1 < 1,
ness of B (see p. 35). In a sufficiently small neighborhood of the point a, all
boundaries under univalent mapping of a domain B onto the disk
boundary points of the domain Blare also boundary points of the domain B. Con
troduce the concept of a prime end. Suppose that we make a sequence of trans
B 1 approaches these points, the corresponding point' must approach the circle 1'1 = 1, that is, the point In accordance with Lemma 3,
Verse cuts Ck , for k=I,2, ... , in the domain B. The cut C1 connects two
sequently, as z €
'1'
we in
distinct accessible boundary points of the domain B and partitions it into two
domains, one of which we denote by B l' The cut C 2 lies in the domain B l' con
of points of the disk
nects two distinct accessible boundary points of the domain B that are not end
.the points 'n
. points
Bn' connects two distinct accessible boundary
Bn ,
B that are not endpoints of C n , and partitions the domain • Bn into two domains. We denote by Bn +1 whichever of these domains does not have points of the ~urve C n on its boundary. The closed domains 8 n , for n =
points of the domain
i
1'\
<1
'0
intersection. If this intersection consists of a single point
I' I = 1, •
pomt
on the circle
the set E is called the prime end of the domain B corresponding to the
)-
1)
"'0'
'0
sequences of cuts defining two prime ends E' and E" corresponding to the same
'0'
Then the domains
G'n
and
G"n
corresponding to the domains
'0'
8n I
and
B"n defined by Consequently, each of _ these cuts have _ a unique common point the domains <; I contains all G" beginning with some m; conversely, each of the _ n _ m domains G,. all I beginning with_ some m. This n contai~s _ _ Gm _ properry carries over
B'n 'and B", but then, E" c B' and E' c B" for all n; that is n n n E ' c E". This means that the prime ends E and E" determined
to the domains
E" c E' and
I
by the cuts C nI and C"n consist of the same points, and this in tum means that a prime end is completely determined only by the point
'0
corresponding to it.
A prime end can also be characterized independently of the sequence of cuts
Cn defining it. Specifically, let us show that the prime end E of the domain B cor responding to the point
'0
on
1'1
I'nl < 1
and
0
=
Suppose that corresponding to
1, ,is a sequence of points zn €
nconsidered abo~L!}ontains all 'm
B.
beginning
B n has the same property with regard to the
Consequently, every cluster point of the sequence
that is, belongs to E. Now suppose, conversely, that Zo €
Iznl
belongs to all
E. Then Zo lies
on the boundary of all the domains B n' In the domain B, let us choose a Sequence -->
Zo such that zn €
'n '0'
;:l;sponding to the points --> The question of the correspondence of boundaries under conformal mapping
:tq:'
7:':' is completely answered in terms of a prime end (see Caratheodory [1913a]):
r·{ .
Theorem 2. Under a univalent conformal mapping
• '
0
1'\ < 1, there exists a one-la-one correspondence 1'1 = 1 and the prime ends of the domain B.
disk
f a domain B onto the unit between points of the
cirel e
It is easy to show that a prime end corresponding to a point on 1" = 1 is independent of the sequence of cuts defining it. Let {C nI ! and {C"n I denote two _point
'0' where
' 0 ' 1' 1
that converge to
Bn' where n = 1, 2, .... Corresponding to them in the disk 1'1 < 1 are the points' n € Gn where n = 1, 2," '. It follows 'that 'n --> '0; that is, Zo is a cluster point of the set of points Zn € B n corre
we denote by E.
corresponding to the domains B n . The closed domains Gn 'also have a nonempry
zm'
.<,pi points zn
1, 2•.••• which are nested in each other, have a nonempty intersection, which
ary points of the domain B. We denote by Gn the domains in the disk _
<1
with some m, anyone of the domains
whichever one of these domains has no points of the curve CIon its boundary.
Let us assume that the cuts C n are always such that E contains only bound
-->
1'1
Since anyone of the domains G
points of C 1 , and partitions the domain B 1 into two domains. Let B 2 denote In general, the cut C n +1 lies in
41
§3. CORRESPONDENCE OF BOUNDARIES
II. CONFORMAL MAPPING OF SIMPLY CONNECTED DOMAINS
40
= 1 is the geometric locu~ of all cluster points
of sequences of points of the domain B corresponding to all possible sequences
Proof. It follows from the definition of a prime end that to every prime end of a domain
B there corresponds a unique point on the circle
'0
1'I
= 1. Let us prove'
the converse, namely, that for every point on 1'1 = 1 there corresponds a rime end of the domain B. Let us take two sequences {Cn I and ICIn I, where P n
=
1, 2,""
sides to
'0
of points on
1'1 =
1 distinct from
'0
which converge from opposite
and to which correspond sequences of accessible boundary points z~
and z~ of the domain B. We may assume that, for each n
=
1, 2,' ", the points
on 1'1 = 1 which contains '0 (see Figure 5). Suppose z,n and z"n are determined by the curves I'n and I". We de 'n
" II. +1 '''+1 ,and n lie inside the arc
Cn Cn
note the corresponding curves in the disk .
1'\ < 1
by A'n and A". For every n, n
we choose. on the curves A'n and A"n arcs with endpoints" n and ("n such that all their points are closer to the straight line passing through " n and they are to the line passing through ,~_
1 ':-1 and
C'n
than
or to the straight line pass
ing through '~+1 and '~+1' Let us connect by a straight line segment the ends of these arcs which lie in the disk 1'1< 1. As we move from C along A'n and n from
'J
along
AJ
to the first points of contact with this segment, we replace our
original arcs with those just mentioned and we replace the straight line segment 1) (A prime end can be defined without bringing in the concept of conformal mapping,
as was done in the definition of accessible boundary points (d., for example, Caratheo dory [1912]). >
with the line segment joining the endpoints of these arcs, which lie in the disk
1'1
< 1.
The cuts thus obtained in
1'1 < 1
are constricted, as n
-->
00 ,
to the
~
_=_..
••R_
~,
..
~,
__
point (0 and have as preimages in the domain B the cuts
en
43
§3. CORRESPONDENCE OF BOUNDARIES
II. CONFORMAL MAPPING OF SIMPLY CONNECTED DOMAINS
42
:i since it is easy to construct a _ sequence of curves C n such that the intersection
determining the
the corresponding domains B n is just this segment. In Figure 4 the segment cd contains only the single accessible boundary point d; in Figure 3 it does not
prime end corresponding to the point (0' This completes the proof of the theorem.
contain any accessible boundary point at all. Analogously the spiral-shaped do . JJlain in Figure 2 has the entire boundary circle as its prime end, which does not contain a single accessible boundary point. SpeCial types of domain. Up to now, we have considered the question of the correspondence of boundaries of arbitrary bounded simply connected domains ,nder univalent mappings. For domains of special types, we can draw a number
of supplementary conclusions. Thus, it follows from the preceding theorems that, H all the boundary points of the domain B are accessible, then a univalent map ping of this domain onto the disk
1(1
defines a one-to-one correspondence be
tween boundary points of the domain B and points of the circle
1(1
=
I because,
in such a case, every prime end of the domain B consists of a single boundary
•
Figure 5 As a supplement to this, we have
n
ous in the closed disk
K,.1
as above. For E to be a prime end of the domain B, it is neces
1(1 ::; I
'0
lie on the circle
of points
J\
in the disk
1'1
Portions of the curves points (I and (2 lie in
J\
Gn'
This means that portions of the curves 11 and 12
in neighborhoods of the points zl ~ points zl and
z2
Let
<j and '\2 in sufficiently small neighborhoods of the z2
lie in B n • This in tum means that the
belong to all the B n and hence to the set E, which contradicts
the hypothesis of the theorem. This completes the proof of the theorem. In Figures 3 and 4, the segment cd is a prime end of the domain in question
=
1, then the relation
(nk --
1'1
= 1, then these points themselves constitute a sequence
(0 and we need to show that for them k--.-w
<1
that terminate at (I and (2'
1'1
lim cp (C nk ) = cp"(C o).
1. On this arc, let us choose two points (I and (2 corresponding to the admissible boundary points zl and z2 of the domain B. Corresponding to the and
that converges to (0' where 1(01 = 1,
lim cp (C n ) = cp (Co)
=
J\
1. If a sequence
n--.co
intersection of the domains Gn used to define E would contain an arc of the
curves 11 and 12 defining these points are curves
=
follows from the definition of ¢((o). On the other hand, if infinitely many of the
tains no more than one accessible boundary point. If E were not a prime end, the
I(I
1(1 ::; I
of points (n of the closed disk
~
contain more than a single accessible boundary point. Suppose now that E con
circle
¢ (() is continuous on "I
contains only finitely many points of the circle
Proof. It follows from Theorems 1 and 2 that, if E is a prime end, it cannot
¢ (() is continu
if we define this function at a point (0 on the
We need only show that
sary and sufficient that E contain no more than one accessible boundary point of the domain B.
=
circle" I = 1 as the limit of its values as ( approaches (0 from the open disk
KI < 1.
Theorem 3. Let E denote the set of boundary points of a domain B defined
by the cuts C
point. Let us show that, in this case, the inverse function z
~
E
denote any positive number. Consider, in the disk
'(I < I,
a sequence of
points (~Ic satisfying the conditions
'0'
ICnk - C~k I~ ~, I cp (tnk) -
cp (C~k)
I~ ; .
Since"nk --. there exists a K > 0 such that, for k> K, we have I¢(Cnk ) - -4.('0)1 'I' E/2. It follows that, for k> K, I¢(( ) - ¢((o)\ < E, and this means that nk
lim _ oc ¢((n k) k the point (0'
=
¢ ((0)' This completes the proof of the continuity of ¢ (() at
<
-.------,----.-.=''''-.-.--=-=-=-=...
---=--=''':~~"-
44
.-;;.~-=;-~-~
~
._.
:-.---
....
"""-
-~-:::~~~..-=-:..-:;--=-~~~'--:-
"'.
-- -_ ,"",,",". -- -
-=-~---=-_·_,u
II. CONFORMAL MAPPING OF SIMPLY CONNECTED DOMAINS
If the boundary of the domain B also has th e property that, for any two distinct boundary points z 1 and z 2' the complex numbers z 1 and
z2' are~ot equal, then the function f(z) is continuous in the closed domain
!!..
To show this, let Zo denote a point on the boundary of the domain B.
If z n € B, where n = 1, 2"", and z n --> zo, then the corresponding sequence I~) converges to a specific point (0 on the circle ,(, = 1 because, otherwise, the sequence I(n I would'have two subsequences that converge to distinct points on
"="""..;;:!:==::.:-;~:
1(1
= 1 and then the corresponding sequences of the points zn would converge
to distinct points on the boundary of the domain B. It follows that (0 is inde pendent of the choice of sequence z n --> Z0 and this proves the continui ty of f( z) on B.
If we apply what has been said to domains bounded by Jordan curves, we ob. tain by virtue of the properties of such curves
Theorem 4.
1
)
1(' < 1
maps
a sufficiently small neighborhood of the point to· Since z '(to) ,j 0 (by the defini· tion of an analytic curve), we can choose a neighborhood sufficiently small that it will be mapped univalently by the function z
dt). The part gt of this neigh
=
borhood lying in the upper or lower half of the t-plane is then mapped into the part
8 % of the neighborhood of the point Zo lying in B and having a common boundary arc with y. The function (= f(z(t)) is regular in gt and it maps gt univalently into the part of the neighborhood of the point (0 = f(zo) lying in the disk 1(1 < 1 and adjacent to the circle 1(1 = 1. By Theorem 4', f(z(t)) is then continuous in gt up to the real axis. By the symmetry principle, it can be extended in a neigh borhood of to across the real axis and, in particular, it is regular at the point to itself and has a nonzero derivative at to. If t tion z = z (t), it is regular at Zo and t'(zo)
=
t(z) is the inverse of the func
-I O.
Consequently, the function
f(z(t(z))) = f(z) is also regular at Zo and f'(zo),j O. This completes the proof
A univalent conformal mapping of a domain B bounded by a
closed Jordan curve onto the disk disk 1(1 ~ 1.
4S
§3. CORRESPONDENCE OF BOUNDARIES
Ii
bicontinuously onto the closed
of the theorem. In particular, if a domain B is bounded by a closed analytic Jordan curve, a function f( z) that maps it onto the disk
Since the above considerations are of a local nature, we can conclude imme
diately that the following more general theorem is valid:
Theorem 4'. If a simply connected domain B has on its boundary a Jordan arc y, no interior point of which is a cluster point for the set of boundary points not belonging -to y, then the function (= f(z) mapping the domain B onto the disk I(I < 1 is continuous in B including the interior points of the arc y and it maps the arc y bicontinuously onto an arc y , of the circle 1(, = 1.
is regular in the closed disk
1(\ ~
'(I <
1 is regular in
Ii
and its inverse
1.
We conclude our investigation of special types of domain by noting that in Chapter X,
§ 1,
we shall use the theory of functions of a real variable to
investigate in addition the correspondence of boundaries under conformal mapping of domains bounded by rectifiable and smooth boundaries. 1) We also point out that the preceding theorems lead immediately to theorems on the correspondence of boundaries under univalent mapping of various domains
In the case of domains containing analytic arcs on the boundary, we have
onto each other. For example, a univalent mapping of one simply connected do
Theorem S. If a simply connected domain B has on its boundary an analytic Jordan arc y, no interior point of which is a cluster point of the set of boundary points not belonging to y, then f(z) is regular at interior points of the arc y and at such points f' (z) ,j 0; that is, the mapping is conformal.
main bounded by a closed Jordan curve onto another is single-valued and contin
Proof. Suppose that the equation of the arc y is z
=
z (t) , where a;S t;S b.
uous, including their boundarie8; in addition, if the boundaries of these domains are closed analytic Jordan curves, the functions representing the direct and in verse mappings are regular in the closed regions. In conclusion, we present another uniqueness theorem dealing with conformal
Let to denote a point in the open interval (a, b). Let Zo denote z(to)' The
mappings. We saw above that if all the boundary points of a bounded domain
function z (t) can be expanded in a power series in a neighborhood of the point
are ac-cessible, then a function z = ¢«() that maps the open disk
1('
B
<1
t = to' If we admit complex values for t, this series defines a regular function in
1) Caratheodory [1913].
1) Profound investigations of the correspondence of boundaries under conformal map ping have been made by the Moscow school of the theory of functions, which includes N. N. Luzin, I. I. Privalov, M. A. Lavrent'ev and others. A detailed survey of the corresponding results can be found in the book Thirty years of Mathematics in the USSR, pp. 324-332 (1948).
46
II. CONFORMAL MAPPING OF SIMPL Y CONNECTED DOMAINS
univalently onto the domain B is continuous on the closed disk ,(, ~ 1. In this cas e, the boundary C of the domain B has the representation z == ¢ (e for 0 ~
t ~
it
)
§4. Distortion theorems
== z (t),
A considerable portion of the theory of conformal mappi"ngs deal with the
217, where z(t) is a continuous function, so that C is a continuous
investigation as to what restrictions the requirement that a mapping be univalent
curve. Furthermore, since log[¢{() - ¢(O)]I( is obviously a regular function in
"I < 1
and is continuous on
harmonic function in
I(I < 1
"I" ~ 1,
imposes on certain quantities connected with such a mapping (for example, the
it follows that arg[¢{() - ¢{O)]I( is a
and is continuous on ,(,
< 1.
modulus of a function or the modulus and argument of its derivative), that
This means that the
is, on the degree of distortion produced by that function at various points in
increment in this argum~t as ( moves around the circle '" = 1 is zero. But
I(I == 1
this means that, as ( moves around the circle
a domain, etc.
in the positive direction,
These restrictions are expressed in various inequalities, the simplest of
the increment in arg[¢{() - ¢(O)] is equal to the increment in arg(, that is,
which will be given in the present section.
equal to 217. We take as the positive direction around the curve C the direction in which arg{z - zo)' where Zo = ¢
to),
.'1,
Our starting point for the derivation of a number of inequalities dealing with
acquires an increment of 271 when z
univalent functions is
completes one encirclement. By virtue of the continuity of the increment in
Theorem 1 (the area lheorem)l). Suppose that the function F{() = (+
~) J Z-Zo
(arg (z - Zo»e =:J (\
00,
al/( + ..• = (+ I.;=1 akl(k is regular in the finite plane, has a pole at (=
e
1(' > 1.
and is univalent in the domain
depending on Zo E: B, the positive direction around C is determined independently
Then,
00
2: n Ian I~ ~ 1.
of the choice of Zo and hence independently of the choice of mapping function
(I)
n=l
¢{(). Theorem 6. Suppose that all the boundary points of a simply connected bounded domain B are accessible. Then there is a unique univalent mapping of
(I
B onto the disk I < I such that three given points z l' z 2' and z 3 on the bound ary C of the domain B indexed in order 0 f their occurrence as one proceeds in the positive direction are mapped into three given points (1' (2' and (3 on the circle
47
§4. DISTORTION THEOREMS
1(' =
1 similarly indexed.
Proof. Suppose that the function" onto the disk
,el < 1.
=
¢(z) maps the domain B univalently
Corresponding to the points zl' z2' and z3 on
lei
around the circle in the positive direction. If we now map the disk valently onto the disk
1(1 < 1 in such a way that the points
Ie I < 1
expression of the following geometric fact: under a mapping of a domain where p> 1, by a function w
=
F{() , a part of the plane w
=
> p,
u + iv of positive
'I
area remains uncovered (the area principl e). This geometrical interpretation indi cates the course of the proof. Specifically, the circle = p, where p> 1, is mapped by the function w = F{() = u + iv into the closed analytic curve w =
F{pe it ) = w{t) , for 0 ~ t ~ 271, which serves as the boundary of a domain of area
S calculated from the formula
= 1
are three points (; , (;, and (~ arranged in order of occurrence as one proceeds
'I
The name" area theorem" is due to the fact that this theorem is the analytic
2"
S
=~
uni
(;,(~, and (; are
=}
2"
IlV'
dt
w (t)
+ W(t)
w' (t) :::-:- W'(i) dt
2"
=
mapped into (1' (2' and (3 respectively (which we know is possible and is
~ [pe it + pe- it 2
achieved by means of a fractional-linear function), then the composite function will yield the required mapping. The uniqueness of such a mapping is proved, as
X
uniqueness is usually proved, by contradiction. This completes the proof of the
pe it [
+2 pe-
00
'\1 ane-int + ;'n eint ]
+..:.. n=1 00
it _
1)
Gronwall (1914-19151.
+ n~neint"1 dr 2
'\1 nane- int
..:..
n=1
theorem.
2 pn
1 _
n
P.
- 'ltp
~
00
_
~ n [ an [2
'It..:..
n=l
2n
P
=--
,_-
-----=
,..--~-,.---
.
:
-
-
._,_0!!2.
-
_. . __ U I I . _ ~ ;
__
.____
_'.
_••
--
-
and this area is positive for arbitrary p
> 1. If we take the limit as
functions that assume no value in the domain
_._.___
_. .
~
- - " ' = " ' " -.."...--.--:----.. . " " - - - - , , _
~
-
...... -
-
..
--
I(I
49
Izl < 1,
then
Ic 21 :s 2.
With this bound for the coefficient c 2 , we can now easily obtain a covering
> 1 at more than p points.I)
:s
Izl < 1
valent in the disk
that disk under the function (2). Then c 1= 0 and the function
DO
!(Z)=Z+C2Z2+ ... = ~ cnz n
Izl < 1.
Then, the function
!p(z)=V'!(zP)=z+ ~ ZP+l+ ...
Thus, (3)
Izi < 1. The univalence follows where zl and z2 are distinct num and hence zf = z~; that is, zl =
(for p = 2, 3, ..•) is also regular and univalent in from the fact that, if we had fp(zl) = f/z2),
Izi < 1,
we would have f(zf) = f(zP = 1. But then, since the series (2) contains only terms with powers
zn p +l , where n=O, 1, 2,.·., we would have f/zl)=."f/z2); that is .,,=1, and
It follows that the function 1
r
c. 1
F(I.,)=-(1\ =1.,-2C
1(I
:s
we obtain the inequality 1e 2 / 2 1 I, so that Jc 2 1 2 with equality holding, that is, with c 2 = 2e i a., if and only if F( () = (+ eial(, that is, if and only if
W=
! (z) =
The function (4) maps the disk
(1
+ ze/J.lz)" =
Izi < 1
z-
2
a.i
e z
2+
.. -
lei ::: ~
with equality holding only when
then the disk Iwl
<~
image of the disk I zl
which issues from the point e- iI1/4, and which would include the point w = 0 if produced
I) See Goluzin [1940] and Alenicyn [1947, 1950]. 2) Bieberbach [1916].
IC 2+ l/cl = 11/c I -lc21
=2
I zl < 1,
(but not always a larger disk) is entirely covered by the
< 1 under that function. The circle 1wi = ~ contains points·
not belonging to the image if and only if fez) is the function (4). I) Koebe's theorem can be put in a different form: If a function fez) such that 1
zl < 1
and does not assume the value c in
then
2
This follows from th1:'application of the preceding theorem to the function f(z)l!' (0), which does not assume the value cl!'(O) in
Izi < 1.
If the. function
w=F(q=C+cxo+ t.l, + ...
(4)
onto the plane with a cut along the ray
in the opposite direction. Thus we have obtained the result
Hence
If' (0) I ~ 4c,
> 1. Then, as a consequence of the area theorem,
:s
II
Theorem 2. If the function (2) is regular and univalent in the disk
Izl < 1,
+.
1) +...
(
and I c 2 1 = 2, that is, c 2 = - 2eia.. However, this will be the case only for the function (4). Thus, we have obtained Koebe's covering theorem:
f. YJ is univalent in the domain
< 1.
{CO) = 0 is regular and univalent in
= z2' which contradicts the assumption.
r
_
n~)-Z+ CII+ C Z
ICIl++1~2.
(2)
n=l
TF2' where ."P
cf (z) A
is also regular and univalent in \ z\
Suppose now that the function
is regular and univalent in the disk
and suppose that c does not belong to the image of
f I(Z)-
maps the domain
'(I
> 1 univalently, if
_
:
._.
------
theorem for the same functions. Suppose that the function (2) is regular and uni
An important consequence of inequality (1) is the fact that jail 1, with equality holding that is, with a l = e ia if and only if a 2 = a 3 = .•• = 0, that is, if F«() = (+ eia.g. This last function maps the domain '(I > 1 univalentlyonto the plane with a rectilinear cut from the point w = 2ea.il 2 to the point w= _ 2e a.i/ 2.
zl
•
This is a sharp inequality_
A similar theorem holds for the corresponding p-valent functions, that is,
hence,
_
If the function (2) is regular and univalent in the disk
p -. 1, we
obtain (1).
bers in
_,0_-=,.--=,_----,
__
,-
§4. DISTORTION TIlEOREMS
II. CONFORMAL MAPPING OF SIMPLY CONNECTED DOMAINS
48
_._
~-:!!'"-
---
(5)
c does not belong to its image, then the
function
)
1) In this formulation, the theorem was proved by Bieberbach (1916]. Koebe(1907] proved only the existence of such a disk as that asserted in the theorem. This covering also holds for suitable p-valent functions (see Goluzin (1948]).
50
These conditional inequalities hold for arbitrary z in
(1' =Z+(C-rl O)Z2+ ... F z)-c
F 1 (z)=
<1
is regular and univalent in Izi
and, consequently,
Ic - aol
51
§4. DISTORTION THEOREMS
II. CONFORMAL MAPPING OF SIMPLY CONNECTED DOMAINS
'z\
< 1.
Let us see pre
cisely when equality can hold in them. Since they were obtained by integrating
. inequalities (7) with respect to r from 0 to r, for equality to hold in (8) and (9)
~ 2. From this we
obtain
for some function f(z) and some z in the disk Izi
< 1,
it is necessary that the
corresponding equality hold in (7) along the entire straight line segment from 0 to
Theorem 3. If the function (5) maps the domain I"~ entire boundary of its im~ge is contained in the disk Iw
> 1 univalently, then the
-
a oI ~ 2.
Another important application of a bound on the coefficient c 2 is the deriva
arbitrary z in Izi
< 1.
f
Izl < 1.
This inequality constitutes a distortion theorem 1)
*
for this class of functions. With regard to inequality (9), which constitutes a
I-eiaz eiaz)" ~ 4 arc sin I z
""'~\'1~1~12\ =C+~2C\l+...
_
the case of this function for suitable real a. Consequently, inequality (8) gives sharp bounds for If'(z)! as f(z) ranges over the class of functions that are uni
rotation theorem,2) we have, for the function (4) with z
(1C+Z) +zC -f(z)
is also regular and univalent in "~I
But this can be true only for
the function (4). By direct verification, we see that equality does indeed hold in
valent and regular in
Then, for
< 1, the function g(C)=
I c21 = Y2 1f"(0)/f' (0)1 = 2.
particular, it follows that
tion of bounds on the modulus and argument of the derivative of a univalent function. Suppose that the function (2) is regular and univalent in Izl
z, and this means that equality must also hold in (6) on the same segment. In
arg f' (z) = arg (1
< 1. Calculation shows that
1 (
_
r
(z)
~\I-2 (1 - zz) f' (z) -
=
+
Iz I
C 4dr ~ y 1 - r"
o
)
22 .
0,
I
<} ~=210 r Iz I
1-
2
l+r g 1- r '
that is, for this function, equality cannot hold in (9) at any z Consequently, since 113 2 1 ~ 2; we have
zr (z)
I t' (z)
-
2r" I . 4r 1- r" ~ 1- r 27
r=lzl·
(6)
If we replace the modulus with the real and imaginary parts in this inequal
O. Later, we
By starting with inequality (8), we can easily obtain upper and lower bounds for If(z)!. To obtain an upper bound, we integrate the second of inequalities (8)
0 with the point z. We obtain
over the line segment connecting the point z
ity and note that
ffi
f.
shall also give a sharp bound for largf'(z)1 (§1, Chapter IV).
(zrf' (z)(Z)) =
r
If (z) I ~ ~ II' (z) II dz I
(Z)) 0 ~ log If' (z)l, 3 (zr f' (z) = r or arg f' (z), or
o
1
Iz I
~
0
= ~ \1'(z)ldlzl ~ ~ "I+~. dr=~ " •.
we obtain the two inequalities
- 1-4+2r 0 log If' (z) I ~ 4+2r r" ~ or 1 _ r" ,
z
(7)
4 0 arg f'() 4 Z ~ -I- 2 ' 1 --"-~-o r r -r
To obtain a lower bound, we take the point on the image of the circle
Izl
=r
. that is closest to w = 0 and we draw a line segment between it and the point
w = O. If we then integrate the first of inequalities (8) along the curve L consti tuting the preimage of that line segment, we obtain
If we integrate with respect to r from 0 to r, we obtain the inequalities I-r f' I I+r -,.-,-, ~ I (z) ~ (I-r;"'
(8)
I argl' (z) I ~ 2 log ~~~.
(9)
l}Bieberbach [1916].
2) Bieberbach [19191.
52
IzI
= ~ II' (Z) II dz I~ ~ ,~--:',. dr
I/(z)1 Thus, in
Izl < 1,
I' (ek+dq
Iz I
(l
0
L
+I
Z
I
1)2'
,In particular for ~ = (';k+l -
we have
,
(1
+_,)2 ~ I/(z) -! 0
1
~_r (1 _ ..\2'
r=lzl·
only if f(z) is the function (4), as is the case
If' I
Now, if we set k
=
iThe number M depends only on the domain B and the set E; it is independent of of
closed domain contained in
and
Z1
Z 2'
B and, if we prove inequality
Izi
E and the boundary of B. This distance is necessarily positive. Let us cover the z-plane with a grid of squares of side d/4. The union of the closed squares of the grid that contain points in E is a closed domain
B'.
We have E C B' C B.
> R.
lie in adjacent squares of the grid and the number n of them does not exceed the number N of all squares constituting
B'.
The distance between the points
';k
1, 2",', n - 1) does not exceed d/V2 < 3d/4. The disks
=c+ C'lC'l + ...
is, for each k, regular and univalent in I~I with (8) we have in Izi < 1 1)\Koebe[19 101.)
< 1.
R, where R is sufficiently large,
Analogously, we have
Theorem 5.0 Let B denote a domain and let E denote a closed set contained in B. Then, there exists a finite positive constant L such that, for every funcJ
tion f( z) that is regular and univalen t in B and for arbitrary points z 1 and z 2 in E,
I/(Z'l) I ~ I/(zt) 1+ L II' (Zt) I·
(3)
Proof. We use the constructions made in the proof of the preceding theorem, again assuming E to be a closed bounded domain. Integrating inequality (2) and using (11), we have
1, 2, ••• , n, all lie in the domain B. Therefore, the function f(tk+ dC ) - f(tk) d· f' (ek)
=
This, itl~conjuction with what we have already proved, completes the
Let zl and z2 denote any two points in E. In the domain B', we can find an = Z2 such that any two consecutive points
Izl
proof of the theorem.
(11) for this domain, it
suppose that E is a closed bounded domain. Let d denote the distance between
';n
and z2' If we reverse the roles
show that this ifl~quality also holds for arbitrary z 1 and z 2 in the domain
must also hold (with the same M) for any set E contained in it. Let us first
n-tuple of points ';1= Z l' ';2"'"
zl
we obtain the other inequality of the theorem. If the domain B
arbitrary points Z 1 and Z2 on the circle
(11)
contained in B. Specifically, every closed set can easily be included in some
=
we have
1, 2" • " n - 1 and multiply the inequalities obtained, we find
)the form of the function f(z) and the choice of points
Proof. Without loss of generality, we may assume that E is a closed domain
for k
< %,
,contains the point 00, we can, by using the fact that inequality (11) holds for
~l ",;:: (Zl) M -= f' (z~) ~M.
I < d,
with \(\
(2)
f' (zs) I ~ m fl - t ~ m N-t = M. I f' (Zl) -=
Theorem 4.1) Let B denote a domain and let E denote a closed subset of B. Then, there exis ts a finite positive cons tant M such that, for every function f( z) that is regular and univalent in B and for arbitrary zl and z2 in E,
.;k
1+1 CI ~ (1-1 C1)3'
(l-{f= m.
So far, we have considered functions that are univalent in a disk. Let us now prove the general distortion theorem.
IZ -
';k)/d,
I
I' (tk+l) I 1+ 4 I f' (tk) ~
(10)
for suitable choice of a.
=
f' (ek)
3
Here, equality can hold for z
and ';k+l (for k
53
§4. DISTORTION THEOREMS
II. CONFORMAL MAPPING OF SIMPLY CONNECTED DOMAINS
~d
\/(~k+l) - I(~k) I ~ (I ~{r II' (~k) I = 12d If' (~k) I< l2Md If' (Zt) \.
Consequently, in accordance
I) \Koebe [1910].)
§S.
II. CONFORMAL MAPPING ON SIMPLY CONNECTED DOMAINS
54
If we successively set k = 1, 2"" ,
11, -
1 and add the resulting inequalities, we
55
CONVERGENCE THEOREMS
contain a neighborhood of the point z = 0 and satisfy the condition B 1 ::J B2 ::J
B 3 ::J .. • •
obtain
Theorem 1 (Caratheodory). 1) Suppose that we have a sequence of functions
If(Z2) - f(Zt) I ~ 12Md (n- I) If' (zt> I ~ 12MNd IF (Zt) 1, from which (3) follows with L
=
12MNd. If B contains
00,
z
we again use the max
imum modulus princip,le.. From this last theorem we have a
point in a domain B. Suppose that every function in a family of functions that are
If(zo)1
< M1 and
l['(zo)1
fn fn ( ' )
< Mr Then,
this family uniformly bounded in B and the condensation principle can be applied to it.
For the sequence Ifn(,)} to converge in
convergence exists, it is uniform inside the disk
to a finite function, it is
1'1
< 1. If the limit function
1'1
const, it maps the disk < 1 onto the kernel B, and the sequence {.J. (z)} of inverse functions ¢ (z) converges uniformly inside B to the function ~n
n
¢(z) inverse to f(').
i '-- \
convergence of a sequence of univalent functions. We present the following definitions. Suppose that we have a sequence of where p> 0, that belongs to all the domains in B n , we de
fine the kernel of this sequence of domains as the largest domain containing z n
B
f n (,)
--+
f(')
in
1'1
< 1.
Let us consider
B. Let us suppose that this is not the case. Then, there exists a disk 'I zl < p, p> 0, that belongs to all the domains Bn' Consequendy, by the Schwarz lemma as applied to the disk \zl < p, we obtain I¢~(O)I ~ lip, where ¢n(z) is --+
n
the inverse of the fun~tion
univalent domains B l' B2 , · · · in the z-plane, each including z = O. If there =
0
from
some n on. By "largest domain" is meant the domain containing any other do main possessing this property. If such a disk does not exist, the kernel of the sequence of domains B1' B 2 , · · · is defined to be the pointz = O. We shall say ' • " converges to the kernel B, and we B, if every subsequence of these domains has n B as its kernel. In particular, if a sequence of simply connected domains B l ' B 2 , · · · , Bn"" that include z = 0 converges to the limiting domain B (also in cluding z = 0) in the sense that all boundary points of the domains B n from some 11, on are arbitrarily close to the boundary of the domain B and all points of the boundary of the domain B are arbitrarily close to the boundaries of the domains Bn , then this sequence has the domain B as its kernel and it converges to that kernel. This applies also for a sequence of domains B n that include z = 0 and satisfy the condition B 1 C B2 C B 3 C ••. or for a sequence of domains Bn that
fn(')'
This means that f~ (0) ~ p. But then, according
to inequality (10) of §4 as applied to the function
we have p 1'1/(1 +
in(,)
1'1)2 :s Ifn(')\.
does not converge to
f(')
fn(')lf~(O) in the disk
1'1 < 1,
Consequently, the sequence of the functions
== 0, which contradicts t he assumption. Since
the same reasoning can be applied to every subsequence of domains among the
81 ,B 2 , · · · , we have Bn-+B.
Now, let us suppose that
f~(O)
that the sequence of domains B l ' B 2 shall denote this by writing B
1'1 < 1
first the case f(~) == O. We shall show that then B is the point z = 0 and that
An important question in the theory of conformal mappings is the question of
such that an arbitrary closed subset of it belongs to all the domains B
onto a domain B n"
Proof. 1 0 • Necessity. Suppose that
a sequence of domains
< p,
1'\ < 1
it') ~
§5. Convergence theorems on the conformal mapping of
exists a disk Izi
n
maps the disk
'necessary and sufficient that the sequence IB n I converge to the kernel B, which is either the point z = 0 or a domain having more than one boundary point. When
Corollary. Let M1 and M2 denote two positive numbers. Let Zo denote a
regular and univalent in B is such that
= f ('), where 11, = 1, 2,'" , that are regular in the disk I" < 1. Suppose that n (0) = 0 and ['(0) > 0 for 11, = 1, 2,' . '. Suppose that, for each 11" the function
--+
11
f(')
!ell,
1\2
~ O. Then, it follows from the inequality
~ Ifn (q I ~f~ (0) ~- ...
, that the numbers f~ (0) are bounded and that the functions
f n(')
are uniformly
1) Caratheodory [1912]; Bieberbach [19131. Marku~evi-:: [19361 showed rhar the condi tions of 'Theorem 1 are also nece ssary and sufficien r for con vergence in mean of z) I to n f ' (z), that is, necessary and sufficient for
It'(
lim ~ ~
n--+OJ B
where
r'n (z)
I
f'
n
(z) -
I' (z) 12 da =
is taken equal to 0 outside the domain B n"
0,
rm
. err
--esC'
----------g;;-.-- -
~:z;m:;:F -;.'-----=-j-:-.:--.;-. - 7 -.=-~---
-.:,:::_--;;.;;--~~~~~";"-
§s.
II. CONFORMAL MAPPING OF SIMPLY CONNECTED DOMAINS
56
bounded inside the disk
1(1
< 1.
the sequence of the functions
fn «()
converges uniformly inside the disk
1'1 < 1.
1(1
<1
"I < 1 univalently onto a domain B containing z = O. B possesses the property that an arbitrary closed domain S* is contained in all the domains Bn from _some non. • Let 8> 0 denote the distance from B* to the boundary of B. Let us cover the z-plane with a grid of squares of side 8/2 and let us consider the domain B* z = f«() maps the disk
constituting the union of all (closed) squares containing points of
C B. and
domains
A.
S·
S. C B.
and
B.
Let
:4.
Then,
t
f'
I,,
.1
and A* denote respectively the preimages of the
under the mapping z = f«() in
is contained in the disk
S*.
1(' < 1.
Let 8 1
>0
1'1 < 1.
Then,
:4.
C A*, and
denote the distance from
S·
- zo I ~ 8 1 for any point Zo E: S*. On the other hand, there exists an N> 0 such that, for n> N, we have Ifn«() - f«()1 < 8 1 on the boundary of A. since the sequence of the functions f n «() converges uniformly inside the disk \'1 < 1. Consequently, for every n > N, Rouch!? s theorem can be applied to the
f«(o)' Let ( denote a positive number such that the circle ,,- (0 \ = ( is con tained in the disk 1'1 < 1. Let m denote a positive number such that If«() - zol > m on that circle. On the other hand, for k> K, we have If «() - f«()1 < moo the circle nk \( - (0 I = (. We again conclude from Rouche's theorem that the functions f «() - Zo for k > K each have a zero, which we denote by (k' in the disk
(01 <(. We have fnk «(k) = zo, and (k = ¢n (zo) for k > K. As k --> ""', the
(k -> ¢(zo)' so that ¢(zo) belongs to the disk (01 Since ( is an arbi
trary positive number, we see that ¢(zo) = (0' Thus, we have shown that Zo = f«(o) implies (0 = ¢(zo); that is, ¢(z) is the inverse of the function f«().
I'-
k - :s (.
f«(), the sequence I¢__ on(z)1 itself converges in the domain B' to the function ¢(z). The function (= ¢(z) maps the domain B' onto a domain A. Since I¢n(z)\ 1 in B n, it follows that I¢(z)! 1 in B'j that is, A is contained in
:s
the disk
function
:s
1'1 < 1.
This last fact enables us to show that B'
In (C) - Zo = (jn (~) - I (C»
+ (j (C) -
ing z = 0 that possesses the properry that any closed subset of it belongs to all domains Bn from some n on; that is, B is the kernel of the sequence of domains B n' It follows that B has more than one boundary point.
contains an arbitrary point Zo E: B·. Thus, every closed domain contained in B n
from some
Let us now take an arbitrary subsequence B
n on. Let us show that B is the largest domain possessing this property.
1
¢ n (z) is bounded on an
sequenc e I¢ n(z)l. Let us choose from the sequence I¢n(z)) an arbitrary subsequence
l¢ nk(z) 1 that
con verges uniformly inside the domain B' to a regular function ¢(z) such that ¢(O) = 0 and ¢'(O)~ O. Since ({J'
(0) = lim ({J~ (0) = lim n-oo
1"
n-1>oo J n
1(0)
=
t,I(O)
¢ '(0) > 0, that is, that ¢ (z) is nonconstant and, consequently,
,B n 2 , · · · , of the domains B n .
fn «()" •. of the functions f «(), which converges to the function f«() , we see 2 n that the function z = f«() also maps the disk 1'1 < 1 onto the kernel of the sub
arbitrary closed subset of the domain B' and the set of such functions is uni formly bounded. It follows that the condensation principle can be applied to the
n1
If we apply the conclusion just obtained to the corresponding subsequence fn «(),
Let B' denote any domain other than B that contains z = 0 and possesses this properry. Then, from some n on, each function
If Zo E: B' and (0 = ¢(zo)'
main B' belongs to the domain B also. Thus, B is the largest domain contain
f n «()
and hence every closed subset of B is contained in all the domains B
c B.
then '(0 I < 1 and, consequently, Zo = f«(o) E: B j that is, every point in the do
Zn).
- Zo has a zero in the domain A*. This means that the image of the domain A. under the mapping z = f.n «(), where n > N, _
In accordance with this theorem,
57
sequence I¢ n (z)1 and the limit function is always the function ¢(z) inverse to
If«()
it foliows that
CONVERGENCE THEOREMS
Since this line of reasoning can be applied to any convergent subsequence of the
to
the boundary of the domain B•. Then, on the boundary of the domain A., we have
__ r~~~~::~::;
¢ (z) is the inverse of the function f«(). Let (0 denote an arbitrary point in the disk "01 < 1. Let (0 (E: B) denote the point
Consequently, the function
Let us show that
"':':'::-':~";-;:':Z"~;'-~~~-i;:':-~~~;-~~':'C"=~~::;:'-~-~::::-:=;::''::;"~",,-
univalent in B'. Let us show that
Therefore, in accordance with Vitali's theorem,
to the function f«(), which is univalent in
S·
S't-~:7~:~-::~~~~~2~---:::'-':;::=~:::_:'~~;;':~-~"~'~,~i££B'--::::':
sequence IB ~.
Bn
-+
nk
I,
and therefore this kernel coincides with B. This shows that
B, so that we have proved the necessity of the condition of the theorem and
have also proved the supplementary conclusions mentioned in the theorem.
It
2°. Sufficiency. Now, suppose that B n
->
B and suppose that the kernel B
.~
I J
,,'
either consists of the single point z
=
0 or is a domain having more than one
boundary point. First, let us suppose
B consists only of the point z = o. If
converge to 0, there would be a subsequence
Ifnk «()I
f'n (0)
did not
such that, for each k,
,__<",~
If'nk (0)1> f,
f>
consists of the point z
0, and
1"
single poi~t. But if {~(O)
Consequently, the kernel B would not consist of a
->
0, then, for all ( in the disk I(j
< 1,
->
0 in
1'1 < 1.
f' (0)
are bounded, because
If'nk
00
and we would
IS uniform inside the disk
1(I :s 1,
1'1 < 1.
This com
we give the following theorem, confining ourselves to domains of the
Theorem 2.1) Let {B
'"l
<1
on, contain an arbitrary given disk
under the mappings z =
Izl < R,
f"k «),
from some k
which would contradict the conver
gence of {B n 1 to B. It follows from what we have proved that the functions f n <0 1'1 < 1. Let us suppose now that the se does not converge at the point (0 such that 1(01
Ifn , «)I
distinct functions [,(0 and
and
If" ,,«) I
that converge in
< 1. Then, there 1'I < 1 to two
f.. «).
If neither f*«) nor f** «) is identically equal to zero, then, from the proof of necessity as applied to the sequences {fn' «) 1 and {fnll «) I, we obtain the result that the functions z
= f*«) and z = f....
kernel of the sequences {B "
,I
and {B
0, f:(O) > 0; f.. (O)
II
n =
I,
1'1 < 1
onto the
that is, on to the domain B. Note that
0, f:.(O)
> O.
n
=
1, 2, " ' , denote a sequence of simply connected
note the boundary of B
"
0 and each bounded by a Jordan curve. De
=
I converges to Ifn«) I denote a se-
by C . Suppose that the sequence {B n
1'1
are uniformly bounded inside the disk exist two subsequences
I
n
a
q uence of functions f n «) such that, for each n, f n (0) = 0, f'n (0) > 0, and f n <0 maps the open disk < 1 onto the domain B n. For the sequence {f,,«)1 to con-
If"k~)I;;:;':lf~k(O)ll1lCJ'\1 that the images of the disk
n
domains each including the point z
domain B (its kernel) bounded by a Jordan curve C. Let
conclude from the inequality
=
to a finite function. Furthermore, in accordanc'e with
Jordan typ e.
otherwise we would have a subsequence
we have {.(O)
disk
us look at the case when the kernel B is a domain having more
Ifn«)!
<1
Vitali's theorem, the convergence
in the open disk ,( I < 1. For the convergence of univalent functions in the closed
This proves the convergence of the se
than one boundary point. In this case, the numbers
quence
1'1
Ifn«) I must
Theorem 1 gives the conditions for convergence of univalent functions only
quence {f «) I.
" Now, let
not converge to a kernel.
pletes the proof of the theorem.
Ifn(~)I~lf~(O)I" I~~".--+O. Consequently, fn«J
IB n 1 does
Again, we have reached a contradiction. Thus, the sequence converge in the disk
< 1.
0 and that the kernel of the other sequence is
=
some domain, so that the sequence
ilCI " • IJ~nk (C) I;;:;.: If,'"k (0) I (1 +Ie/I C1)1 > ~ throughout, the disk
S9
§5. CONVERGENCE THEOREMS
II. CONFORMAL MAPPING OF SIMPLY CONNECTED DOMAINS
58
Therefore, by virtue of the
uniqueness asserted in Riemann's theorem, we have f.«) :; f.... «), and this con tradicts our assumption.
verge uniformly on the closed disk
to a func tion z
=
f«) that vanishes at
1(1 < 1 onto the do- . > 0 there exists a number
0, has positive first derivative at 0, and maps the open disk main B, it is necessary and sufficient that for every
f
N> 0 such that, for n> N, there exists a continuous one-to-one correspondence between the points of the curves Cn and C such that the distance between any point of C n and'the corresponding point of C will be less than f. Proof of the necessity. If f}O sidering the points z
n
=
->
f n «) and z
=
f«) uniformly on
I( I :s 1,
then, by con
f«) as corresponding points on the curves
1(I
1, this establishes a one-to
C
and C for each value of ( on the circle
"
one
continuous correspondence between C nand C with the property stated in
=
the theorem. Proof of the sufficiency. We first prove that under the condition of the theorem, the set of functions f,,«) is equicontinuous on the circle
1(1 =
1. Let
us suppose the opposite. Then, there exists a 8> 0, a sequence of positive integers n , k
On the other hand, if one of the functions f.«), f**«) is identically equal
1'1 :s 1
k
quences {(~
I
=
1, 2," "
and {(~
I of
increasing without bound as k increases, and two se points on the circle
''I =
1 such that when k
->
00,
to zero, the other cannot be identically equal to zero or they would again be identically equal to each other. By applying the necessity of the condition of the theorem, we conclude that the kernel of one of the sequences {B nl
I,
{B nit I
1) RadO [1922-19231. In the case of domains with arbitrary boundaries, the question
has been thoroughly investigated by Marku~evit [1936].
1" I'" Sk - Sk
the part of the arc 0, and
-->
'j
nk (Ck)
'- Ink (C;;) I
We may assume that the sequences I(~ limit a such that
I al = 1
I
points
> 8.
and
a
1(: I
converge to a common
k
because, if this is not the case, we can shift to a sub
nk
lak
I and Ib k Lwhere
«(;), converge, let us say, to points a and b. The
points a k and b" lie on e nk and the points a and b lie on e. On each curve en", we denote by ok the arc a"b" corresponding to the arc (~(~ (which is cons tdcted to the point a as k --> (0) and we denote by T" the remainder of the curve the distance between
e , Since we are assuming that,
a" n"and
for sufficiently large k,
b" exceeds B, there exists for such k a point
e"
on ok at a distance no less than 8/2 from a k and bk · We may assume that e k --> e, where e € because otherwise, we could shift to a subsequence
Ie - al
~ 8/2 and
Ie - bl
We denote this arc by
e is
~ 8/2. Consequently,
note an arc on
e
and the remainder of the curve
0
e - 0'
and less than the distance from
> K,
note a positive number such that, for k one correspondence between sponding points of
enk
and
e by
T.
0'
.• :'.
0'
to
T.
> K, there are no points
T"
in the open disk
T
k
that lies in that disk, Let
<
'i:(':js
:"',fore for k "
that is,
e
k
<
and d
k
and d~ lie on the arc
1
ut as k
Ie
00.
B , Setting nk
f"
=
Now, let Zo denote an arbitrary
< .,,/4. Around this point, let us draw
> K, this arc lies out
Zo lies in the interior of Bnk · The limiting values of 1!/J,,(z)1 asz (k
Ie
that lie in the disk
since there are no points of the arc
Tk
Iz -
zol
in the disk 1z - Zo f
= f k,
< .,,/4
1< .,,/4.
There
> K 1 ' we have
-->
00,
the sequence !/Jk(zO) converges to ¢(zo) - a, where ¢(z) is a
< 1. Consequently, ¢(zo) - a = 0; B in the disk Iz - el < .,,/2 and hence every
clion mapping the domain B onto the disk I zl at is, ¢ (z)
=
a for all
points z €
where in B, which is impossible. This contradiction proves the equicontinuity of e functions oded on
(1)
f n «()
1(1
on the circle
1(1
=
1. The functions
== 1. Reasoning as in subsections
fn «()
are also uniformly
2° and 3° of the proof of the
'ndensation principle, let us show that the sequence Ifn «() I contains a subse ence that converges on some set that is everywhere dense on the circle ,(, d then let us prove that this sequence converges uniformly on
'(I
=
1.
== 1. Such a
ice can be made from an arbitrary subsequence of the sequence If «() I, This
corre
abIes us to show that the sequence Ifn «() I itself converges on
n
1(' =
1.
Specifically, if the sequence Ifn «() I did not converge at some point on , =
<1j,
Since e k Eo" and d k € Tk , it follows that separate the points a" and bk on en" Therefore, corresponding to 0'.
Ie
I ~k (zo) I ~ II Bk 2 "'-1.
I c;.- CI ~I c;.- Ck 1+1 Ck- c I
1+ldk-c
-->
'.~;~<"Pproaches boundary points of the domain Bn
sponding to the points e k and dk • We have
Id;'-ci ~Id;' -dk
¢n (z) - a, where ¢n (z) is the
a positive integer. For k greater than some number K 1
2'~,\side Bnk and
Let K de
e
=
i;: the circle Iz - Zo I = .,,/2. On this circle there are exterior points of the domain B
Iz - el < .,,/2.
and d~ denote the points on
Ie
,:,i;;and hence an arc y lying outside B and having a central angle of 2Tl/m, where m
To show this, let us suppose that, for some k> K, there exists a point k
k> K, every point of the curve en in the disk
k
1
de
IC-Ckl<~.
corresponding to
and its distance from a exceeds .", which contradicts the
!{i:f maxz€o-k !/Jk(z)/, we see that f k --> 0 as k ",f ',point of B that lies inside the disk Iz- eJ
there exists a continuous one-to
Ib-bkl<~,
e
~~;'Iemma 2 of §3 is now applicable. In the present case, M = 2 and
an in
Let
is an arc containing one of the
Consequently, the point on
inverse of the function fn «(). This function is regular in
e nk and e such that the distance between corre e is less than .,,/2 and
la-ak'<~'
ak'
For k> K, we define the function !/Jk(z)
points, inside the arc o. Let." denote a positive number that is less than the
d €
I
that includes e and that is contained, together with its end
distance from e to
Then, for k
0
enk
between e~ and d~ on
is a point of the arc ok'
\fl'::;~ not exceed
terior point of one of the arcs into which the points a and b partition the curve
e,
0'
let us say,
:.,'\i" .. '
e
If «() I. We have nk
lies on the arc
Iz - el < .,,/2
sequence Ifnk «() I of functions satisfying this condition and keep the same nota
a = fnk«(~) and bk = f
k
b",
or
ale
first of inequalities (1). Thus, for
tion for it. Analogously, we may assume that the sequences
61
§5. CONVERGENCE THEOREMS
II. CONFORMAL MAPPING OF SIMPLY CONNECTED DOMAINS
60
de ,
1,
we could choose two subsequences of it that converge uniformly on the
1(1
=
1 and consequently in the disk
e regular in the open disk
1(' < 1
'(I < 1
to two distinct functions that
and continuous on the closed disk
1(' S 1.
"::8u,t this is impossible because, by Theorem 1, the sequence lfn «() I converges in
ak
=
OLE:""
62
--------~--,_____---_7c_~'o_
--
fh-----------
--- --------------
-
n=c=_- ::---
----,--,,--,--
-:_-.j,w-- __'~~..E=t!!!!!.f'WfWF!,jit::~:,:~---
§6. MODULAR AND AUTOMORPHIC FUNCTIONS
II. CONFORMAL MAPPING OF SIMPLY CONNECTED DOMAINS
'I
1<:"1 < 1 to a unique function. Having proved the convergence of the sequence
L
Ifn (<:") I
mapped into a circle and the circle 1zl
on the circle
=
1, we again keep in mind the equicontinuity and uni
form boundedn ess of the functions
fn (<:")
•
to show, just as in subsection 3 of §2 of Chap-.
ter I, that the sequence Ifn(<:")l converges uniformly on the circle \<:", hence on the closed disk 1<:"1
:s 1.
~Jg':_.T'?ift;'!!fI88~tl~@i!#jlOjiU:-
=
1 and
Obviously, the limit function is f(<:"). This
completes the proof of the theorem.
l'
L
2'
63
L 3 • This is true because under these inversions, an arbitrary circle is =
1, being orthogonal to all the circles
across which the extension is made, is mapped into itself. In the domains B l' B 2' and B 3' the function <:" = p. (z) is regular and it maps each of these domains onto the halfplane ':5 (<:") < O. Consequently, p. (z) is regular in the domain B' constituting the union of the domains B, B1' B2 and B 3" However, it is no longer univalent in this domain since it assumes at three points of the domain B' every
§6. Modular and automorphic (unctions
value of the halfplane ':5(<:") < O. If we think of three halfplanes ~(<:") < 0 ob
Among the different functions that map simply connected domains onto each other, of great theoretical importance are the functions that map certain circular
tained by mapping the three domains B l' B 2 and B 3 as lying one above the other and connected with the halfplane ':5(<:")
>0
along segments L;, L; and L~ re
polygons, that is, domains whose boundaries are arcs of circles, either onto a
. spectively, we obtain a Riemann surface onto which the function <:" + p.(z) maps
disk or onto a halfplane. In the present section, we shall study the nature of such
. the domain B' bijectively. On the boundary of B', the function <:" = p.(z) is real. The domain B' is a circular hexagon with zero angles inscribed in the circle
mappings. Let us consider first the case in which the domain B is the interior of a reg ular circular triangle with zero angles inscribed in the circle Izl
=
1 (see Figure 6).
The sides of this triangle are obviously orthogonal to the circle
Izl
= 1. Suppose that a function z = ,\(<:")
halfplane ':5(<:")
>0
00
1
)
the si~ sides of B' into the domains obtained from the domains B , B 2 and B 1 3 by inversions about the corresponding sides. The function p. (z) is regular in each
> 0.·
Let us think of these halfplanes as being distinct and joined along suitable seg
are mapped into the
ments L;, L; and L~ of the real axis with the halfplanes ':5(<:")
vertices of the triangle. Let <:" = p. (z) denote the inverse of that function. Without knowing
this procedure with the domain B'; that is, let us extend the function p. (z) past
of the six newly obtained domains and it maps them onto the halfplanes ':5 (<:")
univalently onto the domain B in such
a way that the points 0, 1, and
Izi = 1. The sides of this hexagon are orthogonal to that circle. Let us repeat
< 0 that we al
ready have. Consequently, the function <:" = p.(z) is regular in the newly obtained
the explicit expres
sion for the functions ,\(<:") and p.(z), we can, by using
domain B", which is a circular duo decagon. It maps B" objectively onto the
the principle of symmetty, ascertain the essential prop
Riemann surface consisting of seven upper and three lower planes connected in a
erties of these functions in a purely geometrical manner.
Figure 6
We denote the sides of the triangle by L 1 , L 2 , and L 3 and the segments of the real axis corresponding to them by L;, L;, and L ~ . The function p. (z) , which is regular in B and continuous on B except at one
suitable mann er along L;, L; and L ~ .
If we continue this extension process indefinitely, we get, on the one hand, a "modular" grid in the z-plane composed of an infinite set of circular triangles with zero angles all contained in the disk
Izl < 1
and having sides orthogonal to
of the vertices, assumes real values on the boundaty of the domain B. Conse
the circle Izl
quently, in accordance with the symmetry principle, it can be extended past the
and consisting of an infinite set of halfplanes connected with each other along
sides L 1 , L 2 , and L 3 into the domains (which we name B 1 , B 2 , and B 3 respec tively) obtained from B by an inversion about the circles of which L 1 , L 2 , and
suitable segments L;, L; and L; of the real axis.
L 3 are arcs. The domains B1' B 2 , and B 3 are also circular triangles with zero angles inscribed in Izl = 1. Each of them is adjacent to B along one of the sides
ing the modular grid coincides with the entire disk 1zj < 1. To do this, it will be
=
1 and, on the other hand, a Riemann surface lying over the <:"-plane
Let us show that the domain B* consisting of the union of all triangles constitut sufficient to show that the set of vertices of the triangle of the grid is everywhere dense on the circle
Izl
=
1. Let us suppose the opposite, namely, that this circle
Contains an arc y that does not include any vertex of any of the triangles consti 1) An explicit expression for the function A(
0
will be given in § 1 of Chapter HI.
tuting the grid. Then there exists a sequence
Ii),
n
=
1, 2, •.. , of arcs of the
'''',
64
II. CONFORMAL MAPPING OF SIMPLY CONNECTED DOMAINS
grid, each arc resting on a y-containing arc of the circle
Izi
=
1, that converges
to an arc L of that circle. Obviously, the arc L is orthogonal to the circle
Izl
=
I and it rests on an arc y' containing y. Then, there are no points of the
domain B* betweeu y' and L. But this is impossible because, for sufficiently large n, the inversion about the arc ln of the entire polygonal domain boundary contains this arc maps the point z
=
B(k)
whose
0 into a point between y' and L,
and hence there are points of B* between y' and L. This shows that the set of vertices of the modular grid is everywhere dense on the circle that the domain B* coincides wi th the entire disk tion /l. (z) is regular in the disk
Izl < I
Izl < 1.
Izl
=
1 and hence
Therefore, the func
and it map s that disk bijecti vely and con
formally onto the infinite-sheeted Riemann surface mentioned above. The function /l. (z) assumes every value other than 0, 1, and
disk
Izl < 1.
00
infinitely many times in the
On the other hand, it does not assume these three values anywhere
in that disk. With regard to the points of the circle
Izj
=
65
§6. MODULAR AND AUTOMORPHIC FUNCTIONS
1, these are all singular
points of the function /l.(z} because an arbitrary neighborhood of each such point
automorphic function with an infinite group. We obtain other very simple automor phi c functions if we con sider a mapping on to the hal fplan e ~ «()
B contained in the disk
Izi < 1,
>0
of a domain
where B is an arbitrary circular triangle all sides
of which are orthogonal to the circle
Izl
= 1 and all interior angles of which are
nonZero. If the interior angles of this triangle have values
TTlm l , TTlm 2 , and
"1m3, where m l , m2 , and m3 are positive integers, then, just as in the case of modular functions, the function (= ¢(z), which maps the domain B univalently onto the halfplane ~ «() > 0, can be extended to the entire disk Iz I < 1. Thi s extension
Izl < 1 except at points that are mapped into and are poles of ¢ (z), and it maps the disk Izl < 1 onto a many-sheeted Riemann surface lying over the entire ( plane. To prove the extendability of ¢(z) to the entire disk Iz[ < 1, let us prove firstthat the vertices of the triangular grid, all of which now lie in the disk Izl < 1, have .DO cluster points inside the disk Izl < 1 (see Figure 7). Let us suppose, to the is regular in
contrary, that
00
Zo
is a cluster point of the set of vertices of the grid and let
Iz k },
contains infinitely many triangles of the modular grid and hence /l. (z) assumes in the subset of this neighborhood contained in and
00.
Izl < 1
all values other than 0, 1,
Consequently, /l. (z) cannot be extended outside the disk
Izl < 1
and this
disk constitutes its entire domain of definition. In this case, we say that the circle
Izi
=
1 is the natural boundary for /l.(z}. The inverse function A«(}, de
fined by any element, can be extended to the (-plane along an arbitrary path that does not pass through 0, 1, or
00.
Here, IA«()
<1
everywhere, though A«(} is
obviously a multiple-valued function. The function /l. (z) is called a modular function and the Riemann surface onto which it maps the disk
Izl < 1
bijectively is called a modular Riemann surface.
This name is due to the connection between /l. (z) and certain quantities encoun tered in the theory of elliptic functions and called moduli. The modular function /l.(z} possesses the remarkable property of invariance
Figure 7 for k = 1, 2,. ", denote a sequence of distinct vertices of the grid that con
under a certain group of fractional-linear transformations of the argument. Specif
'verges to zOo Since these points zk are vertices of triangles I'1 k , for k = 1, 2, ••• , obtained from B by an even number of inversions, that is, by certain
fically, fractional-linear transformations composed of an even number of inversions
fractional-linear transformations z I = Tk(z), where k = 1, 2"", of the disk
of the type encountered above constitute a group of transformations of the argu
lzl < 1
ment under which /l. (z) does not change its values. Single-valued analytic func tions that remain unchanged under a certain group of fractional-linear transforma tions of the argument are called automorphic functions. Examples of automorphic functions with infinite groups are trigonometric and elliptic functions. Next to elliptic functions, a modular function is the simplest nonelementary example of an
into itself included in the group of transformations that leave the function
¢(z) invariant, it will be sufficient to show that the functions Tk(z) map the ver
tices of the domain B into a set of points that has no cluster points in
Izl < 1.
We may assume that the z k are such that the triangular grids with vertices at the points z k for different values of k have no points in common, since otherwise we could shift to a subsequence of the sequence {z k} for which this would be the
66
II. CONFORMAL MAPPING OF SIMPLY CONNECTED DOMAINS
case. Let a denote one of the vertices of the domain B and suppose that the
Iz - al < P,
P > 0, lies entirely in a domain composed of triangles of the grid with vertices at a. The functions z' = Tn(z), n = 1, 2"", map this disk onto disjoint disks K n contained in Izl < 1. Since the functions disk
w= are univalent in
Izl < 1,
Tn (a+pz)- Tn (a) pT~(a)
the disk
these functions map the disk
Iwi < ~
Izl < 1.
=z
+ .,.
with zero angles, which is impossible. On the other hand, if only two of the points
a, {3, and yare distinct, the sequence of these triangles converges to some arc of a circle orthogonal to the circle
series
=
lies entirely in domains onto which
This means that, for each
n,
,z 1 with some point of the domain B. This contradiction shows that the triangular grid entirely covers the disk 1zl < 1.
the disks of
2
z' '- an
::--_-='-
1- anz'
'=
Tn (a), the
§7. Normal families of analytic functions. Applications
An infinite family
points in
Izl < 1.
has a subsequence that converges uniformly inside B to a regular function or to
eian
r
1 -I a II
00.
This proves that the sequence la n 1 has no cluster
00.
The situation is analogous with the remaining vertices of the
domain B. Since each of the points z k is the image of a vertex of the triangle B under one of the functions z' no cluster points in 'zl < I,
=
'ill "" I[(z) I of functions that are regular in a domain B is
said to be normal in B if every sequence of functions belonging to that family
1-1 all = so that Ia n I -> 1 as n --
sults in turn lead to considerabl e further development of questions on the con by Montel, of a normal family of analytic functions (d. Montel [1936]).
az
where a n is a real constant. Therefore T~ (a)
The results obtained above on conformal mapping owe their generality in many ways to the principle of condensation of analytic functions. But these re vergence of analytic functions. To show this, we introduce the concept, proposed
z- a -= e ia n ---_, 1-
1, which again is impossible. Finally,
toO is impossible since all these triangles contain points of the segment joining
converges. Then T;(a) ~ 0 as n ~ 00. If an Tn (z) is obviously obtained from the equation
I:=ll T; (a)1
Izl ""
if a, {3, and y coincide, the triangles must be constricted to a point. But this
radius not less than' T'(a)/4!p lie in K n . Since these disks are disJ'oint and are n contained in Iz'l < I, the sum of their areas is finite and, consequently, the function z'
Tn (z), it follows that the sequence Iz k 1 also has
A family is said to be normal at a point Zo € B if it is normal in some neigh.'
borhood of zo'
If a family 'iJl ""
l[Cz)\
is normal in a domain B, it is obviously normal at each
point Zo in B. Let us prove the opposite, namely that if a family
'zl < 1
belongs to one of the
00.
that there exists a point z I in the disk
z II < 1 that does not belong to any tri angle of the grid. Let us draw a line segment from z I to some point of the domain
center at
B, We conclude that this segment contains points of an infinite set of triangles of the grid since it would otherwise be possible to get from B to zl by means of a
only on the boundary of B. Now, let I[}z) I, n = 1, 2,' . "
finite number of inversions and the point z I would belong to some triangle of the
the sequence Ifn (z)1 has a subsequence
f3 and y, which must a, f3, and Y are distinct,
that the sequences of their vertices converge to limits a,
Izl "" 1.
If all three points
the sequence of the triangles themselves must converge to a circular triangle with vertices a, {3, and y and with sides orthogonal to
Izl ""
1, that is, to a triangle
1[( z) I of
points of the domain B with rational coordinates and arrange it in a sequence
triangles of the grid or lies on its boundary. Let us suppose the opposite, that is,
grid From this infinite set of triangles, we choose a sequence of triangles such
=
Let us take the set of all
IT k I, k = 1, 2,""
1
'ill
functions that are regular in B is normal at each point Zo E: B, it is normal in the domain B. We may assume that B does not include
Let us show now that any point z in the disk
necessarily lie on the circle
67
§7. NORMAL FAMILIES OF ANALYTIC FUNCTIONS
sequence
Tk
and let us denote by Pk the radius of the largest disk with
in which the family 'iJl is normal. One can easily see that the sub
Ip nk 1 can
converge to 0 only when the corresponding points
quence of functions of the family
Iz - TIl < p/2
3JL Since
to a regular function or to
00.
00.
cluster
(z - TIl < PI'
that converges uniformly in
Furthermore, th~ sequence Ifn,/z)\
in turn has a subsequence Ifn,zCz)1 that converges uniformly in a regular function or to
nk
denote an arbitrary se
'9Jl is normal in the disk
Ifn, I(z)l
T
Iz - T21
< P2/2
to
If we continue the process indefinitely, the diagonal
sequence I[n,n (z) I, n = 1, 2, .. " converges in the entire domain B to a regular function or to 00 and the convergence is uniform in euch of the disks Iz - Tkl < Pk/ 2, k,= 1, 2, •.•. Let us show that it also converges uniformly in B. If E is a
68
II. CONFORMAL MAPPING OF SIMPLY CONNECTED DOMAINS
closed set contained in B, it is entirely covered by the disks Iz - rkl
<
Pk/2. By the Heine·Borel theorem, we can choose a finite set of these
disks that provide the same covering. But the sequence If
(z)} con-
n,n verges uniformly in each of these disks. Consequently, it also converges uni formly on the closed set consisting of points of all these disks and, in particular, on E. This proves the assertion.
If all functions of a family
mthat is normal in a domain
B are bounded at
a single point Zo E: B, they are uniformly bounded inside B. This is true be cause all convergent sequences of functions belonging to 311 can converge only to regular functions, and then the assertion follows according to the condensation principl e stated at the end of
§ 2 of Chapter I.
Let us now give some tests for the normality of a family. Theorem 1 (fundamental test for normality (Monte))).l) If a family 311 of func
tions that are regular in a domain B is such that there exist two distinct values a and b which no function of the family assumes anywhere in B, then m is normal in B. Proof. We may assume that a = 0 and b = 1 since, otherwise, we could con sider not the family
:m =
{f(z)l but the family I(f(z) - a)/(b - a)} the normality of
which would imply the normality of 311. Furthermore, it will be sufficient to prove the theorem for the case in which B is a disk because, by virtue of what was said above, we need only show the normality of 311 at each point of the domain B. Thus, let us suppose that the domain B is a disk and that the functions
f(z)
€:m
do not assume the values zero and I in B. Let w
function defined in §6 that maps the halfplane ~ (,)
Izl
>0
=
'11.(') denote the
onto a regular circular
1. The composite functions ¢/z) = A(f(z)) , where fez) € m, can be extended to B along an arbitrary path since the only singular points of the function '11.(') are 0, 1, and 00 and f(z) does not assume any of these values in B. Consequently, the functions ¢ f are regular in B and they satisfy in B the inequality l¢fz)1 < 1. They do depend, however, on the choice of branch of the multiple-valued function '11.('). There fore, let us normalize ¢ z) in such a way that the point ¢ / zo) will, for given Zo € B and an arbitrary function f(z) E: 311, lie in the basic (regular) circular triangle with zero angles inscribed in the circle
r<
=
triangle or in one of the adjacent triangles. We apply to the functions principle of condensation of analytic functions. Now consider an arbitrary sequence Ifn (z)\ of functions
fn (z) €
¢
/z)
the
311. Let us
look first at the case when the set of numbers fn(zo) has a cluster point a other Let Ifn , (z)\ denote a subsequence of the functions f n(z) such that f , (zo) --+ a as n' --+ 00. The sequence I
than 0,1, or
00.
=
=
= iY
B because we would then have ¢(z) const e iY and, in particular, ¢(zo) e , that is, I¢r (zo)1 --+ 1, which in tum can be the case only if Un (zo)} converges nk k to 0 or 1 or 00. Thus, I¢(z)\ < 1 in B. Now, let E denote an arbitrary closed subset of B. On this set, 1¢(z)! Sq, where q < 1. Let q' denote a number such that q < q I < 1. By virtue of the uniform convergence of the sequence I¢ r (z)} on E, there exists a positive number K such that, for k> K, we have nk I¢r (z) - ¢(z)1 S q' -q on E. Consequently, for k> K and z € E, we have nk I¢r (z)1 Sq'· But the function' = /lew) inverse to w = '11.(') is regular in nk Iwl < 1. Consequently, /lew) is bounded for Iwl S q '; let us say \/l(w)1 < M.
Therefore, for k> K, we have \/l(A(fnk (z)) I < M on E; that is, Ifnk (z)\ < M.
This proves the uniform boundedness of the functions f nk (z) in the interior of B. But now we can choose from the sequence If (z)} a new subsequence that con nk verges uniformly in the interior of B to a regular function. Thus, in the present case, we have succeeded in choosing from the sequence Un(z)} a subsequence that converges uniformly in the interior of B to a regular function. Now, let us look at the case in which the sequence Un(zo)} has no cluster points other than 0, 1 and
00.
Then, if 1 is a cluster point of the sequence, let
=.JTJJ
us consider a sequence of functions F n (z) defined (by virtue of the n double value of the radical) in such a way that F n (zo ) --+ -1 as n --+ 00, The functions F (z) are regular in B and they do not assume in B the values 0 and n
1. Consequently, they come under the case just considered with a fore, the sequence
IF n(z)}
contains a subsequence
IFn k (z)}
=
-1. There
that converges uni
formly in the interior of B to a regular function F(z). But then the sequence of the functions [Fn (z)]2 = f (z) also converges uniformly in the interior of B to k nk a regular function. Furthermore, if 0 is a cluster point of the set of numbers
fn(zo), then the functions F n(z) 1) Montel [19271.
69
§7. NORMAL FAMILIES OF ANALYTIC FUNCTIONS
=
1 - f n (z), which are regular in B, do not
assume in B the values 0 and 1, and the limit of the sequence of their values
_______=c=.---:==_ _ =- --;;;""-------~-=- ~~=:::'------~~-l-------------------~~~-~---------~-~~-~-----~--------~
at the point Zo is the number 1. Consequently, these functions come under the
in the interior of B to a regular function or to
case considered above, and we can choose from their sequence a subsequence
generalized normality test.
IFn
k
(z) 1 that converges uniformly in the interior of B to a regular function. The
same is true of the sequence of the functions f (z). Finally, if 00 is the only nk cluster point of the set of numbers f n (z), then the functions F n (z) = l/fn (z) are regular in B, they do not assume in B the values 0 and 1, and the sequence of their
IFnk (z)}
that converges uniformly in the in
terior of B to a regular function F(z). Furthermore, F(zo)
=
o.
It follows that
tions regarding the behavior of analytic functions in a neighborhood of essential singularities and in covering questions. We present some of these applications. Theorem 3.1) Let z
§1
of Chapter I, we could show on the basis of Rouche' s
theorem that all the functions Fn (z) beginning with some n have zeros in an k arbitrary neighborhood of the point zo' which is impossible. Consequently,
fnk (z)
-->
00
uniformly in the interior of B; that is,
Ifnk (z) 1 is
the desired subse
quence of the functions fn(z). This completes the proof of the fundamental normality test. Theorem 2 (generalized normality test). 1) Let
mP
denote a family of func-
tions fez) that are regular in a domain B. Suppose that there exist two distinct finite numbers a and b such that not a single function in the family the value a anywhere in B and no single func tion in more than p points in B. Then
m is normal
m assumes
massumes
the value b at
in B.
denote an arbitrary sequence of functions belonging to
fu:ctions g n(z)
=
P+l/j("J yIn":;!
m.
Consider the
with arbitrary choice of value of the :adicals. These
functions are regular in B and do not vanish anywhere in B. Let 77 1 , 772" •• ,
'" '77 p +1 denote the different values of the (p + 1) st root of unity. Then none of the functions g n (z) assume any of these values in B because otherwise some of the functions
fn (z)
would assume the value 1 at more than p points of the do
main B. Suppose that g n(z)
f. 77 m
n
in B.
The functions h n(z)
=
g n (z)hm n , which
are regular in B, do not assume in B the values 0 and 1. Consequently, in accord ance with the fundamental normality test, the sequence Ihn(z) 1 contains a sub sequence Ih
nk
function or to
(z) 1 that converges uniformly in the interior of B to a regular 00.
The corresponding sequence
1) Montel [1927].
Ifnk (z) I
n
0 or even at the point z
=
Proof. Let us take an arbitrary domain 0 and let us suppose that the functions fn(z)
in that domain. If
Ifn (z)1
the circle
fnk (z) Izi
=
< Izl < (
=
0 itself.
in which f( z) is regular,
f(z/2 n ) constitute a normal family
Ifnk (z)\,
contains a subsequence
converges uniformly inside the domain 0 functions
=
< Izl < (
for k = 1,2,""
that
to a regular function, then the
are uniformly bounded inside that domain and, in particular, on d2. Suppose, for example, that
Ifnk (z)1
< M on
Izl
=
d2. But the
nk+1 Izl = (/2 the same values as f ()z on Izl = d2 +1 . nk Consequently, the inequality If(z)1 < M holds on all the circles Izi = (/2 ; n hence, it holds between them, that is, in the entire domain 0 < Izl < (/2 1 +1, and
functions
f
(z) assume on
nk
this is impossible for the essential singularity z
=
O. On the other hand, if all
convergent subsequences of Ifn(z)! converge only to
Proof. Again, we may assume that the numbers a and b are 0 and 1. Let
If (z)\
Ifn
Then the sequence
0 denote an essential singularity of the function fez). (z) I, where f (z) = f(z/2 n ), for n = 1, 2" ", is not a =
normal family in any neighborhood of z
the function F(z) is identically equal to zero because otherwise, just as in the proof of Theorem 2 of
This completes the proof of the
00
These two normality tests have a number of remarkable applications in ques
values at the point Zo approaches zero. Consequently, the sequence of these functions contains a subsequence
71
§7. NORMAL FAMILIES OF ANALYTIC FUNCTIONS
n. CONFORMAL MAPPING OF SIMPLY CONNECTED DOMAINS
70
-~~_:,,--=-,:;:;;:-~==~-=-_=-:c,=::~-=~-:;;:-~-;-=~_--~~-.
00,
let us show that the point
z = 0 is not a cluster point of the zeros of the function fez). Let us suppose that,
to the contrary, for any ( such that 0 < Zk
= z~OY2nk,
Izkl
> 0, there exists a sequence of distinct numbers z k
where d4 ~ Iz~O)1
. sequence that converges in 0 nulus (/4 ~
\zl < d2.
< (/2. < Iz I < (
The sequence {fnk (z)\ cannot have a sub to
00
since there are no zeros in the an
Consequently, a convergent subsequence of this function,
which is also a subsequence of the sequence Ifn(z)l, converges to a regular func tion and this in turn contradicts our assumption. Therefore, the functions l/fn(z) are, beginning with some n, regular in the domain 0
< Izl < (,
¢ n (z)
and the
subsequences I¢nk (z) 1 converge to the identically-zero function. This implies,
as above, that the function ¢(z)
=
l/f(z)
-->
0 as z
-->
0, so that fez)
which is impossible. This completes the proof of the theorem.
also converges uniformly
1) Moncel (1927].
-->
00,
=
_: - -
= = = = = = = = = = - - .- = = = = = ~
II
U. CONFORMAL MAPPING OF SIMPLY CONNECTED DOMAINS
72
'.::::~""~~~""'~"""""'=."""'-"-="'=~:::~"""'--~'~'~=-""'~.~-""""",~~,:::~=""",~~,..,....~,.",,"'''''-''''''''-"''''~=-=-~
§7. NORMAL FAMILIES OF ANALYTIC FUNCTIONS
g
functions of this kind. The corresponding assertion for the entire class of func
Two other theorems are consequences of this one:
tions f(z)
Theorem 4 (Picard). If Zo is an essential singularity of a function f(z),
z + c2z2
=
+ '"
that are regular in
then f(z) assumes in an arbitrary neighborhood of Zo all finite complex values
covered by the image of the disk =
1zl < 1
is not valid; chat is, we
cannot exhibit a disk of positive radius with center at the origin that is entirely
with one possible exception. I)
Proof. Let us assume that Zo
73
Izl < 1
under the mapping of each such function
because, for example, the function
O. Suppose that there exist two finite num
bers a and b and a neighborhood of zero in which f(z) does not assume either
f
the value a or the value b. Then the functions f rl (z) = f(z/2 rl ), n = 1, 2"" ,
Izl < 1
(z) = I-(l-z)n _
=
Z
+ 2+ C2 Z
•••
Izl < 1
also do not assume the values a and b in that neighborhood. Consequendy, they
is regular in
constitute a normal family, which, by Theorem 3, is impossible.
the value l/n, which can be made arbitrarily close to zero for n sufficiently
2
Theorem 5. ) Let Zo denote an essential singularity of the function f(z).
.J 0
but it does not assume in
: great. On the other hand, we do have the following covering theorems: Theorem 6.° For the family of all functions w = f(z) = z + c z 2 + ... that
Then, there exists a ray I issuing from Zo such that, for every neighborhood of the point Zo
for arbitrary given n
and every sector, with vertex at Zo that is bisected by I, the func
are regular in
Izl < 1,
Izl < 1
2
and assume the value 0 at no more than p points of the
tion f(z) assumes all finite values with one possible exception in the portion of
disk
that neighborhood lying in that sector.
an arbitrary function of the family completely covers the disk
Proof. Again we can assume that Zo
=
O. Let I zl
denote the given neigh
borhood in which f(z) is regular except at z = O. Since the family f rl (z) = f( z/2 rl ), n = 1, 2,· . "
a point z 1 in the domain 0 ing through
zI
is not normal in the domain 0
< I z II < to
take a neighborhood the domain 0 <
n
Izl <
I
Since
mis nonnormal in
f n' (z)
1, 2,. ", constitute in
that assumes the value a in U z . But I
f n , (z)
f n (z)
there is
assumes in U z
I
z'
=
z/2 rl ' . Since this disk is contained in the disk
,
Let us now turn to the covering theorems. In §4, we proved Koebe's covering
Izl < 1,
Iwl < X;
z) = z + C2 z2 + . •• is regular and Iz\ < 1 under this function com
that is, the disk is one and the same for all
I} That there can exist one finite value never assumed by f(z} is illustrated by the function fU} = ell z, which has a singularity at z =0 and yet f(z} j 0 for z j O. 2} Julia [19241.
Since e/> n (0)
=
0, it is also uniformly bounded in the interior of
< 1.
n k
-->
e/> '(0), which contradicts the fact that
ak
-->
O. This completes the
2
~
t:p( > 0
such that the image
('.;' c2 z2 + ••• Center at z
then the image of the disk
pletely covers the disk
Izl < 1.
Theorem 7. ) For arbitrary to in the open in terval (0, 2IT), there exis ts a
This completes the proof of the theorem. theorem, which asserts that, if a function f(
a family of functions that do not assume the value
r proof ot the theorem.
and in the angle indicated above, f( z) assumes the value a in their intersection.
univalent in
Izl < 1
fn (z), for n = 1, 2" . " of the family in ques-. a n . The fun ction s e/> n (z) = f n (z) / a n , for n =
Let Ie/> nk (z)j denote a subsequence that converges uniformly in the interior of the disk Izj < 1 to a regular function e/> (z). Then, e/> I (0) = the disk I zl
.{l/a nk
Izl < to
Izl < 1.
Dormal in
..
the same values as does f(z) in the disk obtained from U ZI by means of the sim ilarity transformation
f n(z) .J
Consequendy, in accordance with the generalized normality test, this family is
t·
Uz , it follows that, for every I
and a sequence of functions
1 at all and do not assume the value zero at more than p points of the disk
Uz of the point z 1 that lies entirely in that angle and in f.
--> ""
tion such that, for every n,
and bisector I and let us
finite number a, with one possible exception, among the functions a function
there exists
will meet the requirements of the theorem. To show this, let us
=0
under
exists. Then there must exist a sequence of numbers an that approaches 0 as
at which it is not normal. The ray I pass
take an arbitrarily small angle with vertex at z
Izl < 1 Iwl < p.
p> 0 such that the image of the disk
Proof. Suppose that no p> 0 with the property described in the theorem
mof functions
< Izl < to,
there exists a
0
that is regular in =
f the disk
Izl < 1
Izl < 1
under any func tion w
=
f(
z) = z +
covers some circular sector of radius pto with
0 and with central angle 2IT -
f.
l) Fekete [1925). For p = 1, the theorem was proved by Caratheodory with an indica , cion of the best value for p. For more on this, d. end of §1, Chapter III.
,
2} Valiron [1927). Valiron proved only the existence of P.. Bermant and Lavrent' ev [1935) found by a different procedure the best values for Pf Jepending on to. This value is given in terms of an extremum of an expression consisting of a function inverse to the modular function.
74
II. CONFORMAL MAPPING OF SIMPL Y CONNECTED DOMAINS
Proof. Obviously, we may assume that «
17. Suppose that no p( > 0 with
these functions constitute a normal family in
the property described in the theorem exists. Then there exist number sequences
_ a /(f3 - a )
la n I and lf3 n I that approach zero as n --> "" such that the difference ep n = arg f3 n - arg a n lies, for each n, between ( and rr and a sequence of functions
It
f n (z) = z+"', every n
=
epn < 17/2,
for n = 1, 2", "
1, 2""
and every
that are regular in
z in Izl < 1,
we have
Izl < 1
fn (z) f.
then, by dropping a perpendicular from the point
an
fn (z) f.
The function
f3 n • If
la n /(f3 n - a n )1
< l/sin
L
If3 n -
On the other hand, if
Furthermore, the numbers
Let lep
denote a subsequence that converges uniformly in the interior of the disk to a regular function
ep '(0),
ep(z).
Since
which is impossible since
nk to a regular function
zj
-->
ep (z)
such that
ep (0) =
O.
must be identically equal to zero because otherwise, we could
< 1,
ep nk(z)
have, from some k
which, however, is not the case. Consequently, ep~ k (0) =
0, which is impossible. Thus, the positive number p mentioned
nk
There exists a positive number p such that every function
z + c 2z2 + •••
that is regular in
Iz\ < 1
covers entirely some neighborhood
Furthermore, we have
Theorem 10. For the family of functions f(z) =z + c 2 z 2 + .•• that are regular in
(z)l
Iz\ < 1
covered by the image of the disk
ep'n k(0) --> ep '(0), it follows that Ij(f3 nk - a nk )-.. ank and f3 nk approach zero. This completes
a given finite nonzero value disk Izl < p is completely
and that do not as sume in that disk
a, there exists a positive number p such that the
Izl < 1
Izi < 1
under an arbitrary function in the family.
This theorem also follows from Theorem 8 if we take for K an annulus con . , . a. 2) talnmg t h e pomt
the proof of the theorem.
Theorem 8.1) Let K denote an annulus R 1 < jwl < R 2 and let p denote a positive number. Then, there exists a positive number p such that the image of the disk Izi < 1 under each of the functions w = f(z) = z + c 2 z 2 + ••• that is regular in Iz I < 1 either covers the disk Iwi < p or assumes every value w in the annulus K at no fewer than p points 0 f the disk Izl < 1.
1)
of radius no less than p with center at the origin.
are bounded. This means that
Izl < 1.
1/(f3 n - a n )
f(z) =
they constitute a normal family.
the family is uniformly bounded in the interior of the disk
1
Theorem 9.
1a n /(f3 n - a n)1 -< 1 -< 1/ sinE. Let us now consider the functions ep n (z) = [[(z) - a n]/(f3 n - a n), where n = 1, 2, •.•. These functions do not assume the
Izi < 1. Consequently, ep n (0) = - a n /(f3 n - a n )
Izl < 1
< 1.
(z) I that converges uniformly
contains a subsequence!ep
As a consequence of this theorem, we obtain
it follows from this
same construction that If3 n - a n I > Ia n I. Consequently, we again have -
values 0 and 1 in the disk
on, zeros in
ep n (0) =
in the theorem must exist, and the theorem is proved.
a nI > Ia n I sin (; that is, -
ep n >- rr/2,
ep(z)
Since
0 as n -.. "", they are uniformly bounded in the disk \ zl
the disk
0
Izl < 1.
show, by using Rouche's theorem, that the functions
to the line segment
connecting the points 0 and f3 n, we see that 1f3 n - a n I is greater than the length of that perpendicular, which is equal to Ia n I sin ep n and therefore not less
than Ia n I sin a. Consequently, we have
-->
f~llo':s tha~ ~n(z)l
in the interior
and such that, for
a n and
75
§7. NORMAL FAMILIES OF ANALYTIC FUNCTIONS
4 .It
Proof. Let us suppose that the theorem is not true. Then, there exist se quences la n I and lf3 n I such that a n
-->
0 as n -.. "" and
1f3 n I
belongs, for each
n, to the interval (R 1, R2) and a sequence lfn(z)l of functions fn(z) = z + '" that are regular in 1zl < 1 such that, for every z in Iz\ < 1, we have f n (z) f- a n for every n and
ep n (z)
f n(z) = f3 n
for no more than p - 1 values of z. But the functions
= [f'(z) - a n ]/(f3 n - a n ) do not assume the value 0 in n
Izi < 1
and they
't"
~
:lit *:~
assume the value 1 in that disk at no more than p - 1 points. Consequently, 1) Landau [1922]. 1) Mantel
2) A more profound study of me modulat function led Bermant [1944] to a number of
[1933].
generalizations and sharpenings of Theorems 8-10.
.
~
::=-=--=---===-~-=:;;=--::-~~--:::~.=s=.-~
-=--
--~'-
---_._~----_._-~-
.~:~-~~~~~';~,,;;~:;---,~'-~~ -.~~--~-~.~~-~:"~:.._. ~;;:~~~:~'
~~~~=-=~'=~ ,
§ 1.
:~~~~_-::~,::~';.':~~~~,--=~~-~:
DOMAINS BOUNDED BY POLYGONS
77
sponding angle are parallel, we assume the value of the interior angle to be O. On the other hand, if the sides diverge at a vertex A k
= ""
but their extensions meet
at a finite point A ~ forming an angle of TTak directed toward the vertex A k, we take TTa,.
CHAPTER III
-TTak. With these conventions, ak can also assume values from - 2
=
to +2. Of course, the cases ak
REALIZATION OF CONFORMAL MAPPING OF
SIMPLY CONNECTED DOMAINS
=
2, -1, and - 2 should be understood as limiting
Under these conditions, if the domain B is mapped univalently onto the half plane ~ «)
> 0 and if points
ak,
k = 1, 2,'"
,n, on the real axis of the
(
'plane are mapped onto the vertices Ak , then, in the case in which these ak are
all finite, we can show by the reasoning usually employed here l ) that the inverse , function z
§ 1. Conformal mapping Qf domains bounded by rectilinear
=
f«) is determined by the Schwarz-Christoffel formula n
I(C) = C ~
and circular polygons Although in Riemann's theorem we had a geometrical principle for
n (~-
ak)"k-
I
d~
+ C'i
(1)
k=1
f~rming
analytic functions (specifically, to every simply connected domain was assigned a univalent function mapping that domain onto some canonical domain), neither in this principle nor in the rest of Chapter I, did we touch on the question of more or less explicit construction of these functions. In the present chapter, we shall look at some questions of this nature. Let us now look at domains of certain particular forms, for example, domains bounded by rectilinear or circular polygons for which we can examine the analytic
where C and C' are constants. In the case in which one of the vertices, let us say Ako ' is mapped by this inverse mapping into "", the same formula holds for
f«) except that now, the product constituting the integrand is over all k = 1, , 2, .•. , n excluding k = k o• Of course, formula (1) remains valid when the domain' " B is mapped onto the disk \zl < 1. In this case, the points ak, the images of the vertices Ak , are points on the circle I(I = 1. We note that, in accordance with Theorem 6 of
§ 3, Chapter II, we can choose
only three of the points ak arbitrarily. The remaining points ak and also the
nature of the mapping functions rather extensively and can obtain appropriate
,:constants C and C' are uniquely determined by the domain. 2 ) The absence of a
explicit analytic expressions for them. The basic tool for studying these functions
,general method for determining the remaining unknown constants decreases the
in the special cases mentioned is the principle of symmetry.
':'Practical significance of the Schwarz-Christoffel formula. Furthermore, it is only
Domains bounded by rectilinear polygons. Assuming that this case is already familiar to the reader, we confine ourselves to formulating the generalized condi
exceptional cases that we can evaluate the integral (1) in closed form in terms f elementary functions. We cite as very,simple examples of the application of
tions under which the final results hold and to deriving the results themselves.
formula (1):
Specifically, suppose that a simply connected domain B in the z-plane does not
IS the interior of a regular n-sided polygon with center at = 0 and with a vertex at z = a > O. Suppose thi t Z = I«) is a function mapping the disk 1(1 < 1 onto the domain B. Suppose tbat [(0) = 0 and ['(0) > O. Then,
contain
00
and is bounded by a rectilinear n-sided polygon with vertices at the
points AI' A2 ,' • "
An.
Suppose that the angles at these vertices are respectively
(a) Suppose that B
TTa h TTa2, •• " TTG.,.. Let us regard these vertices as accessible boundary points of a domain B. Therefore, if several angular boundary points fall at a certain point in the z-plane, there will be just as many vertices. We shall also consider the case when one or several vertices lie at
76
00.
Here, if the sides of the corre
1) Of course, we use here the results established in Chapter II on the existence of Such an inverse function and its behavior on the boundary. 2) (We have Kufarev's method [1947b] for detennining the points ak, which is con venient in certain cases.)
§l.
III. REALIZATION OF CONFORMAL MAPPING
78
uk = e2TTkiin
vertices A k
for k = 0, 1,"', n - 1. From (1), we obtain C
~
-~,:k
de__ .<),,,,
I(~)=C C ~
i~~
(2)
o
1;(;
where C is found from the condition [(1) = a.
f'"
for k
ae2TTkiin
=
0, 1,···, n - 1, where
U
:.q.~I;
l
and continuous on the closed disk
I"
,
is a positive
f'
Ie,)
I_~n
,c
I(~)=C.\
o
and
1+2 (I+~n) n
d '-
C~_
(3)
(l+~n)2In
Ie,)
=
Ie,)
that maps the disk
"I
<1
00,
O
I"
(~) f'(q
f'"
3
-:r
<1
and also can be extended past an arbitrary arc y of
the type described above into 1'1 > 1. Furthermore, one can easily show that it possesses the important property of invariance under an arbitrary fractional-linear
.
00,
we
then {fI, (I = {f,
cf(C)+d
'I.
Let us examine the behavior of ak
1[, 'I
at the points
n
do not include the numbers 0, 1, 2 and that
second point of intersection Al. The function TTa k
are the values of the
Domains bounded by circular polygons. Now, let B denote a simply connected
=
through t = O. According to the symmetry principle, the function
»I /
domain in the z-plane with boundary consisting of a finite number n of circular arcs or straight line segments. In the latter case, the segments can be regarded
the point
as degenerate circular arcS. We denote the endpoints of the arcs, that is, the
In particular, it is regular at the point
vertices of the polygonal domain, by AI' A2l
that point. Consequently,in a neighborhood of
A n and we denote the values of
the interior angles (with respect to B) at these vertices by
TTa 1> TTa 2 , •• "
Uk
CL
lying in \"
=
Ie,}
1"
< 1.
<1
to the part of the neighborhood lying in \ (,
inverse to one of the mapping functions.
k
I > 1.
itself and has a nonzero derivative at
(f«()-Ak)"-k_b' f(C)-A - l(\,-ak ftom which we get
Let us
ak
I
We may assume that the domain B is bounded. We can always arrange fot this to The domain B can be mapped conformally onto the disk
(C[C') - Ak )/
k is therefore extendable from the part of the neighborhood of
TTa n •
be the case by means of a suitable fractional-linear transformation.
= C(Z - Ak)/(z - A~»l/CLk maps
0 in the t-plane that lies to one side of a line passing
([C') - A ~
l
t
.the part of a neighborhood of the point A k that lies in B onto the part of a neigh borhood of the point t
interior angles (with respect to the domain B).
• ••
k = 1, 2,·", n. Let
circles on which the boundary arcs lie, forming angles with vertex A k, have a
k=1
Here, the
Uk'
no circular arc of the boundary degenerates into a straight line segment. The
~ n:('-ak)"k-I~; +C ak'
(4) t
which is known as the Schwarz invariant or the Schwarzian. Obviously, it is a
us suppose first that the numbers
examine in detail the function z
(I"f'(~) (q)~
f 1(q = af (~)+b
then, for the function
onto B in such a way that [(0) =
with analogous values of the constants
> 1 in
is regular (with nonzero derivative) on y.
{I, q=
If the
obtain the formula
I(C)=C
all lie
transformation of the function [C'); that is, if
Above, we have assumed that the domain B does not include
z
can be extended past y into the domain \,\
C = uV4.
domain B bounded by an n-sided polygon does include
ak
Now, let us look at the expression
regular function in
tions. From (1) we have
On an arbitrary arc y of the circle
on some circular arc. Consequently, in accordance with the generalized symmetry ptinciple, the function
number, in the direction opposite to the point z = O. Suppose that z = [C') is a function mapping the disk 1'1 < 1 onto the domain B and that [(0) = 0 and (0) > O. Then, we can again determine the constants ak from symmetry condi
1'I :; 1.
= 1, the values that this function assumes between adjacent points
such a way that
(b) Suppose that the domain B is the z-plane with cuts along the rays issuing from the points
I" = 1 corresponding to the k = 1, .. " n. The function [C') is regular in the open disk 1'1 < 1
We denote by uk, k = 1," " n, points on the circle
the constants uk are easily determined from symmetry conditions. Specifically, we have
79
DOMAINS BOUNDED BY POLYGONS
,=
)+
f«() -A~ = (~_ aktk (Cl f(t) -A k
uk, we have
-+ 0 , ... , bJ-r-
+...),
cl
=F O.
§ 1.
III. REALIZA nON OF CONFORMAL MAPPING
80
DOMAINS BOUNDED BY POLYGONS
If we set up the expression (4) for this last function and keep in mind that it is equal to
If, (I
{I, q =
we get, after some simple calculations, the expansion in a neigh
.!- (1
{I, where
-
ai)
q= ~C-ak)'
+c Ckak +...,
order poles at the points Uk.I) Therefore, the function
I( \ < 1
n
ak)2
if these assumptions are not satisfied, we can still show that (8) is valid by re
originally defined
,on the convergence of univalent functions (Theorem 1, §5, Chapter II). Thus, equation (8) is proved for an arbitrary domain bounded by a circular
phic function with poles only at the points ak. The principal parts are determined
polygon. It represents a third-order differential equation in f«(). Its right-hand
at these poles from formula (5). The difference
(~) = {j, ~} -
ember includes the unknown constants ak and
1: (~(l-ai) (C +Ca,,)'
an expansion of the form (7) in a neighborhood of
(6) 00.
Since a
"
~ Ck=O,
a!(C)+b It (q = c! (C) d $. canst
+
k=1
is real on each of the arcs y cons idered above and since
f 1 (0) f-
00 ,
the point (=
08;
h «() = h (1/().
that is, in a neighborhood of (=
11 (C) = If, (I
{t, This means that
If, (J
1(1 < 1
In particular, it is regular at it has an expansion
00,
f 1 «()
k=1
k=l
n
n
k=1 tbat the constants ak and if"rem 6,
has the form
q= ~~ +...
§ 3,
k=1
Ck must satisfy. Furthermore, it should be noted that
,Ii
,.'
= 1 by virtue of Theo
ak
and
Ck
.
Similarly, we can also show that, when the domain B is mapped onto the
.~!;upper halfplane ~«()
But then the dif
I (I
Chapter II. In the case of a circular triangle, this is sufficient to deter
'l'.mine all the constants
(7) 00.
(9)
Ckak=O,
.three of the constants ak can be chosen arbitrarily on
at the point (= 0). But then,
has a zero of at least fourth order at
1:" (l-ttO+ 1:"
~ (1-<10+ ~ Cka:=O,
co+ ~1 + ...,
where elf- 0 (by virtue of the conformality of the corresponding expansion for
~
it follows
from the symmetry principle that this function can be extended from the disk by taking
the coefficients of 1/(, 1/(2,
00,
member of (8). This gives us the three conditions
suitably chosen function
I" > 1
= 1, 2, ... , n. There
and 1/(3 must be equal to 0 in the corresponding expansion of the right-hand
k=l
has no finite singular points at all. Let us examine its behavior at
to the domain
Ck, for k
are certain relationships between these constants. Specifically, since If, (I has
Ck) ak
n
is 0, 1, or 2 and
',placing the domain B with suitable circular domains and then using the theorem
and then extended to the entire plane, represents a meromor
F
Uk
that there are no straight line segments on the boundary of the domain B. However,
(5)
If, (I,
(8)
Ckak .
Our assumption up to now has been that none of the
Ck is an unknown constant. This expansion shows that If, (I has second
in the disk
. ) 1: (~(I-a:) ~C _ +C
k=1
borhood of (= ak:
81
> 0, the inverse function z
=
f«() also satisfies the differ
ference (6) has a zero at the point 00; that is, this difference is regular on the en
ential equation (8), where, this time, the ak are points on the real axis corre
tire plane including
sponding to the vertices A k. The only difference is that if one of the vertice s
00.
Therefore, it is identically equal to zero, so that we have
. A k, let us say A k 0' is mapped into 00, then the summation in (8) is over all k 1, 2,"', n excluding k = k o•
the identity
=
Now, let us look at the nature of the differential equation (8). Since it is a 1) We note that, if a finite point ~0.1 ~o \ > 1, is such that f( ~ 0) = 00, then in a neigh borhood of '0 we have f(~) = b_tl(~- ~o) + b O +"', b-Oi 0 so that ~Jis regular at
,= '0'
tbird-order equation, its general solution has three arbitrary constants. On the
It,
other band, if
, ~:: ~'.'
f. «()
is a particular solution of equation (8), then, obviously, the
III. REALIZATION OF CON FORMAL MAPPING
82
§ 1.
fraction (af* (t;;") + b)/(c[* (t;;") + d) which already contains three arbitrary constants
DOMAINS BOUNDED BY POL YGONS
83
Example. In conclusion, we present a simple but important example when we
(al c, blc, and d/ c), is also a solution. Consequently, this is the general solu
can make a complete determination of the mapping function. Suppose that we are
tion of equation (8).
required to map the disk
Although we need to know only one particular solution of equation (8) to find the desired mapping, the finding of that one solution is difficult because of the
I(I < I
onto the domain B consisting of the interior of
an n-sided circular polygon with center at z
z
=
=
0, with one vertex at the point
1, and with all the interior angles equal to 1Tq, where 0::; q ::; 2. Suppose
nonlinearity of the equation. However, the problem of solving equation (8) can be
that the mapping function z = [(0 is normalized by the conditions [(0) = 0 and
reduced to solving a second-order linear equation, owing to a single remarkable
f ' (0) > O.
property of the function
If, (I.
Let wI and W2 denote two linearly independent solutions of the linear dif ferential equation
w"+q(qw=o with given q«(). It is easy to show that WIW~
(10)
C
(C) = wf
,r, Y1 'Yj ~=
{lj,q= -2v
I(I < 1,
0
< arg
(
<
21T/n onto the circular triangle bounded by the two straight line segments exrend ing from the origin z = 0 to the points z = e 21Til n and the circular arc contained in the boundary of the domain B. According to the symmetry principle, the func tion executing this mapping will provide the required mapping of the disk
I(I < 1
onto the domain B. It maps the points e21Tki/n, for k = 1, 2"", n, onto them
- W2W{
= const = c (the left-hand
member is the Wronskian of the two solutions). Therefore, if we set TJ = w 2 1w 1> we have d 2 _ 1_ 'Yj'
To do this, we consider a mapping of the sector
W
W
selves so that Uk
= e 21Tk i/n,
= 1,2, ., .. ,
for k
n. Furthermore, by virtue of the
uniqueness of the mapping, we have the functional equation [(e 21Ti /
in 1(1 < 1. We see from this equation that the expansion of f«() in a neighborhood of (= 0 is of the form OJ
-w~ =2q(q.
f (C) = ~ ChC
nH / "
C1
k=O
It follows that the ratio of any two linearly independent solutions of equation (10)
n () =
21Ti n e / [«()
l!,
and that the expansion of
> 0,
(I is of the form
is a solution of the equation
{lj, C} = 2q (C).
This shows that, if we denote the right-hand member of equation (8) by R«(), set q (0
=
R(0/2, and solve the differential equation
(10) with this q «(), then
tqe ratio of the two linearly independent solutions yields a solution of equation (8).1) Equation (10) with this
R«() belongs to the most intensively studied class
of linear differential equations with variable coefficients, known as equations of the Fuchsian type. However, the use of this advantage is made difficult by the fact that we do not have a general method for determining the constants
ak
and Ck •
~ dkC nH .
{I, q =
(10' )
k=1
. On the other hand, since, in the present case, 1
! ~.(l-~~) 10
11
--l(1 _qq) ~
dC "'" , - Ok
2
k=l
V _1~
k=!
n [ nC 211- S
(n _I)'n-s] q d nCII-I = -T(I-q ) dccn-I =2(1-qq) (CII-l)2- en - l '
1
it follows from (8) that the summation 10
V
1) Also, equation (l0') can be reduced to equation (10) in another way, specifically, by remembering that 2
{"'1,
q=
d 1 ) -2H de; Y;;' (
and setting I/V~ ::: w. Then, if wI is a particular solution of equation (l0), the corre sponding particular solution 7) 1 of equation (10') is obtained from 7){ = l/wj.
Ck
"'" ek=!
ak
in that equation must include only terms with powers
("k-2;
that is, this sum
reduces to c(n-2/«(" - 1). But 1[, (I has a zero of at least fourth order at
00.
Therefore, C = -(n/2) (I - q2). Therefore, equation (8) takes the form
{I,
q=
n' 2(1 - qq)
en -
i,.,.
2
n ••
(11)
~",.,,-~-"--~
=~~
'~~~~--'----~~~-'----'-'-'-'-'--~~~;""'~--'-" '-'_.~~~~~~~~;;;,..;:~
§ I. DOMAINS
III. REALIZATION OF CONFORMAL MAPPING
84
BOUNDED BY POLYGONS
85
and this solution is, as one can easily see, linearly independent 'of (16) for 0
The problem of finding f«() thus reduces to solving the linear differential equa
<
y < 1. Thus, we have for equation (15) the two linearly independent solutions
tion
nl en-I w" +-4(1 - q'i) ,rn 1\1 w= O.
F
(12)
In this equation, we make the substitution
el/tlF
w=(l:n-l)p v ,
e=~n,
l)'i ev" + 2
(e -
1)
(e (2p +
+ (p (p _
~) -
1-
(1 -
~ )) v'
+ p( ~ - 1) + 1 /2)
~) e
V
=
I-q
1
1J.=~2--n' ~=-2-' 1=I- n
,
~:
1
I = ~ t H (1 -- t)T ~"l (1 o (14)
f3 > 0,
which converges for
(15)
Y-
f3 > 0,
co
~
~ k=O
1·2···k
This is the familiar differential equation of Gauss. Solution of it with the aid of a
I.e;:' =0 cn~n
F (
IJ.
~
't"
"(,
e)-
which converges in the disk
ro
X(-I)ktkekdt=
shows that it is satisfied by the so-called hypergeo
metric series I) -1
1(1
+ r:-:r a~ e a(a+I).~(~+I) 'i + 1.2'1("'(+1) e +..., <1
for all a and
f3
V
~ k=O
a(a+l) ... (a+k-I)
1·2 .. ·k
I
X ~ t[l+k-l(l_t)T-H
(16)
o
Xr
second solution of equation (15), we make the substitution v = ~I-'YVI' The re
~
a(a+l) ... (a+k-I)
1·2· .. k
r (1- ~) ek _ r r ()'+k) -
(~+ k)
(~)
r ()' r ()')
~)
F (IJ., ~' 1. e)
1
F(IJ.,
~'
"(,
e)=r(~)~g)_~) ~
tH(I-t)T H(l-etr/Xdt.
(18)
,(I
Let us use this formula to transform the function (17). Then, for 0
I(I < 1, 1) For more on the hypergeometric series, see, for example, Smimov [1953]. p. 381 of English tranSlladon.
V
And, consequently,
a- y + 1 = a', f3 - y + 1 = f3' , and 2 - Y = YI . Therefore, this equation is satisfied by VI = F(a', f3', y', ~'). Then we obtain for equation (15) the second solution
v=et-lF(IJ.-"(+I, ~-"(+1, 2-"(, e),
00
dt·e k =
k=O
and all y> O. To obtain a
sulting equation can also be put in the form of the Gauss equation if we set
Let us expand the factor
1= \' t~-I(1_-t)H-l V -a(-a-I) .. ·(-a-k+l)
o
power series v =
I~\ < 1.
and
elf/X dt,
", (1 - ~t) -(1, in a binomial series and then integrate term-by-term. We obtain 1
"()v' +1J.~v=O.
(17)
"grals. Consider the integral
it takes the form
qe -1)v" +«IJ.+ ~+ 1) e-
n , ell) en)
The geometric series in (17) can also be expressed in terms of definite inte
-~1f --~))v' n 1--q(l-q_~)v=O +-2 2 n
1
1+~I' e).
is a solution of equation (11).
or, if we set
I-q
I-q 2'
(I -
( 13)
Let us choose p in such a way that the coefficient of v in (13) has a root ~ = 1.
I)
C-q +~ 2 n'
e)
I I-q pr l q +_L -2 ' I+ \ 2 n ' (C) = C F q __!.. , I-q -2-' [-- -~ 2' Il, n '
f 0.
This yields p = (1 - q)/2. With this choice, equation (13) takes the form
e(t-I)V"+(e(2-q-n
I-q 1 -q ---~ -2- , 1 - n' 2 n'
Returning to the variable s (and w, we conclude that the function
where p is a constant. We obtain
(e -
e
we obtain
:s q < 1 and
----
- ~---
=
_ =.-,,= __ 'On_ . _,:,;.,~~u:::::_-_=-~;_;-";::-n=:~,i:-~~:::.,~,,,"-~-_-.
III. REALIZATION OF CONFORMAL MAPPING
§ 1.
n --
86
1
f
-,
__ q +_1
~i';,~:'::"C.,,_~.-,-::.,:;,:;'::::o-::.-=: -;.~;~~",="_;:'~~~:::::::::=:_::.:_::;_:~~~:",;,,,,",":.;'.:':::::-.-:::::._._
\zl < 1
function ~ = [-;'V"z) maps the disk
_q-_l+..!_q-_I_..! (I-t) 2 1I(1_~lIt)2 n dt
bijectively onto the many-sheeted
Riemann surface constructed from the interiors and exteriors of the circle Iz I = 1
~t 2 o (~) = cC'-cI --q~+~'lc---q-_----'-1----;lc----'i-_~I.---.I~ t - -2- (1_t)-2--n (I _~IIt)-2-+'ndt
in a manner similar to the construction of the modular surface in §6 of Chapter II. (9)
On the other hand, the function z
=
[n (~) can be extended in Bn along an arbi
trary path and it assumes, when this is done, only values in the disk
o
Let us see what we get if we take the limit as n --->
Since the desired mapping function satisfies the conditions [(0) =
Furthermore, by making the substitution
d/O -
(1 -
and [( 1)
= 1,
can, if we begin at the point ~ = 0 with an initial element of the form c 1 ~ + ••• ,
~ nt) in the integrals in
points 0, 1, or
00.
From formula (21), we have
(19), we obtain another form of the function [(~) for real values ~ and hence for all complex values of ~ in the disk I~'
f(Q=~
1
< 1: [fn(~I/nw=~
1 -+IIdt t 2
o~
«(I -
q+1
1
q-I
1
t -2 - -
o~
~ y.t (I--t) (I-'t) 101 t- I / n dt
\'
~nt»-2
t) (I -
(20)
ndt
,II
t l / n dt
\'
q-I
1
Iz I < 1.
The function [[n(~l/n)]n
00.
be extended along an arbitrary path in the ~-plane that does not pass through the
1. This is the final result.
it is obviously contained in formula (19) with c T=
=0
87
DOMAINS BOUNDED BY POL YGONS
I =C (1 +
A )11 nil
,
(22)
o ~ Y t (I - t) (1- Ct) where
1
\' n (t l / n -
q+1
I' o~ 11t ' A I1m n= 1m 1
((I-t) (1_~lIt»-2
in which the dependence on n is simpler. We note that none of the integrals in
n_ro
o~
1
I/ II) dt
0' y
t
t) (I -
(I -
~t)
=~I------
t-I/ndt
y t (~ -
210g t dt
\'
(I - t) (1- 't)
n-ro\,
formulas (19) or (20) can be evaluated in closed form in terms of elementary func
t-
\'
dt
o~ 1It-(I- t) (1- ~t)
t) (1- 't)
tions. The most interesting case is that of q = O. In this case, equation (20) becomes
This limiting operation is uniform on an arbitrary finite chain of closed disks not containing 0, 1, or
I
t l / lldt
\'
~
til (~) =~o
Y t (1 -
In view of this, formula ( 22) yields the results that the 1
\'
t) (I - 'lit)
I
- t
- II
.1 Y'
(21)
1 \'
00.
function
o I
dt
~
~ 1It(l-t)(I-~nt)
f(C)= lim
o
2 log' dT
(I-,)(I-~')
_d~
[fn(~I/n)ln=Ceo YT(l-,)(I-',)
(23)
n_ro
with [(~) now being denoted by [n (~). Let us point out some very important properties of the function (21) and of its inverse [n- I (z). On the basis of the same considerations as in §6 of Chapter II, the function ~ =
[ ; I(z),
can be extended
the entire disk \z I < 1. It represents in that disk a meromor
to
defined inside the n-sided polygon with zero angles,
phic function that assumes only values in the domain B n' which is the z-plane with the points Gk = e 2TTki/ ", k
=
1, 2, ... , n, deleted. More precisely, the
being regular in the disk I~I
< 1,
can be extended from \ ~I
path in the ~-plane that does not pass through 0, 1, or function satisfies the inequality
I[(~)\ < 1.
00
<1
along an arbitrary
and that the extended
Let us denote by B' the image of the
I~\ < 1 under the mapping z = [(~). This image is a domain in the disk Izi < 1. Let us show that the inverse function ~ = [-1 (z), which is necessarily regular in B', is regular in fact in the entire disk I z 1< 1. To do this, let us
disk
clarify the boundary of the domain B I . The function z = [n (~) maps the disk
88
III. REALIZATION OF CONFORMAL MAPPING §2. PARAMETRIC REPRESENTATION OF UNIVALENT FUNCTIONS
1'1 < 1 onto a domain one arc of the boundary of which is given by the equation Iz - 1 - i tan (17/n) I = tan (17ln) with jz I < 1. Consequently, the function z = f n ('lin) maps the disk 1'1 < 1 univalently onto a domain bounded by the curve Ivz- 1 - i tan (17ln)1 = tan (17ln) with Iz I < 1. As we let n approach this 00,
I = TT with I z I <1, which is also the boundary of the domain B. If we shift to the t-plane by setting t = log z, then B' is mapped single-valuedly into a domain B" with boundary determined by t he system of equations It - i17k I = 17, k = ± 1, ± 3,' .. with ~ (t) < 0, that is, into the left half plane with cuts along semicircles. The function t = log f('), which is regular in 0< 1'1 < 1 maps the circle "I = 1 onto the boundary of the domain B" in such a 1log z -
curve becomes the curve
iTT
,=
way that an arbitrary continuous image of the circle with the point 1 excluded is the boundary semicircle of the domain B". The inverse function' = f-1(e t ),
Izl < 1 along an arbitrary path. Therefore, it is regular in Izl < 1 and !ep(z)! < 1 in Izl < 1. But then, just as in the Schwarz lemma, we easily obtain the result that \1/16cl S 1. From this we obtain the inequality Icl?: 1/16 with extended in
equality holding only in F(z)
,=,= ff,= 1
~{t) < O.
This shows that the func
< 1. Also, it follows from the extension process that -1 (z), does not assume the value 1 in Iz I < 1 and that it assumes the value 0 only at z = O. The function f('), on the other hand, can, by be tion
(z) is regular in Izi
,=
ginning with an element defined in a neighborhood of
,=
0, be extended along an
arbitrary path that does not pass through the points 0, 1, and passing through
,=
0, its extension through
,=
00.
an original element of the function. We easily obtain from (23) the expansion
in a neighborhood of
,=
f(t) =
, '2
16+ 32 +
...
q ~ 1, is regular in the disk \ zl the disk
(24)
q?:
1, is regular in
Izi < 1
and that it does not
assume the value 0 in 0 < Iz I < 1. Let c denote a number such that F (z) ~ c
Iz I < 1.
+ .. "
and has no zeros in the domain 0
,=
under the mapping
where
< Iz\ < 1.
F (z) completely covers
< 1/16 but not always a greater disk with center at
,=
O.
A more profound study of the function (21) leads to the following theorem on the covering of segments: Theorem 2. 2 ) Suppose that the function F(z) = in the disk
Izl < 1.
z9
+"', for q?: 1, is regular
Then the image of that disk under the mapping' = F(z) com
pletely covers some segment of arbitrary prenamed slope that contains the point
,=
0 and is of length no less than A = 817 2 /rn0 4 = 0.45 .... The number A
cannot be increased without additional restrictions on F (z). §2. Parametric representation of univalent functions In the present section, we shall give a parametric representation exhibited by tation, we can find the complete solution of a number of extremal problems in the theory of univalent functions. This we shall do in the following chapter.
The following is a simple application of this last function f(') to one of the covering questions considered in §7 of Chapter II. Suppose that the function
in
I"
<1
Izl < 1
Then the image of the disk
C 1 ZHI
Loewner for an important class of univalent functions. 3) Thanks to this represen
0. 1 )
F(z) = z9 + cI Z 9+1 +"', for
This result, which is the analogue
Theorem 1. I) Suppose that the function F (z) = z9 +
But along a path
0 is possible only by means of
= cf-I(eio.z q ).
of Koebe's covering theorem for nonunivalent functions, can be formulated as
which is regular in B", is, by virtue of the symmetry principle, extendable, JUSt as in §6 of Chapter II, to the entire halfplane
89
Then, the composite function ep (z)
= f(F (z)1 c) = z 9/16c
+ . .. can be
Let B denote a simply connected domain in the w-plane including the point w = 0 but not the point w
=
00.
Consider the set of domains obtained from B by
making a cut along any Jordan curve that is contained in B, that has one end point on the boundary of B, and that does not pass through the point w = O. Let us call these domains the domains (s). If the domain B is unbounded (that is, if 00 is one of its boundary points), this cut can extend to
00.
Then, in contrast with
the preceding situation, we understand by "cut" a curve defined parametrically as 1) In (23), we need to make the substitution T = sin 2¢ for 0 < ¢ <17/2. Then calcu lation of the coefficients reduces to evaluation of elementary integrals and integrals that reduce to the integral
follows: w = Wet) for w (til) for a
as t < b,
S t' < til < b,
where w (t) is continuous in
and w (t)
---+
00
as t
-->
" ""2
~
o
log sin 'f d'f = -
; log 2.
as t < b,
w (t') I-
b. One can easily show that it
1) Caratheodory [1907]; see also Goluzin [.1948].
2) Bermant [1944] and Goluzin [1948].
3) Loewner [1922]; see also Goluzin [1939d, 1949cl.
..,._
-II
-= .;:
..
90
-,._-
._~-
>_~,~Y;'
~--_~?i~
,..;,
_,~
~._.-
.._?---:~:-
_c4_..
,-,~~~~~~~~:~~_,~~:~~Y'~:,;_e'~ti~~s~'j;;.:;~~::;~~:z;=,:::"':"',":"_,;::~;W.;~:~~~L~jt..~~'=-~;;::;;:~~~~~,.;;:.t=::;:~l";4.3:,
is possible to approximate an arbitrary simply connected domain functions w = !(z),
! (0) =
a'
r' (0) > 0
Izl < 1
that map the disk
a'.
To see this, note that the domain
a'
and consideri~g the circular images in
Jordan curve C contained in
a,
a'
a'.
Assuming
a'
a
interval 0
<1
t
< to.
Furthermore, one can easily see that, for every t in the inter
< to, the function w = f(z, t) maps the disk I z I < 1 onto the disk < I with a cut along a Jordan arc, that is, onto a domain that, with respect to the disk I wi < 1, is a domain (s). Let us show that, for every z in the domain I z 1 < 1', the function fez, t) for 0 < t < to, satisfies the following differential
I wi
and
but lying entirely outside B'. We
continue along it from the boundary of B to a point c €
~
val 0 ~ t
bounded by a
a
(2)
Like the function g(z, t), the function fez, t) is a continuous function of t in the
can be approximated first
we connect the boundaries of the domains
B' by a straight line segment contained in
I(z, f) = g-l (g(z, 0), f)=e- t (z+ ... ).
<1
a by mapping the disk Izl
by domains bounded by Jordan curves contained in onto
Along with the function g (z, t), we define the function
univalently onto
the approximating domains converges to a function mapping the domain I zl onto the domain
which, on the basis of what has been said, is possible.
contained in
w = 0 with domains (s) in the sense that a sequence of 0,
91
§2. PARAMETRIC REPRESENTATION OF UNIVALENT FUNCTIONS
111. REALIZATION OF CONFORMAL MAPPING
a and including the point
.."E;:~--""'~---;;;;:.ti:.:i:i~:i;:£7:=:::i:~~::=~~~
C and then proceed from c
equation, known as Loewner's equation:
along the curve C to a variable point d. If we let d proceed around C until it returns to the point c, then, in accordance with Theorem 1 of §5 of Chapter II,
-_II +kf
Of
ot -
the domains (s) so obtained approximate the domain B'. By applying a diagonal
(3)
l-kf'
I k(t) \ = 1.
process, we can obtain the existence of the required sequence of domains (s)
where k = k (t) is a continuous complex function of t such that
approximating an arbitrary domain B' C B including the point w
remembering that the equation g (f(z, t), t) = g (z, 0) implies (iJg/an (af/at) +
=
O.
. iJg/iJt = 0 we see that the function g(z, t) satisfies, for all z in the disk
Now, suppose that we are given a domain (s) obtained from the domain B
and all t in the interval 0
by making a cut L. By shortening the cut L, beginning with the endpoint lying
is the domain B with the complete cut L; B to and, for t'
< til,
g(O, f) = 0,
denote a function that maps the domain
I z I < 1,
For every z in the disk
S t < to
giO, f)
Iz\ < 1
> 0,
O.:=;;; f
w = h (z, f', f") = g-t (g(z, t'), t") = et' - I " (z
< fo,
I z I < 1,
univalently onto the disk
according to which 1jJ' (0)
< 1.
~ (f)
=
~el
in 0 ~ f
~
=
zl
IzI < 1
Consequently,
< 1.
g(z, t) into that endpoint of the cut lying in
=
Iz\ < 1
by the function w
=
Iwl < 1
obtained under the mapping of
h(z, t', t") and let B t "
point A(e'I) is the endpoint of the cut
J';!Ji'"
const,
'~::
5t"tll
5 t' • t" on the circle \ wi
=
in =
h(z, t', t") obvi Iwl < 1, and the 1.
It follows immediately from what has been said that
1) for fixed t' = t, the arc B t
2) for fixed til
trh;
denote the arc
til
on the circle 1z\ = 1 corresponding to it. Then, the mapping w
~:
< fo,
1
ously maps the point A(t') into the endpoint of the cut
< t",
The form of the parameter t is at our disposal.
Let us choose it in such a way that
1 mapped by the function w
the disk
The strict monotonicity follows from the
Schwarz lemma as applied to the function ljJ(z)=g-l(g(z, t'),t") for t'
=
B. Let 5t , ,t" denote the cut in the disk
(1)
the function (Ht) is a positive, strictly increasing, continuous func
S t < to.
with a cut along a Jordan arc.
(5)
Let us adopt the following notation: let A(t) denote the point on the circle
Izi
+ ... ),
Iwl < 1
+ ... ).
this function maps the disk
for such t' and t", it is continuous in the disk
this function is a continuous function of t in
by virtue of the theorem on the convergence of univalent
tion of t in the interval 0
< til,
For arbitrary t' and t" such that t'
functions. Therefore, if we set
in
(4)
To prove (3), consider the function
Let
univalently onto the domain Bt •
g(z, f)= ~ (f) (z +a2 (f) z,
og 1+ kz
af=azz I-kz'
~
to' The domain B o is the domain B without the cut; t
the domain B t' is properly contained in B t'"
w = g(z, f),
the interval 0
~
Izi < 1
a second equation of Loewner:
og
in B, we obtain a family of simply connected domains B t depending on a real parameter t that varies continuously in the interval 0
< t < to,
Then;
=
t, the arc 5 t
' • t"
' • t"
is constricted to the point A (t) as t" is constricted to the point A (t) as
t
-->
t;
-->
t.
92
III. REALIZATION OF CONFORMAL MAPPING
§2. PARAMETRIC REPRESENTATION OF UNIVALENT FUNCTIONS
Let us show that under both these limiting operations, the arcs B t'. t" and
5 t , ,t" are constricted simultaneously h(z, t , , t ")
Iz I =
circle
past the arc B*t'
,t"
1 onto the domain
parts of the integral in formula (7), we obtain
to the point A(t). If we extend the function
complementary to B t ,
Iz I > 1,
,t"
1 (z, t") _ [
log +I~ ~'\
with respect to the
we see that the function w
=
h (z,
-
(e iB '
+1 (2, t')) + l3 (eeiBiB"" + 1 (z, t'))] - 1 (z, t') c
ffi eiB' _ 1 (2, t')
~
l, t ")
X 2~ ~ ffi (<1> (ei~) dO,
maps the entire z-plane with a cut along the arc Btl. t" onto the entire w-plane with a cut along the arc ~ 't" and along the arc
S; 't,;"
symmetric to 5t
, t"
about the
where ()' and ()" are points in the interval (a, (3). If we divide (9) by ( 8), we
s; ,
obtain log/(z, t")-Jog/(z, t') =_[ffi(eiB'+/(Z, t'))+l3(eiB"+/(Z, t'))]
point. This is proved on the basis of the principle of condensation of analytic
elB ' -
t" - t'
functions and the fact that, if the z-plane with a single excluded point is mapped Since a and
univalently by means of a function w = ¢(z), where ¢(O) = 0 and ¢'(O) = 1, then
f3
1 (z, t')
elB" -
e'
tions that we are considering, the same will be true of
function h (z, t', t") converges to z uniformly on each closed bounded subset of
by taking the limit in (10), we obtain the differential equation
iJ log!
the z-plane that does not include the point A(t). In particular, it follows that A(t) t
< to.
and e if3 , where a<
II> (z) = log 11 (z, is regular in the open disk Izi
<1
f3 < a+
2rr,
iJl
(Jf
then the function
z
I
z1
=
interval 0
g
z
< t < to
I
k (t) = A(t) .
< t < to·
<1
=
0 and g;(O, 0)
> 0,
univalently onto the domain B with cut L is defined
t-+O
i
e&
-+ z d6.
where g (z, t) is that solution of the differential equation
eI8~ __ z
iJg iJg I +k (t) z -iJ-t = iJz- z -=-I--'-"""k"";'(t"""")-z
If we now replace z with the value {(z, t'), we obtain
'a
I 1 (z, t") -1.- \' ffi (II) ( iO» ~~ +1 (z, og I(z, t') - 21t ~ e e,9- 1 (z,
that satisfies the conditions g (0, t) t') dO
t')
=
0 and g; (0, t)
> O.
If the domain B is not the entire finite w-plane, the function
(7)
.
w=g(z, to)= lim g(z, t)
II
=
=
g(z, O)=limg(z, t),
"
For z
1+ kf I _ kl' k
as the limit
Iz I < 1
~
=- f
such that the function g(z, 0), where g(0, 0)
which maps the disk \zl
1 except on the arc (a, (3), on
1.C ffi (<1> (ei8» 21t ~
10 11 (z, t', t") =
+1
function k (t) that is continuous and equal to unity in absolute value in a suitable
which it is negative. On the basis of what has been said, if we apply the Schwarz formula to the function (6), we have in
(t)
and ()". Consequently,
This result shows that corresponding to every curve L contained in B is a
(6)
and continuous in the closed disk Izi ~ 1 and
its real part is zero everywhere on the circle
).
Thus, equations (3) and (4) hold throughout the interval 0
W(0) ~ 0,
t', t"),
(10) •
that is,
operations, the arc B t ' ,t" is constricted to the point A(t). If we denote the end ia
t')
~=-A(t)-I'
What we w-e concerned with at the moment is only that, under both limiting points of this arc by e
1 (z,
approach the same limlt, arg A(t), under both limiting opera
¢(z) == z. On the basis of what has been said, under both limiting operations, the
~
(9)
II
circle Iw I = 1. Since one of the arcs Bt ' •t" and 5t '.t" U ,t" is constricted to the point A(t) in both limiting operations, the other must also be constricted to the same
is continuous throughout the interval 0
93
t -+ to
0, this yields ~
t'-t'=--2k ~ ffi«(lJ(eio»dO.
maps the disk
(8)
1
zl
<1
onto the entire domain B. On the other hand, if the domain
B has as unique boundary point w = 00, then, for any M> 0, the closed disk Iwj ~ M lies entirely in all the domains B t for t sufficiently close to to and,
II
If we now apply the integral theorem of the mean to the real and imaginary
consequently, for such values of t on the circle Iz1
f:
;l;
=
1. Therefore, we also have
§2. PARAMETRIC REPRESENTATION OF UNIVALENT FUNCTIONS
III. REALIZATION OF CONFORMAL MAPPING
94
Iz/g(z, t)! ~ 11M in
jzl < 1
g(z, t) -+ as t
-->
Let us turn now to the function !(z, t). Since !(z, 0)
since, in this case,
g; (0, t) -+ CO
CO,
to when 0 < Izl < 1. This means that to has to be "". Now, let us look =
< t < to of the dif
ferential equation (3) that satisfies the initial condition !(z, 0)
=
Izi < 1
function !(z) mapping the disk
e-tg(z, t) converge as t
=
--> ""
lim!(z, t)=f(z, to)=g-t (g(z, 0), to). In particular, if B is the disk Iwl
~ "l ~\2' 11J I =
t
1-+00
With regard to the inverse function z
condition. This was Loewner's result.
curve L that extends to "", we have, for arbitrary finite w, Iim~elg-t(w,
t)=w.
On the other hand, if the domain B is the entire w-plane excluding the point "", then, in addition to the differential equation (3) and the initial condition
! (z,
0)
=
z, we have in accordance with (2) and (12) g (z, 0) = lim ~et! (z, t).
(12)
1-.. 00
1-+ 00
In accordance with inequality (10) of §4 of Chapter II as applied to the function
f3- 1 e- t g(z, t), we have in the disk (l
Then, by setting z
=
~el I z
+I
Z
Izi <
plane with cuts along a finite number of Jordan arcs that do not pass through the
I ~el I z i 1)2::=:;; I g(z, t) I::=:;; " I
~ef I g-t (w, t)
..
1..-_ "' ". ~ I w I ~
point w
" ••
~el I g-t (w, t) 11
I
'/
"
(13)
•
w
I
such that tv
--> ""
and f3e tv g-l (w, t)/wo
1= -->
1.
(14)
a .;, 1. If we choose from the
¢ (w), we conclude on the basis of
(14) that I¢(w)! = 1 and, since ¢(O) = 1, ¢(w)
contradiction proves (12) for arbitrary finite w.
parameter t such that, as t varies from 0 to "", the point w (t) successively de
=1.
tions (3) and (4) with all the supplementary conclusions hold for the corresponding functions !(z, t) and g(z, t). The function k(t), however, admits a finite number
sequence of the functions f3e tvg-l (w, t)/w a subsequence that converges every where in the w-plane to a regular function
w (t), for 0 < t < "", where w (t) is a piecewise continuous function of a real
Under these conditions, all our reasoning above obviously remains valid and equa
Now, if (12) fails to hold for some finite point wo, there exists a sequence
{tv
=
corresponding to higher values of t bounds a domain B t of the same form as B o·
1-+ 00
But then it follows from the two inequalities (13) that, for arbitrary finite w,
I
w
scribes all the boundary curves in such a way that, for any value of t, the curve
lim Ig-t (w, t) 1=0.
lim ~et g-l (w, t)
0 and have no points in common with each other except possibly end
tion for the boundary L of the domain B 0 can be represented as a single equation
I ".
=
points. Suppose also that at least one of these arcs extends to "". Then the equa
The first of these inequalities yields f3e t Ig-1(w, t)1 < 4[wl and consequently,
1-+00
This result can be immediately generalized somewhat. Specifically, suppose that the simply connected domain B o has no exterior points and is, in fact, the w
1
g-l (w, t), we obtain for arbitrary wEB t
,. "
z and it follows from
t -+ 10
(11)
1.
g-l (w, t), let us show that, for any
=
=
where !(z, t) is a solution of the differential equation (3) satisfying the initial
that satisfies equation (4) also satisfies the condition
lim e- g(z, t) = (1
< 1, then g(z, to)
g(z, 0)= limf(z, t),
0, then the function g(z, t)
=
(15)
what was said above that the function g(z, 0) can be represented as the limit
cut L is such that, as we move the point w along it toward "", this point ap proaches asymptotically a ray issuing from the point w
z <1
1 I
1-+ 10
to a
onto the domain B'. For example, if the
z. If B has more
than a single boundary point, then, in addition, we have in the disk
e-tw, converge as t - .... "" to a domain B' as kernel (in the sense
of §5, Chapter II). Then", the functions w
z, we conclude from
what was said above that !(z, t) is a solution in the interval 0
at the particular case in which the domains B t, subjected to the similarity trans formation w'
=
95
But ¢(wo)
=
a';' 1. This
of points of discontinuity of the first kind corresponding to values of t at which the point w(t) jumps from one boundary arc to another. These results are vivid examples of the reduction of geometrical problems to purely analytical ones. For example, the conclusion that we have just drawn can be formulated in the form of the following theorem, which will play an important role in what follows:
Ill. REALIZATION OF CONFORMAL MAPPING
96
§2. PARAMETRIC REPRESENTATION OF UNIVALENT FUNCTIONS
Theorem 1. Let B denote a simply connected domain without exterior points
that includes the point w
=
0 but not the point w
=
obtained from (18) by dividing by wand integrating with respect to t from 0 to t.
Suppose that the boundary
00.
Let us solve equation (20) by making successive approximations:
of B consists of a finite number of Jordan arcs. Then there exists a complex function k(d of real argument t for 0
St <+
00
Suppose that the function w disk
Izl
_rt \+kW n _1 dt
that is continuous except at a
finite number of points of discontinuity of the first kind. Suppose that !k(t)!
= fez), where f(O) = 0 and f '(0) > 0, which maps the
< 1 onto the domain B, can be represented in the form (16)
t-+oo
ot =_ 1.
of
1 +kf I-kf
(o
00
tions of
z
)
aWn _ I + kW n_1 at --- W n I-kw . n··1
Therefore,
I(z, O)=z.
aWn _
aWn_I
I at
Conversely, we have
=
Theorem 2. Let k (t) denote a function that either is continuous or has a
St<
at
I
I +
I kW n_1 (Wn - Wn_l) I -- kW n_1
1+lzl
,,;;;; IWn -
00.
Wn_1 - Wn- 2 2Wn_l (1- kw n_l ) (1-- kw n_2 )
-
Wn_l \ I _I z I
+ IWn
that is,
1+ k (t) W W
(18)
k--7(t-+-)-w
-;-1............
that satisfies the initial condition wi t"'O = z is a function w '" fez, t) such that f(O, t) = 0 and f~ (0, t) = e -t which, for each t in the interval 0 S t < 00 is a
z in
regular univalent function of
the disk
Izi < 1.1)
t-
oIWn-wn_11 at
Wn _2 1 (l
obtain, for 0
< t < to (where to <
IW n -
(19)
00
- AI Wn -
-
Wn_2 \,
,,;;;; eAtoB ~ \ W n_l -
Wn _2\ dt.
o
(eAtoBt)n -\ 1.. _1\1
•
=
2, 3, ... ):
C= C(z),
from which the asserted uniform convergence of the sequence t
_ r 1 + kw
limit function w = fez, t) is regular in
dt
j l-kw
w=ze
(20)
0
< Izl·
Iwl
Iwnl
follows. The
< 1 and continuous in 0 < t < =
~uppose
and for two points
z\
Then the functions fez
and l>
Z2
that, for some
in the disk
Iz\
t\
00.
z. Furthermore,
Izi
< 1. Let in the interval 0 < t\ <
Let us show that the function fez, t) is univalent in
us suppose the opposite.
Izi
Izl
Obviously, it satisfies equation (18) and the conditions wi t=O
If(z, t)1
1) However, as Kufarev [I9'47c} has shown with an example, the function w '" f(z, t) does not always map the disk < 1 onto the disk < 1 with a cut.
Wn_21,
t
w n_ll
Iw n -wn _l1,,;;;;C
z is obviously equivalent
to the integral equation
I+ B IWn_l -
WIl _l I ~-BI 'W n _l
Integration leads to the following inequality (with n =
1
00),
exists and is also a function with these properties. Proof. Equation (18) with initial condition wi t=O
21 z _I z I )2
where A and B depend only on z. If we multiply this bye-At and integrate, we
Furthermore, the limit
I(z)= lim etl(z, t), 1(0)=0, /,(0)=1,
_l -
= A IWn - 'U'n.l
Suppos e that lk(t)1 == 1. Then that solution of the differential equation
ow
converges uniformly inside the disk
Wn
Izi < 1 and hence in the interval 0 < t < to' where to < 00. From (21), we have
that satisfies the initial condition
-Of = -
(21)
=
(17)
finite number of points of discontinuity of the first kind in the interval 0
n=l, 2, ....
,
> 0 for Iwl < 1, it follows from (21) that Iwnl S 0, 1,···. This means that all the w n = wn(z, t) are regular func in the disk Izi < 1 and that wn(O, t) = 0 and w: (0, t) = e- t • Let us
for all n
show that the sequence of the functions
where fez, t) denotes that solution of the differential equation
0
Since ~ «(1 + kw)/(I - kw»
Izi
I(z)= lim ~et I(z, t),
.\ l-kwn _1
Wo=z, wn=wn(z, t)=ze
1.
=
97
00
< 1, we have f(z\, t\)
t} and f(z2' t) have the same value at t
=
=
f(Z2> t\).
t\. From an
easily proved theorem I) on the existence of solutions of equation (18), it follows that [(z
1>
t)
=
99
§3. VARIATION OF UNIVALENT FUNCTIONS
III. REALIZATION OF CONFORMAL MAPPING
98
[(Z2, t) everywhere on the interval 0 ~ t ~ t1' For t
=
in a neighborhood of z
=
O. If we substitute into this equation the expansion for
gto (z, t) and equate coefficients of like powers, we obtain the system of equations
0, this
It remains for us to prove formula (19). To do this, we substitute into equa
n-l c;.(t, to)=(n-l) en(t, t o)+-2 ~ pocl'-(t, to)knl'-(t),
fez, t) and write it in the form
n=2, 3, .,.,
yields Z1
=
Z2'
tion (18) the function w
=
iJ (log f iJt
+ t)
1'-=1
From this system, we obtain, in view of the fact that gto(z, to)
2kl =-l-kf' t
_
2kl dt
\
~ l-kf
c n (t, to) = -
(22)
.
S. Consequently, for Izi < 1, we have I[(z, t)1 ~ l Mze- , where Mz depends only on z for I z I < 1. This shows that the integral (k[/(1 - kf)) dt converges uniformly inside the disk I z I < 1. From this we con But the function et[(z, t) €
e t [(z, t) converges in the disk
--> "",
to a univalent regular function [(z) such that [(0) = 0 and
f'
(0)
Iz I < 1
> O. This com
Furthermore, since gto (z, 0)
f (z,
=
<
00.
=
+
+...),
lim cn(to)= lim cn(O, to)'
'0 ..... 00
10 ..... 00
00
2)
e-~ k ('t) d't,
(23)
o 00
C3 = -
2 ~ e- 2' (k 2 ('t)
o
O. From
.
pocl'- ('t, to) kn-I'- ('t) d't;.
fez, to), by setting
C2 = -
We define
+ 2k ('t) C2 ('t,
CO
» d't
00
en
o
0
= - 2 ) e-2~k2('t)d't+4( ~ e-·k('t)d'tY.
(24)
These are Loewner's formulas, which we shall need in what follows.
~~''L iJf -t- iJg to = 0. iJl dt
h
In particular, we obtain
gto if (z, t), t) =! (z, to)
we have
1-'=\
Cn (to) = cn(O, to)' c n =
gto(z, t)=f(j-'(Z, t), t o)=et - to (z+C2(t, t o)Z2+ ...).
This function is regular at the point z
,
to) = e- 10 (z c~ (to) Z2 !(z)=z +C2Z2+ ... ,
the function [(z) referred to in Theorem 2. To do this, we consider first the func tion [(z, t) for 0 ~ t ~ to, where to
n-\
we obtain
pletes the proof of the theorem. 2 )
In conclusion, let us show how to obtain the coefficients in the expansion of
to
2e(n-t)' ~ e-(n-l)~
This formula enables us to calculate successively the coefficients c n (t, to)·
.fo'
clude on the basis of (22) that, as t
z and hence
cn{to, to) = 0, for n = 2,3,"',
Integration yields
_ I og 1(2)+t 2 -
=
at
§3. Variation of univalenl functions
Consequent! y, in accordance wi th (17), we obtain the equation iJg 10 _ iJg 10 1 k (t) z -(ft-7Z z 1 - k (t) z
+
In solving extremal problems of conformal mapping, we often use different
variations of univalent functions. A fairly general type of such variations is
given by the
I) One gets the result from a suitable inequality of the form
I
iJ ifliJt- Is) I ~ MZ If I-It,I
I I
where f1 = fez 1, t), h = f(Z2' t), and M z depends on z for z < 1. 2) Equations (17) and (18) were studied by Kufarev [1946, 1947, 1947a, 1947c, 1947d, 1948). In the article [1943), he srudied the question of parametric representation from a more general point of view.
Theorem. I) Suppose that a [unction w
=
fez) such that [(0)
=
0 is regular
and univalent in the disk I z \ < 1. Suppose that the [unction w* = F(z, A) is a regular [unction o[ both z and A in the region r:S \ z I < 1, I AI < AO and that, [or
I)
Goluzin [1946r).
"_:.,_ ..._:.1..
_£
U, __ '-~
•
• ••
..
'"~.~.~~--~_~-~'-~-~-=--~7~~:"===r='-
-~
H!":::----
__ ... ~.- ----.
.__. _. c-._:-_-_ . -_~~~~==:
~~
~ --.:~\'{~~;::!:-~~~~~.!3~~~~==='~~~~~''''=:~=:i$'':==--'=.:~7.;:=~:-:-i£.=:.;.
each A in the interval 0 < A < 1..0 , it is univalent in the annulus r:S I z
I < 1.
pose that, for every z in the domain r < I z I < 1 and small A,
F (z, A) = j (z)
+ ~.q (z) + 0 (A
101
§3. VARIATION OF UNIVALENT FUNCTIONS
Ul. REALIZATION OF CONFORMAL MAPPING
100
- .. ,---=;~-----
Sup to Iz' \ < 1 and I z' \ > 1 are mapped respectively into the disks I z \ < 1 and \ z I > 1 since \ z I 1 on Iz' I 1. But then, for small positive A, the function 0=
0=
00
2 ).
~ ),''l', \z')
(1)
If we combine the image of th.e annulus r < I z 1 < 1 under the function w*
w*=F(z'e,=1
(4)
, A)
< I z' \ < 1 univalently onto a doubly connected domain whose outer boundary is the boundary of the domain BA· Since the function small A, a simply connected domain B A including the point w* = o. For the func (4) is regular in \ z' I < 1, we conclude on the basis of a familiar theorem that it tion w* = (z), where (0) = 0, which maps the disk I z \ < 1 univalently onto , maps the disk \ z' I < 1 univalentlyonto B A' Thus, the problem of determining BA' we have in I z I < 1 ~ the function w* = f*(z) which maps the disk I z \ < 1 univalently onto the domain BA will be solved if we can construct functions ¢n (z') satisfying conditions 1) 2 F (z, A) with the domain interior to the image of the circle
r
IzI=
maps a rather narroW annulus r'
r, we obtain, for
r
j* (z) =1 (z)
+ Aq (z) -
AZj' (z) S(z)
+ AZj' (z) S U)+ o (A
(2)
),
4). where S(z) is the sum of terms with negative powers of z in the Laurent expanTo construct the functions ¢n (z'), we substitute the function (3) into F(z, A) sion of the function q(z)/z[' (z) in the annulus r < I z I < 1. and expand the result formally in a series of powers of A; The symbol 0(1.. 2 ) in equations (1) and (2) denotes a quantity woose ratio to 1..2
co
is uniformly bounded for all z belonging to an arbitrary closed subset of the disk
I z I < 1. Proof. Our problem will be to find the function w* = theorem, such that
{* (0)
=
{* (z) referred to in the
where 111 0 (z') = f(z'), and, for
1
~ "''l',(z')
z=z'e,=1
2) ~(¢v(z')) = 0 on the circle
E
for v = 1, 2,"';
(1
+ Ancen (Z'»,
A)
I
>'=0
+'1'(') z z cen (') z
A) \
'),=0
, ,- I 1 d n F ( ze-
- zl (z) [ n! dAn
l:s
z'f'
(z')
'1 ),=0+ cen (Z}J,
,I\.) (z')-~ v
(6)
In particular, for n = 1, this gives
F' (z' 0)
I
(1)1
(z') = z'!' (z') [ Z~f' (~')
]
-\-
'PI (z') •
(7)
and that vanishes at z' = O. But, for r ~
Iz' \ < 1, we
have the convergent expansion
p, where r:S p ~ l/r are mapped by the function (3) into curves close to the circle I z I = p and the derivative dz/ dz' differs on I z' I = p from 1 by a
F). (z', 0) _
quantity of the order of A. Therefore, for small positive values of A, the function
-I
I=
1
'
E j.'"
_",
When these conditions are observed, for small positive values of A the cir
(3) maps the annulus r ~
),=0
n-l
4) when w~ substitute (3) into F (z, A) for sufficiently small A, we obtain
cles \ z'
),''1' (z')
= __
I :s llr;
1/r for sufficiently small A; a function that is regular in the disk \ z'
an
1 an E),''P(z') F(z'e'=l' nl d).n
1, 2, ... , have been chosen so that
3) the series I':=IAV¢)ZI) converges uniformly in the annulus r:S Iz'
I
n-l
(3)
I z' I = 1
,A)
n-l
=1iT dAn F(z'e'=l
0:>
1) they are regular in the annulus r ~ \ z '
E ),''1' (z') , ,,=1 "
n(z)--l n d,nF(ze ~
F (z, A). Specifically, we set
(5)
n> 0,
1 a'n
' _
this by making a suitable substitution for the variable z in the function w* =
=
A) = ~
co
0 (depending on the value of A). We can arrange for
where we assume the functions ¢)z'), v
co
E ),' F(z'e,=1 'P,\Z') ,
z' \ < 1/r in such a way that the subannuli corresponding
z'f' (z')
,.,
(I) 'k
(8)
~ cllz.
-
k=-co
If we define
COI(Z')=I
~
..
-I
c(l)z'/I+ k
k=-C0
~
\1)
k
c _J
....... z'k
k=-o:>
(9)
102
2) with n = 1. We also satisfy in part condition
we satisfy conditions 1) and
4):
specifically, the function c;l>1(Z') is regular in the disk
Iz' I < 1
103
§3. VARIATION OF UNIVALENT FUNCTIONS
III. REALIZATION OF CONFORMAL MAPPING
la"l~ (n~I)~' Ib"I~(n-:W'
since the
n=l, 2, ... ,
bracketed expression in (7) has no negative powers of z' and <1>1 (0) = O. (We note that the function ¢ 1 (z') is precisely determined up to a purely imaginary constant
and if 00
term by the conditions listed.)
CD
Let us suppose now that the functions
¢.)z'),
I z'l
<1
,,=1
for v = 1, 2, ... , n - 1, are
defined in such a way that 'conditions 1) and 2) are satisfied and the c;I>.)z'),
for v = 1,2"", n - 1, are regular in Iz' I < 1, we have the expansion
then
.. , + a,,_lb l \ ~ WAB AB AB 2AB + 3 (n_I)l! +. ... + 7i222 (n+ 1)2
I Cn \ = \alb"_1 + a2b ,,_2 +
with c;l>v(O) = O. Then, in r:::;
2
,,-I
E
"'P,(z')
d" F(z'e'= 1 -n!I dA" z'f' (z')
,
A)
I
! ( (, )z
'k k'
-
1.=0 -
If we take
-1
~
(j',,(Z')=-
-I
and c;l>n (0)
=
O. Furthermore, condi
¢n (z')' (Here, and in general, ¢" (z') is determined up to a purely imaginary constant term.) If we choose ¢" (z ') succes 1) and
~ ,.~~Bm U2 + 312 +... )= (n ~~.
( 11)
k=-oo
I z' I < 1
2) are satisfied for
sively, for n = 1, 2,' ", according to the rule given above, conditions 1) and 2)
where C = 8 (Z-2 +
r
2
functions ¢ (A)k, we get the conclusion of the lemma. Let us make a convention on notation. For any Laurent series x(z) =
1': =
-00
c n z " we define
(X)
IIx(z)ll= ~ Ic"llzl"· n=-OO
We still need to investigate the convergence of the series appearing in condi tion 3), on which the satisfaction of condition 4) depends. To do this, we need to prove the
Lemma. Suppose that the function ¢(A) has the expansion ¢(A) = l':;'=l a the coefficients in which satisfy the inequalities Ian I :::; I/(n + 1)2 for n = 1,2,.... Suppose that the kth power of this function has the expansion [¢ (A)] k = 1': =1 a~k)An
When such majorizing series converge, they possess the obvious properties
II XI (z) +"1.2 (z) II ~ II XI (z) II + II X2 (z) II, II Xl (z) X2 (z) II ~ II Xl (z) 11·11 X2 (z) II·
"A n
where k is a positive integer. Then the coefficients in this last expansion satisfy the inequalities la~k)1 :::; Ck/(n + 1)2, n = 1, 2,"', where C is an absolute constant. Proof. If the coefficients in the expansions of the two functions
(j' (A) = ~ a"A", n=1
satisfy the inequalities
00
~(A)= ~b"A" ,,=1
,
+ •.. ). If we apply this inequality successively to the
will be completely satisfied and condition 4) will be formally satisfied.
co
22 ~)"
1 I 1]
(:+:)' + ... + Int']' .(n-r~1+2)'
+3'·
then the bracketed expression in (6) will contain no negative powers of z' and, tions
1
(10)
(n)
(~)Z'k+ _~ Z'k' ~
"~~-oo
[I I
00
k=-oo
hence, c;l>n (z') will be regular in
(A) HA)= ~ ("All
(12)
Let us investigate the convergence of the series ~:~' =1 A ¢)z'). Let us show V
that the inequalities
,
DM,-l
1I(j'.(z)II~('1+I)'-' v=l, 2, ... , hold in the annulus r :::; I z'1
::;
llr for suitably chosen constants D and M.
To prove the validity of (13) for v of the quantity 4
1\ ¢1 (z')1\
(13)
=
1, we need only take for D the maximum
in the annulus r:::;
I z I : :; I
Ilr.. Let us seek, by choos
ing M suitably, to satisfy inequality (13) for all v. Let us suppose that inequal ity (13) holds for v = 1, 2,"', n - 1 and let us find a bound for
II¢" (z') II·
To
104
III. REALIZATION OF CONFORMAL MAPPING
§3. VARIATION OF UNIVALENT FUNCTIONS
105
do this, we consider the function (10), wbich we denote by r(z'). We define ~
F (z, ),) =
II'tn(z') II ~ B
AkIZ k).,./.
(4)
k=-oo.oo 1=0,00
This series converges for bave
II 'tn (z') ,,~ I z't' (z') 1
r
< I z I < 1 and
I
II
I Akll r
k
~
. '1.1 dAn ~ I Akll k,l 'I. -
~
,
Ik)"le
n! dAn
v7:1 (v+I)"
I
Ae
~I
v=1
n
00,
l+j
J=O
k, I
(if we replace >.AI with A\ where R = max
fl. I
I
is a regular function for r:S
ll
=r 1/z'
'I.
nldAn
Al (
A'
'n.
In particular,
"
A=O
-I ~
(15)
(n)
~ le k
k,=-oo
f' (z ')/1.
, Ik
DMII-l
liz I ~2(n+I)2'
(17)
Furthermore, since the majorant series of the second sum in (11) does not exceed the majorant series of the first sum, for
Iz 'I < 1,
it follows that (17)
yields the inequality
DMn-l
I Cl'n (z') 1\
"
Y~_)J I,=o-(n-I)ldAII-I __1_ d n- I (~_A_V )J I """,(-v+1)2 '=0
on the circle
Iz' \ =
'I.,
(13) holds for all v in the annulus r:S
I.-:
n-I
v)1 = ~ "UP' c.
= tAl'¢)z
~ ('I.
+ I)'
r and hence in the annulus r:S
For M chosen independently of
.=1
is the coefficient of An -I in the expansion
\• =1
Tberefore, this last sum in (6)
DMn-l
..0.1('1+1)2
( " A ~ ~+ 1)1
IAI < Ao.
II'tn(z·)II~~.
~V (-v +A'1)2 )JI
n-I
.=1
and
M (tbough independent of 'I.), less than DMn- /2(n + 1)2. Thus, for such M on the circle I z' I = r, we have
we may assume tbat l + j ~ 2 and, obviously, that l:s n. Furthermore, since
-l_~
Iz I < 1
M since tbe series (14)
1
are equal to O. Therefore, when summing with respect to k, l, and j, n-I
DC
ber in (16) is, for sufficiently large
Now, the terms with j = 0 and j = 1 for 1=0 and the terms with j = 0 for
l? 1, l';'
I kl + 1)9 rkM-It---xr
with large values of M is of the order of 1/M 2 • Consequently, the right-hand mem
--.w- v~ = I (V+I)2/ bO (
(16) I
2, this sum will be equal to
I z' I = r,
v=1
tz'
~
~ I Akll (t
A=O
k Ii Di I d n I j!MF 'I.! lii.." A
+ l)~ rkM-I- J I k 11~)JCj JI
where N is independent of z' . If in the last sum we drop tbe requirement that
A=O
n-I
=B.! I A k/l rkM l! I
J
and this series converges for sufficiently large values of
IklD ~
n n-I I d I Ak/lr M n!dA nA e
~ I Ak/l (l l+J~2
n-I k
I k JJDiGi "I ( J. n-I+W
~
NMn
('1.+ I)'
k, l.
n-l Ikl.E AVII'Pv(Z')llj
n-I AVDM V- 1 Ikl~ _ _
~I
=B!I
Iz
I·i
IAk/l(l+ l)jrkMII-I.i" Ik/iDJCi J![(l+I)(n-l-fl)J2
I A=O
1, we obtain on the circle
1 d n_
I rkM n
kl,
'I' (z') r
kn
"
1 dn
Then, by (13) for v = 1, 2, .. "
!
k.l~J
k, 1
~ I z't' (z')
II 'tn(Z), II ~B
=8!
0 <. A < Ao. From (0) and (12), we
II '1'f 'I.!1 dfd n ~ Aklz'k)"/e v=EII Av v n
~ IA
~
k,I.J
(18)
I z ' \ ~ 1/r.
we conclude by induction that inequality
I z ' I :s II r.
This shows that the series
') converges uniformly in the annulus r $. \ z ' I < l/r if
IA I < 11M,
and this proves the validity of condition 3) and hence of condition 4) .
A,
¢)z') satisfy all conditions 1)- 4) and function (4) constructed from them does indeed map the disk Iz ' I < 1 univalently onto the domain B A' This function [*(z') is, for every z' in the disk Iz ' I < 1, an analytic Thus the functions
we obtain, by applying the lemma, !<~ll:s c/(n -l + 1)2. Therefore (15) yields, for z' on the circle 'z' I = r,
h~~
/
is regular and univalent in the disk
function of A in a neighborhood of the point A = O. We have the expansion
dition
OJ
/* (z') = ~ )!«P
y
(z'),
=
f; (0) =
fez '). Furthermore, keeping (1) in mind, we obtain from formulas
m
zj' (z) - . ' -
1:w :J.
residues A k f(zk)2/z k[' (Zk)2, k
are finite points in the
• 'W m
=
1,"', m, because, for distinct points w'
'\'1
A ( f (Zk)
k=1
m
Iw' -
oW'
I(I -
(w -Wk) (w -Wk)
AI.. - A max '\'1 A I w'w' 1-1-1 Wk II w'
w',w"/::1
k
;f
+ w' I)
fez), where f(O)
=
0 and
f' (0) =
--:;=:,----'--'--
=
m
k
f~Zk) )2+ 0(1.. 2). (Zk)
zd
m
f* (Z) =/ (Z) + A.'- f k=1
'\'1
•
A/ (Z) .'- A k
(z) _ f (zk) -
(Zk) k .zk!' (Zk)
k=1 m
+ AZj' (z) '\'1 A .'-
k=1
(21)
;~
(23)
satisfying the condi
(f (Zk) )2 zk!' (Zll)
k=1
m
.'-
k~1
f* (z)
m
Ad (Z)I
_ AZj' (Z) '\'1 A (. f
f
(22)
1, we need only divide (22) by (23). We then have
"\"1
,~~!
0(1..11 ).
-1
Here, we have
tion [' (0)
m
" Ak! (Z}2 + A .'--;;-; ..
12 _1-Zk:ZkZ +
If we again wish to obtain a variational function
1
w* =/* (z)=/(z)
(f(Zk»)i ~ Z-Zk
A k zk!' (Zk)
(2-(Zk) z,,/' (Zk).
k=1
1, is
)i----.!.L Z-Zk'
k=1
.'-
z I < 1 and that it maps that disk onto the domain B. If the points w 1> ••• ,wm are all exterior points to the domain B then, for any positive p less than the distance between any of the points w k and the boundary of B, we conclude that, for small A> 0, the function (A = 0) regular in the disk
Ak
.'-
k
!
m
- Az!' (z)
1*'(0)= 1 +1.. '\'1 A (
fore, for sufficiently small A, the last expression in parentheses is positive, and hence w* (w·/) 1= w* (w "). =
I
I w'-wk \·1 w'-Wk I '
where the maximum, taken over all pairs w / and w", is obviously finite. There
Now, suppose that the function w
zkf' (Zk)
_,_ _
-LAzj'(z) '\'1
t\
m
;?-;
Ak/(Z)I (Zk)
.'- f (z) - f
W'~'+Wk(~'+W'»)\
k
m
1* (z)=/(z)+A "\"1
domain,
Ak
1, ... , m, with
k=1
and w" in this
m
=
Consequently, the formula yields
is univalent in the infinite domain consisting of the exteriors of the m circles
1:
f (zk)
1,"', m, it is obvious that m
For arbitrary given p > 0 and for sufficiently small A> 0, the function (20)
k=1
'"
~_I
k zf' (z) f (z) -
-.'-
=1(W'-W')(I+A),+A
"
has only simple poles in the disk \ z I < 1 at the points z k' k
(20)
S (z) -
w* (W'') I
=
k=1
Ak
w-plane.
I w* (w') -
Akf (Z)2
...
..!!....!fL _ '\'1 A f (z)2
k=1
=
fez k), k
The theorem that we have proved is applicable to this function. Since the function
m
p, k
=
k=i
sider first the function
=
such that w k
m
Let us give some examples of the application of this theorem. Let us con
WI'"
I<1
* '\'1 W =/(z)+A .'-
From (19), we obtain (2). This completes the proof of the theorem.
where A and A k are complex constants and
'
1, 2, ... , m, then (21) can be rewritten
«Pdz') = q (z') - z'j' (z') ( s (z') - S ( zl, ))
Iw - wkl
< I Z I < 1.
If z 1> ••• , Zm are points of the disk \ z
•
w*=w+AW(A+
and that it is normalized by\the con I
and univalent in some annulus r
(7) and (9)
IZ \ < 1
1. On the other hand, if the points wI"", w m all lie in tHe inter
ior of the domain B, then, for sufficiently small A> 0, the function (21) 1s regular
(19)
y=O
where «1>0 (z ')
107
§3. VARIATION OF UNIVALENT FUNCTIONS
III. REALIZATION OF CONFORMAL MAPPING
106
k
(
)II--.!.!L Z - Zk
f(Zk) )i Zk~ zk!' (Zk) I - ZkZ
+ 0 (Ai).
(24)
lei__"lII{.lIIIlnll!.~'lfl_.ll1IIl"
~-------
mi..... r-FI1.-!WiIIMliIilU~~ _ _~ ~ ~ ! Q l N ~ ~ ~ i l i i : _ I I I _ " ' £ _ ~ ~ ~ _ 9 M i (
III. REALIZATION OF CONFORMAL MAPPING
108
§3. VARIATION OF UNIVALENT FUNCTIONS
f(z), then, for arbitrary
w
From these formulas we can easily obtain the corresponding variational
trary A and for sufficiently small A> 0, the function
wk either lying inside
formulas for the function
m
II
F(~)=~+l1o+~1 + ... , which is univalent and regular in the domain ( =
B or external to B, for arbi
Formulas (21) and (24) are the desired variational formulas. I)
=
00
I (-I > 1
(W-ak)
w*=w+AAwk;:1
except for the pole
(27)
II
aod which does not vaoish in that domain. To do this, it is sufficient to
(W-wk)
k=1
apply (21) and (24) to the function
will, as was shown above, map the portion of some neighborhood of the boundary ; of B that lies in B onto a doubly connected domain and it will map the boundary
I
F(;)
!(Z) =
itself into the boundary of some simply connected domain BA that includes w*
=
O.
Furthermore, the doubly connected domain referred to above will lie in BA' Since
and set
.F. (~) = (We note that F * (()
to
in
(I I ) = ~ + 116 +': It[ +
the points al"", am do not lie in B and are mapped by (27) into themselves, , . 'll'
it follows that, for sufficiently small A, they will lie outside BA. Therefore, the function w* = (z), which maps the domain 1 z I < 1 onto the domain B A does
I. C
r
I (I > 1.)
oot assume in
Simple transformations lead to the following formulas:
F* F ({) = F (~)
*
+ A"'"~ A m
m
l: A
(q=F(~) +A k=1
k
k
+ AF (~) k~1 '\1 A (--f k CkF' (C
F~~;~(~k + O(A~),
k
Wk
of the domain
~
) )
k
_I=--+- 0 (A ~), 1 --- C:.C
under the mapping by the function w
k=1 m
II A o= A,
(26)
=
and that does not assume in
=
A k = A -m=--- II (Wk-W.,) .,~I
~=;:'k
F«().
Then, we find the formula for
[* (z): m
II
f«() tha t is regular and univalent in
Iz I < 1
certain given finite values a I •...
IT
am in
I<1
(f (z) -- 1 (zk»
m
_ ).Az/' (z) '\1 Akl (Zk) "'" Zkf' (Zk)! (z - Zk) k=1
I z I < 1.
Specifically, if B is the image of the disk \ z
__
k=1
we can construct univalent varied functions with the same property, that is, not •• "
(I (z) - ak)
f*(z)=!(z)+AA!(z) _~=_I-
..•• am, where m ~ 1. Following the lead of the preceding variational formulas,
assuming the values a 1>
(28)
(Wk-a.,)
,,=[
As our next example, illustrating the theorem on the variation of a univalent function, let us consider a function w
l:w~kwJ,
W*=W+AW(A o+
in formula (25) are arbitrary fixed points all exterior to the image
1(1 > 1
nT,
m
+ ),~F' (~)k= l: A (--.!'(kF'. tC(C ) k
all lie inside B, that is,
it is convenient to put (27) in the form
.
I
Wk
Wk=!(zk),lzkl
)~ _ A~F'~) ~ A (_ F ((k) ) _C_ k~1 k CkF' (C k) C- C,~ m
where the
(25)
m
k)
the values al'" " am' This function is constructed in
when
F (C k ) F (C)
P (C) - F (C k )
(C k )
Iz I < 1
..the same way as before. In the case in which the
k=1
m
Iz I < 1
109
under the mapping
+ AA
1) Formula (24) was first stated by Schiffer [1943].
Ii,e
m
Z2/' (z)
l:
_
Akl (Zk) k=1 Zkf' (Zk)' (l-Zk Z )
+ 0 (A 9).
(29)
§ 1.
ROTATION THEOREM~
111
Chapter II, we shall give in the present section the corresponding sharp bounds on their arguments. In particular, we shall establish the final formula for the rotation theorem for the class 5 consisting of a sharp bound on the argument of the deriva tive. The parametric representation of univalent functions given in § 3 of Chapter
CHAPTER IV
III will provide us with these bounds. 1) Let us denote by 5' the subclass of the class 5 that consists of functions [(z) having a representation of the form
EXTREMAL QUESTIONS AND INEQUALITIES HOLDING IN CLASSES
OF UNIVALENT FUNCTIONS
f (z) = lim etf (z, t),
(1)
t-+et:>
where [(z, t) is a solution of the differential equation
oj _ _ fl+kj atl-kt'
(2)
fez, O)=z
(3)
satisfying the initial condition
§ 1. Rotation theorems As we noted in Chapter II and as will again follow from Chapter V, extremal
that, for fixed t, is a regular function of
questions for various classes of univalent functions are of prime importance in
z
in
I zI < I
0, and [: (0, t) > O. Here, k
such that
I[(z,
t)\
the theory of univalent conformal mappings. When these questions are answered,
that interval, [(0, t)
it turns out that the requirement of univalence of the mapping influences the in
continuous function that is equal to unity in absolute value in the interval 0 .:s t
crease in various quantities characterizing to some degree or other the amount
=
=
In
k(t) is an arbitrary piecewise
<
00.
In order to find a bound for any of the quantities that we shall consider below
of distortion of the domain being mapped. In §4 of Chapter II, a number of extre
for the entire class 5, it will be sufficient for us to do this for the subclass 5'
mal problems dealing with inequalities involving the initial coefficients of the
because, as was shown in §3,of Chapter III, an arbitrary function [(z) of the class
mapping functions, and covering and distortion questions were solved by
S can be approximated by functions [n (z) such that [n (0) = 0 and [~(O) > 0 each
an elementary procedure. In the present chapter, we shall concern ourselves
of which maps the disk
specifically with these questions, this time using more powerful methods, for
Jordan arc that goes out to
example, the method of parametric representation of univalent functions and the
by functions [n (z)/[~ (0) E: S '. When thIS approximating procedure is carried out,
method of variations. 1)
Iz I < 1 00
univalently onto the w-plane with cut along a
and doe s not pas s through w = 0, and consequently
... the sequence of bounded quantities for the approximating functions converges to
For convenience, let us adopt some notational conventions regarding certain classes of univalent functions with which we shall be dealing in what follows. Specifically, let us denote by 5 the class of functions [(z) = z + C2z2 + ... that are regular and univalent in the disk
I"
I z I < 1.
Let us denote by I, the class of
functions F(<) = (+ a o + a1(-1 + ... that are meromorphic (with pole at 00) and univalent in the domain > 1. As a supplement to the sharp bounds on the modulus of a function belonging to the class 5 and the modulus of its derivative that were established in §4 of
1) The reader wishing to become more familiar with extremal questions regarding univalent functions is referred to the author's articles [1939d , 1949cl.
110
~:
the same quantity for the function [(z). A bound for arg [[ (z)/ z] and arg [z[' (z)/ [(z)l. We divide the identity
aj(z, t)-_f( t) l+k(t)j(z, t) at z, l-k (t)j(z, t) , by [(z, t) and separate the real and imaginary parts. This gives us the two iden tities
alj(z,~=_ \f(z, t)\
~at
l-Ij(z, t)1 2 Il--k(t)j(z, t)12'
1) See Goluzin [1936] and a number of subsequent articles by the author in Mat.
Sbornik, for example, [1939c, 1946cl.
(4)
-
----
-
-
-
-
~~~=~-----~-----~~-----=~'----'~-----=~-----'--O=-------.::i~c,_z---~~-'=~-=,;;;;;;;;;_~c-"-------'=~=OO
IV. EXTREMAL QUESTIONS
112
oarg/(z, ot If(z, t) I
It follows from (4) that increases from 0 to quantity
If(z,
00.
t»
23 (k (t) I (z, k (t) I (z. t)
t) _
-
II -
(5)
12
I zI to
decreases monotonically from
t)1 instead of t. Now, from (5), we have
(wbere we write for brevity
f
d t arg/l ~, .
21/1 . .,.
2d
If we integrate this witb respect to t from 0 to I
zI
I(z)
III 00
I
+ I zz II" 1-1
1
~Iog
z
(10)
+
arg k (t) arg/(zo, t)= - ; . Since all these operations are reversible, we have equation (10) for the func
(6)
, tion f(z, t) defined from tbe function k(t), which is a solution of equation (2) evaluated at the point z = Zn'; Here, as t
t ~'
+
determined from
or, consequently, with respect
log (1 +
arg/(Z) = \ d arg/(z, t) •
?L"
Z
IZol>/(I- \zol> .
-> ""
the rh!ht-hand member approaches
Consequently, the upper bound for arg [f(z)/z]
given by inequality (6) is tbe least upper bound. In tbe same way we can sbow
1that the
Consequently, it represents that branch of tbis multivalued function thar approaches
lower bound for arg [{(z)/z] given by inequality (6) is the greatest lower
1bound.
Since the function
O.
Inequality (6), which we have proved for tbe class S I, holds also for the
g(\.,)=
Let us show that the bounds for arg [f(z)/z] given by inequality (6) are
:s t <
00
1
+zC
<"/~\/l
and for wbich f(z, t) defined by
for arbitrary t such that 0
:s t <
00
t» = -If(zo, t) I and given z 0, where Iz 0 \
is
(1 I)
,=
1
,I
arg zl' (z)
I
(z)
I",:::: log 1 + I z I ~
. 1 -I z I'
(12)
!which holds for an arbitrary function belonging to the class S and for an arbi
satisfied, then from equations (4) and (5), which are equivalent to equation
'oj trary point
(2), we have for z = z 0
01/1=_1/11-1/11 iJt 1+1111 '
... ,
'following inequality: 1) (7)
< 1. If (7)
~+
z I < I, belongs to tbe class S when ,; f(z) does, by applying inequality (6) to this function witb z, we obtain the
equation (2) and condition (3) will also satisfy tbe condition '::5 (k (t)/(zo.
1~11\
wbere z is an arbitrary number in the disk
sharp. To do tbis, we sbow that there exists a function k (t) of absolute value which is continuous in the interval 0
I(C+~) -/(z)
~
entire class S by virtue of what was said above.
z in
1z I
< 1.
By arg [zf' (z)/f(z)] we mean tbe branch defined by tbe
equation
oargl _~_
(8)
~-1+1/12'
If we integrate the first of tbese equations, taking obtain
l/l
which, on the basis of (9), can also be expressed in terms of t. Then, k (t) is
00
->
d
+
Here, the argument is defined by the formula
z
2
1 _ I <,II •
arg/(Zo, t) =log 1 I Zo I -log I III Zo l-izol 1-111'
f = arg eff,
to 0, we obtain, by virtue of (1) and (3)
Iarg--z
o as
(9)
. from whicb we get
t
from
If I -t I Zo I I-J/II=e I-Jzo/2
darg/=-
dt
Id t arg ell ~ - 1-111" If 1
113
cally from I z 0 ,I to 0 as t increases from 0 to "". Furthermore, by eliminating ,; dt from equations (8), we have
and k instead of f(z, t) and k(t» or, eliminating
dt with the aid of (4) and keeping in mind that arg
to
§l. ROTATION THEOREMS
; From this we determine I fl as a positive function of t that decreases monotoni
0 as t
This enables us to take as our independent variable the
1
~=--'-------=~-;'---'-----===~-----="==----'------=~-===~=--==--'-----
If(z 0, 0) I =
\z0
I,
we
1) Inequalities (6) and (12) were first given by Grunsky [1932]. For informacion on ';their ~eneralization to univalent functions in multiply connected domains, see Goluzin LI937J .
-
-----------~-~
----_._._---:-----=--~---
..
-,...-=-~~..,.
="'""~"",:~~~->=~=""~"'~. '~.=""""""",.,,,,,,,,,,,,,==.=---~-=;=,==~,,,~~~~,,,,,,~, ,,,._~-.,,,,
§l.
IV. EXTREMAL QUESTIONS
114
I+z(
.
()
Consequently, for functions f(z) in the class S, we have l )
O. The bounds given by inequality
-->
2 I Z 1 Z
(= - z, we again obtain (6). atg
f' (z).
f' (z).
+
ot -
from which we get
-:;.1
iJt
for
I z !;;::: Y2'
I
12
(13)
I' _
(I +2kl- kIP) _ (I - k/)2
01 (z, t)
-
iJz
2~
«1- k/)2)
-
II-
,_
d t argl -
2S' «(1- kf)') 11
1..e12
-
dill
I""
z
'
in the class
kl If
5, we have
o
for every p S
,,,.,,
p
1/J2,.
The sharpness of the bound (13) (for arbitrary z in that is continuous in (0 St
~1 sin2arcsinlfl=2IfIVI-I/I~ I
I
1/\ ~Y2'
for
J
II I - YI_1/12
for
4d
II I
1-III (1-//1
•
1/1~y'2'
for
< 00)
I argj' (z) I ~ \ y 4d III
.:1
«J -
- 2 \I kl)2)
for
1-1/1 2
-I
II I
1/1-=Y2'
III ~ Y2' Iz I S 1/J2,
..3M. -, =
.
Sin
-1/\
=
arg(1- kl)
o
for
1/\~Y2
I
=4 arc sin I zl,
for arbitrary t in the interval 0
I
for
\/l~~,r-' ,2
for
1/1;;:::Y2
I
\
largj'(z)l~ .) YI_1/12+
\/1 ~ Y2'
or, what amounts to the same thing, the condition
-Y2
S t < 00 and given z
I
(14)
= z 0 in the disk \ z
I < 1.
From (14), we obtain by simple calculations
III (\/\- tV 1 - 1112) 4d
I
for
.
I
2)
I V I -If [2
-1-I=k/1 2 = sm2arg(l-kl)=
I z I ~ 1/y"i, C
in fact) can be
for which f(z, t) defined by equation (2) and the
Iz I
l'2
Iz I < 1
condition f(z, 0) = z satisfies the condition
\
~ _I_
Integrating this with respect to t from 0 to 00, we obtain for
~
rJ.= arc cos p,
seen by showing that there exists a function k (t) of absolute value equal to 1
II- k/1
2d
is obvious
)
I arg/' (p).[ =4 arcsin
13 «1-kf)2) I =jsin2arg(l-kl)1 2
Idt arg I I ~
I z lSI/Vi
.
C I elC1z I(z)= ~ (I~. ia~\8 dz = z+ C2 Z2 + ... ,
But
Therefore,
for
from the fact that, for a function
and, consequently,
and, for
I z I ~ y2 '
I arg f' (z)1
That this inequality gives a sharp bound on
I 2kl - k 2p (1-kl)2'
01' _ _ I'
.
for
In an equally simple manner, we can find a bound for
Specifically, from (2) we have
oarg/' =
I
f 4 arc sin I z \ I arg j' (z) I ~ ~l 7t + log I _I
(12) are also sharp because, when we apply (12) to the function (11) at the point
A hound for arg
I Z 12
(argl (z)\~7t+log I-Iz/!'
0
zl'(z) C C I(Q-/(z) arg I(z) =.) d arg V' (z) (1 _I z I') =.) d arg ~i- , o z that is, the branch that approaches 0 as z
115
that is,
C+Z) I (- --I z
-z
ROTATION THEOREMS
Izi
\
r
_ Y2
II I 1/1(1-1//2 ) ' 2d
kl=
I±Y21./12-1
\
')
.1=t-Y21/1 2-1 2
-l
1
for
1/1~y:r'
for
III ~ y~(
I
1) Inequality (13) was proved by the author (see Goluzin [1936] ). That it represents a sharp bound for z > IIV'2 was shown by Bazilevi'C [1936a].,
I I
116
§ 1.
IV. EXTREMAL QUESTIONS
±.
Here, we can take either of the signs terms of
I fl,
On the basis of this equation, we have for arbitrary real
Now that we have found kf expressed in
pendent of t,
let us substitute the result into (4) and (5). Then, from (4) we obtain
as a continuous monotonic function of I fl and, consequently, I fl as a continu ous and positive function of t. From (5) we then have a representation of
ffi
t
i l
I' + log -.-I ,-._
This shows that the upper bound for arg least upper bound for arbitrary z in
1
f' (z)
for
I(z)
'U vI
I I Zo I;;::: Y2'
Since given by inequalities (13) is the
I
I
I!)
we
(15)
I
z 1 < 1. Therefore, by shifting from f(z) to the function F(')
const, we immediately obtain sharp bounds in the class
I arg F' (C) I ~ -log ( I -
I.
=
I 'I
f' /
\ log F' (C) I ~-Iog
< 1.
(19)
We note that we can show
e.
F (0 € I. in the domain
(1 - I elll)'
(20)
Furthermore, this inequality, from which one can get the inequality
dif 1 -
:1".
1/
rfr~ IF'(C) I ~
I~I_'
constituting the distortion theorem for the class
(21)
Ie 12
I
olog 7 2kl -a-t- = - ( l - k f ) I '
from which we get
(18)
'> 1 we have the sharp inequality
(16)
f,
S and all z in \ z I
In the same way we can show that for the functions
We note that, by modifying slightly the preceding derivations, we can some times obtain stronger results. For example, if we divide equation (2) by ferentiate with respect to z, and then divide by f, we obtain
(17)
by a procedure like the one used above that inequality (17) is sharp for arbitrary
F '(,):
n-r)
----=:
~) + log (1 -I zit) \ ~ log ~ + \: i
which also holds for all fez) €
l/f(1/ ,) +
for arg
\ log I
,~~
which gives sharp bounds for arg [z2f ' (z)/f(z)2] at an arbitrary point z in
(e-IO log zl'I (z)(Z)) ,,;:::: log II + Iz I -I z I'
apply (17) to the function (11) with' = -z, we obtain another inequality: 1)
:l
9
2d II I - I I + Iz I ~ 1-1/1 1 - og 1-1 z I
I
~rJ
If
~-log(1-lzl),
-..::
e is arbitnlry, this is equivalent to the inequality zl' (z) I I + 1z I log I(z) ~log l-Iz\'
We note that, reasoning as above, we can prove the inequality zl/' (z)
from 0 to 00,
S we have in \ z I < 1
Consequently, for all fez) €
z I < 1. In the same way we can show that
t
I ZI
the lower bound given by inequalities (13) is the greatest lower bound.
arg I(z)'
2d III . (e-10 d log I') 7 .:;:;:; - I=T7T1
ffi (e- iO log zl' (Z)) -e \
I for I zol~'Y2' ZO
11:
which we assume inde
we obtain
For the function (1) constructed from this k(t), we have, as we can see from
the proof of inequalities (13),
argj' (zo) =
e,
If we integrate both sides of this last inequality with respect to
arg [f(zo, t)/zo] as a function of t. Knowing arg kf, we find arg k(t) and thus k (t) itself.
4arc sin Zo
117
ROTATION THEOREMS
I.,
will be derived later by a
different procedure.
olog 71'1
I-o-t-
"
21 II
= II-kill'
Eliminating dt here and in equation (4), we arrive at the equation
d log f'
I
I
I= -
(We give the following definition: suppose that a functional II: is defined
~. on a certain class of functions g(z). The set D of values w that the functional w = II: assumes on the given class of functions is called the range of values of the functional II: in that class of functions.
2d I I
I .
I-I/I~
1) Inequalities (18) and (19) were obtained by Grunsky [1932] by a different procedure.
_ u __ ._._"
•.
_ _ u •• _ . _ _
~ _ . _ .
-"
~ _ - ~ - - - - . - - - - - - ~ - "
118
-----._,.-:--
~
..-- _
.
_ _ .. _ .
.."',
~~.
' _ ._ _ '
._"'
' _ . _•• _
.•
...,,,,,. .-
~!
-
._.. .'... __... '~
... "..".. ..... _- . ..
.-'- _"_.,,
.,.,~~._
,,_~
~
"' v_'-"
. . "'-
_,._~_':.~
As will be shown in §3, p. 133, inequality (20) determines the range of
2.
_~,_
. __
.~.".~._
_~_
,_ _._
...
_~~
' ' • .._."' __.,_
.. __._",._.,__.
__.n
~_~
__..
._, .............
""""'"'--~....,~."....".........""'~.-.~ _'.~_",_·~_.~_"
_,.~_,."
~.~~:~~
~_
. ,.'
__.
.
_.
.n·._._..'_'._.._._·.. .. .
."",."",..,....,~""""'''~~-.,..
~'''._._.~_·,.~_
...
!!'!!.. _ ~ . _
,'~".,..,..
~'''_''
_·_'
....
"".-''''''"'
where we set
By the same reasoning, we can
C, =
show that inequalities (I8) and (19) determine the ranges of values of
I Zv'
F (C) = f
I
(-H
j
F (C) E ~.
log [z['(z)/[(z)] and log [[(z)lz] respectively in the class S. For more on this We introduce the expansions
point, see also Lebedev [1955aL)
co
§2. Sharpening of the distortion theorems
k.I=1
k (t)/(z, t) ~ k l-k(t)/(z,t)= ~bk(t)Z,
present sharpened forms of certain inequalities constituting distortion theorems for the classes Sand
2.
<1.
When we substItute these expansions into (2) and identify the resulting equations
co
Consider an arbitrary function fez) € C' defined in accor,dance with formula
(1) of §l, where fez, t) satisfies in the interval 0
ak. I
the differential equa
00
tion (2) and initial condition (3) of §1. For the function fez, t), we have a num
Iz I < 1
ber of interesting relationships. Specifically, let us take in 1,"', n and let us write for brevity [v
2 ~ bk (t) bl (t) dt
0 for k =1= I, J... for k-I
{
~ bk (t)bl(t) dt= o
arbitrary
(4)
(k, 1= 1, 2, ...),
o
co
[(zv' t). By using the
=
=-
2k
(k, 1= I, 2, ...).
(5)
Formulas (5) express the important property of orthogonality of the functions
alv
__
at -
+k
f v
I (t)/v 1 - k (t) lv'
b k (t) for k
"1=1, ... , n,
a [e-i/ Iv - lv, ] - 2 -kl-v . -kl v'
at 10 gl.fv'zv-Zv' I-kfv I-kl., , a _. klv (kfv' \ at log (I - fJv')
(where k=k(t». If zv=zv" be understood to mean
= 2I-
=
1, 2, ....
We shall now prove twO theorems dealing with the applications of formulas
we obtain by direct calculation the relations
kfv'
I-
(2) to inequalities.
.
( 1)
kl.,)
,Theorem 1. Let F denote a [unction belonging to 2. Let a vn v" [or v, v = 1,2,"', n, where n> x , _ 1, denote real numbers such that 2 v v I =1 a v v IX v'v is a pos itive quadratic form. Then, [or arbitrary C::v ' v = 1, 2, ... , 'n, the' inequalities
the expression ([v-[v,)/(zv-zv') in (I) should
II II _ ---ln
f' (zv)'
If we integrate equations (1) with respect to t from 0 to "" and keep (1) and,
,
I
V
'I \ _
\<1" .'
cve v,
~
II n
v,
v I F (C )
'11'=1
(3) of § 1 in mind, we obtain the following integral formulas:
log
F(C.)-F(C y
"II
_
Cv'
v')
--2S~. klv' dt I - kl I - kl ' 0"11
log ( 1 -
eve., = -
Goluzin [1948b, 1951a].
2
SOO
u
I _klkl
v
'
l v' ) (k I _ kf, dt,
v
v
v
F (C ,) l<1 . v'
Cv'
(6)
n\ n
'II,
'1/'
(2) I ) ---
c"
= ~
00
1)
I
with respect to C::V and C::v ' , we arrive at the formulas
role in questions concerning bounds.
=
Iz
k=!
) Let us first show a number of relationships that hold
in the parametric representation of univalent functions and that play an important
points zvfor v equations
(3)
co
As another application of the method of parametric representation, we now 1
~ ah, I C-!lt.'-I, lel>l, lC'I>I, ~
log F(C)-F(C')= y y,
'11'=1
I
["V, v'
1- cve"
1'I
> 1, where we should understand the [actor IF' (C::) I in the second product when C::V = C::v ' .
hold throughout the domain
Proof. Let us first prove inequalities (6) for the functions F(C::) where [(z) €
S I . In this case, if we set
=
....
......
.,."..,.-."."..,.
'-'._"~~~_~'_"~.~_<~_~_~_
119
§2. SHARPENING OF DISTORTION THEOREMS
IV. EXTREMAL QUESTIONS
values of the quantity log F ' UJ in the class
!Z'o_
~
_, _
1/[(II (),
120
IV. EXTREMAL QUESTIONS
I k
f
for F(C) E: ~.l)
kfv =Xv+iY.., '1=1, , .. , n,
Proof. Just as above, it will be sufficient to prove (7) for the functions F(C) = 1/[0/(), where f(z) E: 5 I . But, in the present case, we have from the
and take out the real parts in (2), we obtain
log I
F (C,) -
p (C v')
t, _ Cv '
I=
S 2 (XvX"
first of formulas (2)
00
-
•
-
n
YvYv') dt,
0 co
---!:-I = C.C
log j 1 -
2
-
v•
'Il,
S(XvX" + YvY.,)dt.
1.1" log
!
log f 10
I g
I
F (C,,) = -
'v -C.
,. _" .. ~,
.. ~
II - ---J-I_
o
C.C v'
s 0
(tv> -
P (C,,)
Cv- Cv'
j = log
[!
XvX v' dt,
1.1" log
F
0
'-1
+ const, where [( z) E: 5, and conse
I.
This completes the proof of the
=2
fJ
Iv
o ."
kj~
•
_ I dt
'1 -
1.J.
kf
(k
Jv
')
l'I"I--kf,' l-kf"
dt.
\ '
~'=l
=
1, "', n,
Let F denote a function belonging to ~. Then, nn'
~. ~
'I
.-::I:t:II,lv' og I
F (C v) - F (C:')
C.--C:, I~
1) The expression log F (C)C -=-~, (C') Wilh
'
~
and
~
I
in the domain
I{; I > I
always
refers to that branch (single-valued in this domain) of the mulrivalued funclion
log
n
~-
v=1
Theorem 3. Let nand n' denote two positive integers. Let Y IJ , v
n
~
o
and Y~" v'= 1,' " , n " denote n + n' complex numbers. Let (IJ' v = 1,"', n, and (:" v I = 1,' .. , n " denote n + n' complex numbers in the domain I(I > 1.
~ IvI.,logF(C.)-F(C,)
C -c ,
!
In an analogous manner, we can prove the more general
Theorem 2. Let n denote a positive integer. For v = 1, "', n, let Y)I denote arbitrary complex numbers and let (IJ denote complex numbers in the domain 1(' > 1. Then
",'=1
n
co
~2~
determines the range of values for the summation on the left •
v'=]
quently they remain valid for the entire class theorem.
F (C,,)
As will be shown in §3, inequality (7) is sharp for arbitrary YIJ and (IJ' and
proved for the functions F (C) E: ~ represented in the form F{(,} = 1/{(110, where !(z) E: 5', obviously remain valid for arbitrary functions FCC) E: I that can be
II{Ol ()
.=1
completes the proof of the theorem.
We then immediately obtain inequalities (6). But these inequalities, which were
represented in the form F (()=
J
From the second of formulas (2) applied to the last integral, this yields (7) and
n
~ a v, v,XvX" ~ 0, ~ a v, v' YvYv' ~ 0.
.., '11'=1
(C) -
v
We multiply these formulas by the numbers alJ IJ' and sum each of them with with respect to v, v' = 1, ... , n. Then, rememberi'ng that for numbers a)l)l' that , satisfy the conditions of the theorem, we have n
'\1 kf)9 ."-IVI-kf, dt
\' ( 0
Cv -C·"
',v'=1
11 - ~ I+ 4SY V"~ dt. C,C.,
kf.,
',v'=1
0::>
P
kf
1.IV'I_kj, l_kf.,dt=-2
and, consequently,
co
4
n o o n
r" =-2 J ko 00
o
log
F(C;)-F(C,,)
"I'~~l
Then, by adding and subtracting, we arrive at the following formulas
F (C,) -
121
§2. SHARPENING OF DISTORTION THEOREMS
! l;r.,IOg(l----l-) C.C
v,.' = 1
v'
(7)
F(C)-F(C')
C-C'
,
that approaches zero when at least one of the POintS {;, ~
,
approaches
For {;,. ~' , this quantity becomes equal to log F' (0 and then it can be understood as the value of that branch of the multivalued funcrion thaI vanishes at {; = 00.
00.
122
~ (If
y-
with equality holding for the same function F«(). n -
TI'
1
I
._,
from (9) for n
1
V 'v,v' log 11 --_-) Y"r.,IOg(l---;:=;-). ~ \ C'lI Cv' ",,,'~1 .... "vCv' v,'Y'=l
we write inequality (8) for F 2 «() €
123
§2. SHARPENING OF DISTo.RTION THEORI;idS
IV. EXTREMAL QUESTIONS
(8)
=
n'
=
I and
1(11 = I({ I =
(Inequality (ll) also follows
p > 1.)
The applications of formulas (4) and (5) are effectively used. Theorem 4. Suppose that fez)
~2' replace nand n' with 2n and 2n' ,
P (Q = _1_
and set YlI+v=-Y v and (lI+V=-(V for v=I,''',n and Y~'+v =Y~ and (~'+vo= )', , . -':.v for JI = 1,"', n , we obtain the
t(i),
ICI> land'Jog
.
=
~';' =ICllZll €
S. If we set
F (C) - F (t')
C'
c-
00
Corollary. Let nand n' denote positive integers. Let Y v for v = 1,"', n and y~, for v' = 1"", n' denote complex numbers. Let (v' for v = 1,"', n
and (~, for v'= 1,"', n', denote complex numbers in the domain
1(1) 1.
- k,l=l 2: ak,IC-kC'-I, ICI>I, le'l>l
Let
where the ak, I are rational functions of the coefficients c k, then, for arbitrary
F 2 denote a function belonging to ~2' Then,
v "-V'v"'.1og [F (C v) "v= 1.'=1 F (C v) + F
values of the complex variables Xk (where k
!
+ C:'1
" v (C.') C. - C'
i
I
CvCv' + 1 " ,-, CvCv' + I ' v ," log C - _ ( "- ,v,v' log ,-, . " v'=1 vC., v, v'=1 C,-C v' V
~
~
(9)
z/(l ±z)2. Proof. We can confine ourselves to functions fez) € tions (5). Therefore,
We mention some special cases of Theorem 3. In the first place, if n
F (C') 1,,;:::: -log
F (C) -
C-C'
~
with equality holding for the function F «() = ( + e i
n' = 1
Second, if n = n = 2, Yl we obtain the inequality
(1 _l-) pi , ag €~,
=
,
Y2 '" 1, Y2 '"
k.I=1
>:-1,
n
where a is a real
':.2 =-':.\1
and
1
+
k
+ pi"1
001
't}k (t) X k dt = 2 ~ k!t=lbk (t) b l (t)Xkx1dt.
0
k=1
l
2
n
1
00 n
=
1, 2," "
=
(13)
corresponding to them are real. Therefore, equality holds in the =
1,2,·..
•.. , n, are real. This completes the proof of the theorem.
which, if F«() is an odd function, yields the sharp inequality of Bazilevi'C [1951] and Milin (see Lebedev and Milin [1951]):
1"(~~ Ill·
~:,';
Schiffer [1948] proved the sharpness of inequalities (12) for arbitrary complex values of xk'
'",(
As a generalization of inequality (12), let us prove, under the same condi
1
I~ log 1+ 71; I - pi
=
± 1 for 0 < t < "'", so that the functions bk (t) for
conditional inequality (13) for these functions when the values of x k' k
p2
(F(C)-F(C')) (C+C') C')
k, 1=1
k,~lak, lXkX 11 ~ 2 ~
z/(1 ± z)2, we have k (t)
1-
I log (F (C) + F (t')) (t -
y
0
On the basis of (5), this yields inequalities (12). In the case of functions fez)
':.2 =-':.1,
(F (C) - F (C'» (F (~P-{-C'») (C C')21 log (F (C) - F (- C'» (F (- C) - F (C')) (C - C')2 ~ 2 log - - 1
I
oon
~ bk (t) bi (t) Xkxl dt = -~~ (2: bk (t) x k dt
and, consequently,
(10)
, ) ' ) ' ) ' , ) ' ,
Yl
oon
TI
2: ak, IXkX, =.- 2 ~
\
constant (Goluzin [1947 ]).
.,
=
then (8) gives the sharp inequality (the subscripts are
II og
5', In this case, the
ak,l have an integral representation (4) with the functions bk(t) subject to condi
started with the area principle.)
I = p > 1,
x:
I
Theorem 3 and its corollary were first obtained by Lebedev [1951], who
1(11
1, 2,··· , n with n ~ 1),
ak,lxkxI ~! I II . (12) k,I=1 k=1 ,r- - - - - . : ."""". . . - - " . , - - - - -,-, -- 11 11' In the case of real values of Xk, equality holds in (12) for the functions fez) = F i (C:.) Cv
I
and = I(~ omitted for ()
=
n
TI
(11)
tions,
m
I
n."
~11~lak'IXkXI'=:;; 1, n
2:
1, and
2,"', m and l
=
1, 2,"', n.)
where m
2:
Xk
{m
t1-
l~ II
k-
.
-/-, 1X '1 I
2
Let us prove two mor~ theorems for functions in the class
2
n
2: 2, on the circle
II I
1'1 = p, C.
Theorem 6. Let n denote an integer =
such that
1,
C.
--\-)2(n-l)
=
'I
(14)
where A is positive and depends only on nand £v'
except coincidental ones (v
=
v' ).
=
1,' . " n
Proof. In (6), let us take for av v' tWO systems of numbers a; v'
,
"
and
a~ v'
F(Cv)---F\~"'1
n
v'
-a., v'
c. - c.'
"i,v'=l
II ~. n
1 _ _1_
a.~. 'Y,+ct "y ,
(15)
p, we have
€,,~I===pn(n-I)
III ,€
-
c•. 1 === A1plt(It-I),
- V-:f..'
I . .*.' III
1 -- -1- === A 2 e.e-.,p2·
- - :::::e: mm
1 e.e~,p2
~l
I
£ b
••• ,
£v.
Keeping all this in mind, we obtain from (14)
2n(n-l) Mn(n-I)
.+xn_1Xn===(XI
+.. ,+xn)~'
~
~
A A
n(n-I)
1 2P
(
1 - 1
2(n-l)
) 2' p
(XIX~ +, ..+ xn__1xn) === 2 (n ~ I) ~ (x, -
We now have the following sharpened bounds on the modulus of a function belonging to the class S: X,,)2
v-:f..'
Theorem 7. Let n denote an integer
are both positive, we can set ex~·,.'
ex.,"=== "
= p.
which yields inequality (17). This completes the proof of the theorem.
x~+ ... +X~+XIX~+" nil
=
Here, A 1 and A 2 are positive and they depend only on n,
In particular, since the forms
x~ +... + x~ -
1'1
y'
c.e.,
v, .'=1
1, 2,'" , n.
'Y-:;=''''
III 1 -
one by the other. We then obtain as one of the inequalities the following: I
1'I
.,=;t.v'
satisfying the conditions of Theorem 1 and let us divide the resulting inequalities
,
V =
(17)
nIF(c.~)--F(z,.ql~2n(nl)h1n(n I).
V-1'.' Furthermore, on the circle
III €,~ -
a.
"
Proof. We apply inequality (14) for 'v = (v', V = 1,"', n, and Denoting by M the maximum on the left-hand side of (17), we have
'I
range over all pairs of values v, v'
denote distinct numbers
(v
p, p> 1,
C
I
11
=
2: 2. Let
1,"', n. Let F denote a function belonging to 2. Then,
1 )2/n
II 11- ~II-,. C,
p . ..
V::F V'
(r \
1'1
=
I (I - y v=~~:nF(e.c)I~Ap
1,"" n, where
In the derivatives, v and v
n
1, v
=
2.
2
I~ (1 -
kvi
on the circle
p > 1, we have
F(C.)= F(C.,)
_
y-:f. y'
and arbitrary 'v' v
p, v = 1, .• " n, this yields inequality (14). This completes the
proof of the theorem .
and xt are arbitrary complex numbers (for k
Theorem 5. For functions F(t;;) €
I'v 1=
with
2 1 Xk 1
125
§2. SHARPENING OF DISTORTION THEOREMS
IV. EXTREMAL QUESTIONS
124
{ -
distinct numbers such that
=== I, V, v' === 1, ..., n,
I Z 1=
I
n-I
for
k v I = 1.
2. Let £v'
V =
1,"', n, denote
Then, for functions fez) €
v· === 1, ..., n.
v=l, .. n
where A is finite and depends only on nand
'1:1=- v',
S on the circle
< 1, z ) I '=:;;~'?I'" min \f(€v
for v === v·, V,
r, where r
2:
(18)
(v.
Proof. Inequality (18) is obtained from (17) if we apply the latter to the func
We then obtain the inequality n
2
2
FCC.)=FCC.,) I~ II (1- _1)2-,. II \1- ~\I-,. C, c.' _ IC,I C,C., II .*v' \ v=1 v*v' 2
(16)
tion F(D = Ilf(z), where' = liz, and to the points £v = llz v and then replace (v with '£ v • We note that the exponent 21n in the right-hand member of inequality (18) cannot be replaced with a smaller number without additional restrictions on the
IV. EXTREMAL QUESTIONS
126
§2. SHARPENING OF DISTORTION THEOREMS
function fez). To see this, consider the function
the form
z fez) =
II
_
II (I-2hZ
lD (C) = log F
)2/n
(~~
=~
(Co)
-+ log (- Co) + log ( 1 <Xl
=
h=1
In
I z \ < 1,
n
. 1
a,
+ ~ ~m( n ~ h=1
_
tk~
n k=1 ~
I•
I - 2 h z
n k=1 ~ II -
2hZ
EkZ
~
Iz I =
Consequently, as z moves around an arbitrary circle
where 0
< r < 1,
in the positive direction, the function arg fez) increases mono
r,
<Xl
CQ
T
1..
This means that fez) €
s.
p211
Now,
p )2n _ ( pi ) n;r --log 1 - Tfor°
11=1
If we now let p approach 1, this yields <Xl
If(s,r) 1= n
r
II II - £kE,r 12/n
~
!
Ar
11=1
(I ~ r)2(n
h=1
Now we have, for 1
where A is a positive number depending only on n and (v' v = 1," ., n. This proves what was said regarding the sharpness of inequality (18). =
2, we obtain from inequality (14) the following inequal
I:
I
P (~1) -
F (~2)
C1 - C 2
I::>: 1- ~ 0--
I.
This theorem, which>
was proved by the author, 1) has found numerous applications. Here, we mention the direct elementary derivation (given by Milin)2) of inequality (19) and also of inequality (10).
P (Co)
C- Co
I.:;:: ~ Ian (Co) I = ~ v;l\ an (Co) I· '1"""::1 "'-,n ~ """ I ~ In ~ 11=1
y n
11=1
ro
<Xl
~ 1/ ~ n I an (Co) 1 n=1
which, as we let
I(0 I
approach
=
..
•
!
11=1
n I ~ 1211
~ y'-IO-g-(1-1 L/2) .log (1 - m") 1(\
Ilog P (C~ CCo) I~ -
=
I
p, yields
log ( 1 -
p\)' ICI = ICol = P> 1.
(10')
we now replace the absolute value with the real part taken with a minus sign,
Specifically, the function (() regular and univalent in 1 <
->
J
ro
<Xl
P (q
(1- I C~ Ill)'
(19)
p2 ,
by functions belonging co the class
IIog
n I an (Co) I~ ~ -log
2
where (1 and (2 are arbitrary points on the circle 1(\ = p, where p> 1. Ine quality (19) is called the theorem on the distortion of chords under mapping of
I(I > 1
It
n=1
~ I ( :s;; ~ n
11=1
we have
ity for F(() €
l..
11=1
(Xl
" n I an (~o) [2
Iz I < r
a closed Jordan curve. But then the function w = fez) maps the disk
We note that, for n
ro
"., l (._ ~ )n
we have the inequality
conically by an amount 217; that is the point w = fez) describes (one time around)
< r < 1,
0
11=1
= rei¢;o
< r < 1.
+ 10g (_ C) _
~~W-oL-~";"-n2. (_p )2n) I Co I •
S -- ' I t (_ ~ n I anpIn
2
1
where z
univalently for any r in the interval 0
Zo·)
Therefore, the area S of the bounded ddmain bounded by the image C of the I P circle I z! :: p, where 1 < P < I(0 I, is ~qual to
)
=2 ~m(I+~Z)=2~ 1_1~12 >0
the domain
"" cn n=1
we have
iJargj(z)
for 0
~ an (Co)
127
=
log [F(() - F((o)), where 1(0
I> 1,
is
\ (I < I (0 \, and its expansion in that annulus has
1) Goluzin [I946e}.
2) Lebedev and Milin [1951], inequality (11.1).
we obtain inequality (19). We note that inequality (11) can be derived in the same
••
---:--:-:__ ~=--:=-_~==:·~-""",-~_~---=-_===·::;;c.=-~---==:_-_::::::--=_=--==_-_---==-~~--,".;-"",-,--"",_~",,~==,,_=_-=~~,,~=,~=.:;·=:;=c.=-::-
-~· ,-~"",=-:::-_c_-,:,::;:::-_-=-~--'--'-",::::;::",,,,,,,-~---,;;;:::~~,~~,>;;~·_~:
__:c.
§ 3.
IF has the same value for functions that differ from each IF in the subclass 1° of functions F«()of the class I such that F«() ~ 0 in 1(' > 1. That this maximum is attained f~'lows from the. normality of the class 1°. Suppose Since the quantity
Extrema and majorizations of the type of the dislortion theorems
other by a constant amount, this maximum coincides with the maximum
The method of variations also enables us to establish a numbet of results of the distortion-theorem type and to calculate completely the extremal functions. 1) Furthermore, in the investigation of extrema of certain quantities, we ate able to
tha t w = F «() E: I
ascertain their ranges of values.
Theorem 1. Let n denote a positive integer. Let Yv' v
° is one of the ~'xt.rema.lcf~ctions. Let us suppose that the
image B of the domain
We shall stop here t6 go into only one theorem in detail.
I(I
> 1 under the mapping w
=
1, " ' , n, denote
(25) of
I
have the sharp inequality
F* =
n
~ F(C.)--F(C.,») ~ ( ffi\( ~ 1,1.,log C.-C" ~-.:. 1,1.,log 1., • .,'=1
." .,'=:;;-1
Equality holds in (1) only for functions de fined in the domain
I(I
1 \
ce)'
§3
n
.,
w~
wo)
(F. woHF., C. - C.'
+ 0 (A
B
)
)
~
(F.-Wn~!F.,-Wo)
T.T.,
)+O(A
2 ).
Since Wo can always be chosen in such a way that the last sum is nonzero, we
(2)
obtain IF. > IF for small A> 0 and suitable arg A l ' But this contradicts the extremality of the function F«(). Consequently, the domain B cannot have Let us now turn to the varied function (26) of §3 of Chapter III with m
(2' )
F (t.) - F (C.')
C.-C.'
I~-
n
"\"1 - I ~ 1.1.,og " .'=1
(1 ---=-, 1)
IF. =IF- Am ( A1 " (3)
A -
1 "II.
~
~
,,'=1
(
FO)B
T.T; CoF~
-
1
F~
Fo)(F.,-f'o)
A ( 1
=
For this function we have
C:F~
n
~
FO)B
CoF~
~
T.T.'
co-c.
C~,F-., _
Co-C.'
F. - F.,
"11',,'=1
+~(C;JB
n
'11,.,'=1
~ T•T., (F. 7:I
n
Proof. Let us find the maximum of the quantity
IF=ffi( ~ 1.1., log F(CC)=[:C.,»)
1(I > 1.
•. and with (1 replaced with (0, where
C.C._ which defines the range of values for the summation on the left. Thus, for arbi trary given Yv and (v' v = 1, 2," " n, this range of values is a disk.
i
C.F~
C.'F~')
T.T.' ~--=-~~."-l
+O(A
i
),
'II, ,,'=1
(4)
(Here, all the ratios representing indeterminate forms must be replaced with the
""
for given Yv and (v' v = 1, 2,"', n, and for F«() ranging over the entire class
1
[Here and in what follows we shall write for brevity, Fv instead of F«().]
> 1, we have the inequality
n
J
(l-AA
. exterior points.
.-1
~
(F.-F.,)
'II,v'=1
n
1.1.,og I" .:. .'=1
T.T.,log
=IF-Affi(A 1
> 1 onto the entire w-plane with a cut along
.ffi (~l.log (w - F (~,» )=const.
I(I
n ~
> 1 by the
the curve defined by the equation
Furthermore, for
by using formula
n
.=1
I(I
(
" .'=1
n
These functions map the domain
ffi
v
F(C) - F(C ) ~( 1 ) ~ 1.Iog C-C.· = - .i.;,l.log .1-CC~ .=1
1°
of Chapter III with m = 1 and with wI replaced with Wo.
(1)
equation ~
F «() has an exterior
=
. point wo. Then, let us form the varied function F. «() €
n complex numbers, not all zero. Let (1' "', (n denote distinct complex num For this function, we have
bers in the domain I(I > 1. Let F denote a function belonging to I. Then, we n
129
§3. EXTREMA AND MAJORIZATIONS
IV. EXTREMAL QUESTIONS
128
ratios of deriv'atives.) If we replace the last term in the large parenthetical ex
I.
pression following ~ with its complex conjugate, we conclude on the basis of the arbitrariness of arg
A 1 and the extremality of F «(), that the quantity in the
parentheses following lR after we divide by the common factor A 1 must be equal
1) Goluzin [1947 ].
f;'
/
A l' we would have IF* > IF' which would contradict the extremality of the function F «). This leads to the
to zero because otherwise, for suitable choice of
n
ffi ( •.
condition
n
~IT.T" + "~=lT.T"
~
C+C. -C .'po ··C-~., C+~") •P' •C-C. F. -F..
~ O.
n
n
'~~F~s
131
§3. EXTREMA ANp MAJORIZATIONS
IV. EXTREMAL QUESTIONS
130
!
'II, ~/=1
1.1.' (P.-Fo) (Fv,-Fo)
+
!
qF~
n
If we denote the left-hand memb er of this inequality by 8
T;h'
", '11'=1
~ i;'(v' C6 -C.
i.o
on the circle
C~,P'., _
Co-C.'
!
T.T.'
v, v'=1
~C,,-I P -F
C;;Cv-1
. .'
Since ~ in this equation is an arbitrary point in the domain
'0
1'1
> 1,
c,~:=~(,.)).
I"
=
1, we have
A(q=B(~)~O.
It follows from this result that, if the rational function A (() has zeros on the circle
1'1 = 1,
all these zeros are of even multiplicity because the increase
in arg A (0 as , moves around a small circle with center at any of these zeros will, by virtue of the principle of symmetry, be equal to twice the increase in arg A (,) as 'moves around an arc lying in
with h > O. From this we obtain, for small h,
I"
> 1 and this last increase is
equal to an integral multiple of 2". Therefore, the function
'a
real on
0/'(00)= 1 +h.
I"
=
VA'W
1. Turning now to the differential equation (5) (with
is regular and
'0
replaced
with '), we note that twice the sum in its left-hand member can be represented in
By means of the function 0/('), let us define the following function in the class
F J (~) = F,I"(~\ (0/ (C)) = F (~) _ h (F (~).....I-~F' (~) 1J--CelG +ce~6) I
~o for different
to vanishing of the right-hand member of ( 6). Thus, on the circle
+ h) (~eia + Ce~"8) + 2h
~o.
(6)
real values of (). The quantity IFe attains its maximum on the interval 0 ~ () ~ 2" at () = O. Therefore, (dIF/d())!ezo=O, which, in expanded form is equivalent
if we
1'\ =
t=~ (q=~ - h~ ~ +~::a + o (h 2).
do this, let us again consider the function Fo = e-ieF(eieO €
1. To show this, let us consider a function t = 0/(1,) where 0/(00) = 00 and 0/' (00) > 0, that maps the domain 1'1 :> 1 univalently onto the domain I t 1> 1 with a small radial cut issuing from the point t = e- ie. This function is obviously obtained from the equation
teta + t~ia = (1
tI
M "( 1-
,
Let us show that the right-hand member of this equation is equal to zero. To
(5)
replace with " we obtain the differential equation that the extremal function F (,) must satisfy. We denote the right-hand member of this equation by A (,). Let us show that A «) ~ 0 everywhere on
1.
A(t)-B(t)=l~ (,
C.F~ C.,F~, -----
n
=
we have 8 (,) ~ 0
On the other hand, on this circle we have
F. - Fv'
", ,,'=1
the form
n
(! F.~F .=1
+ 0 (h2).
r
Coosequently, this equation can be written in the form
For this function,
IF, =IF- hffi(
1'I
(0,
i
n
ry
T.T..
! v= 1
", .,'=1 n
~ T.T.'
+ 'II •
.,'=1
C P' C+C. -C P' C+C.,) v 'C-';._;~, v'C-C.,
+ O(h
F 1.
k'
=
VA (~) .
v
If we solve it we get the following dependence of F(') on ,: 2
) ,
n
!
.
where C= e ie . Then, by virtue of the extremality of F( 0, we obtain, for all , on the circle \"
T.log(F.-F)=-
v= 1
=
1
which holds in
I" > 1.
~ y' A(q ~C =C(~),
(7)
§3. EXTREMA AND MAJORIZATIONS
IV. EXTREMAL QUESTIONS
132
-',
If we take an arbitrary boundary Wo of the domain B and denote by (0 the
corresponding point on
1(I
= 1 under the mapping
and woo In this sense, equation (7) also holds on
:a on the circle
ffi (C(C)) =
\(1
1(I
=
1. Also, since
~ (aCa~C) ) = ffi (iCC (~)) = ffi (i VA (C)) =
= 1, where
WF
(= F -I(w), we conclude by
passing to the limit that equation (7) holds (with Wo instead of F) holds between
'0
C.-C
.=1
=CC,.)
for given Yv and ~' v = 1, 2,"', n, and for F(() ranging over the entire class completely covers the circle
0,
n
".
1
_
!
!W!=-
.J.
(9)
i,i.,}Og(I- • CC ,,'=1
Let us now consider the quantity W; = F.p
(1 -cr) 1)1 II
= C (~) will have a constant real part on
1(1
~ (i' log (~, -
C) -l.Jog ( 1 -
.=1
(8)
C~
I(I > 1
I(I > 1 and hence it cannot attain its maxi > 1. From this we conclude that the left-hand side of (8) is a constant everywhere in 1(\ > 1. Then, by considering its behavior at (= 00,
IWI=-
and
WF(C),p
I
if F(()€
1
~ 1,1.,jOg(1-~)
For n
e is an arbitrary real number, we II
!
l.i"IOgF(CC~=~~C.,»)~-
'1'=1
which, by the virtue of the arbitrariness of
e,
lI.
1.1,.tog(1-
cd ..).
'1'=1
implies inequality (3). But this
also shows that the set of values of the quantity
Wl~-
!
'. ,'=1
i,l,'IOg(I-~),
=
1, inequality (3) takes the form
\ log F' (q I ~ -log (1 -1,1 12 ),
(12)
and inequality (1) takes the form, with YI = e ia
ffi (e 2i
obtain the following sharp inequality,
!
This shows that the set of values of WF as
theorem.
C.~.'
supplementary conclusion regarding the equality sign. Furthermore, if we replace
'II,
I.
F (() ranges over the entire class I also covers the circle (11). As p ranges
(The sum on the right is, of course, real.) This proves inequality (1) with the
rn(e 2i6
WF(pC) ,
C,C., which, therefore, is the range of values for WF • This completes the proof of the
II
II
=
(11)
II
1, .. " n. We then obtain for the extremal function
1,"', n in (1) with Yveie, where
i,T,'log(I-~)
p
Let us now set ('" (v' in (2), multiply both sides by Yv' , and sum over
lI.'~= 1
!
from 1 to 00, the corresponding set of circles (11) covers the entire disk
we have proved equation (2') for the boundary of the domain B.
IF = -
(10)
pC.-pC.,
.'.'
" ~'= 1 p2C,C,' But from the construction of the quantity (10), we can easily see that
where F(p()/p €
we conclude that this constant is equal to zero.
I(I > 1
v• .,'=1
F(() ranges over the entire class I, the values W = WF • P cover the entire circle
1(\
Thus, for an extremal function, we have proved equation (2) in
.,." log F (pC,) - F (pC.,)
~
n
because it is a harmonic function in
mum or minimum in
~
with the same Yv and (v and with given p> 1. From what we have proved, as
~))
~ 1. But this means that the sum on the left
side of (8) has a real part that is constant everywhere in the domain
=
1.1,' logF (CC~
",.,,'=1
(= e ie , it follows that the right-hand member of
~( F,-F+~ i,log i.log
Yv for v
---1
II
II
=
n /,-/
I,
equation (7) and, hence, the right-hand side of the transformed equation
v'
133
!
-;
~a ~
; .
(13)
Here, equality holds in (13) with ('" (I only when Lcf. (2')] the function F(() maps the domain
I (I > 1
onto the w-plane with cut along an arc of the curve
defined by the equation
ffi (ei
= const.
(14)
:§. 'fP""!-'..
_"
::':::--7':3"~:::=:i;';-
::.:;:~:::;;.~~-~,.-
~.~=~,.::::==-==",::'="''''=------------_'_--
§3. EXTREMA A~D MAJORIZATIONS
IV. EXTREMAL QUESTIONS
134
But equation (14) defines a logarithmic spiral with asymptotic point at w
=
Fe(l)
./
I
F(C1Tc.::.::./P(C.) ;::d
I el-c.
such that rays issuing from that point intersect the spiral at an angle Tr/2 - a. In
135
1 -7 .
(17)
the special cases a= 0 and a= Tr/2, this spiral degenerates respectively into a Here from what was said above, equality holds only for the function
circle with center at the point Fe(1) and a ray issuing from Fe(l)'
ei~
F(C)=C- T +const,
For a= 0 and a= Tr/2, we obtain from inequality (13) the distortion theorem!) for the class
I: 1
~ I ~ I! ~ \ F' (t) I ~
~=
_1
(15)
I
1Tff2
=
F e() that maps the domain
I(I
> 1 onto the w-plane with cut
along an arc of the circle with center at the point w
Fe(l) and the greatest
=
lower bound is attained only by a function Fe() that maps the domain
1(I
>1
Let us apply inequality (1) in the case n = 2, first for Y1 = e = -
ffi (e
I
log
log P (~l:
P(C¥l=~.(C.»)~--}log [(I-
=[(c
2 )
I~
ia
and Y2'" e
ia
ie i ~ where a is a real number. Then let us add
the results. We obtain 2ia
-~
c:
m·)(1-cr)] .
~!)],
I IS) (1 - \
log[( I -
arg C1
+ arg C
2•
inequality (16) will not be sharp because,
I (I
equations of the type (2'), map the circle
=
1 onto an arc of a logarithmic
spiral with asymptotic point simultaneously at Fe(l) and at Fe(2) or ( for a= 0)
With regard to the complete answer of the question that we have touched on here, we present the following theorem:
We give yet another application of Theorem I. ie; a and Y2
(11= (2,
is impossible.
Here, F e(1) obviously can be chosen arbitrarily.
=
and
onto an arc of the circle with center simultaneously at Fe(l) and Fe(2), which
onto the plane with cut along a segment of the ray issuing from the point F e( 1)'
and then for Y1
e'a.1= i
if it were, the extremal function Fe() would, as follows from the corresponding
From what has been said above, the least upper bound is attained for (= (I only by a function w
Furthermore, for
Equality can hold in (16) only if the functions Fe() €
1
Theorem 2. Let a denote a real number and let (I and (2 denote distinct complex numbers outside the disk
IF = ffi (16)
1
2
maps the domain
1(1 > I
(e2ia log F (~I! ~c~ (e.) )
(18)
onto the entire w-plane with cut along the curve defined
by the equation
w
I for which equations of
1(\ > 1.
1(1 > 1. Then, among the functions F e()€ I,
the function maximizing the quantity
=
p1-!:-p.
+ PI-:;P.
(e(c+it) e-
ia
+
e-(c
+ it) e- ia ),
(19)
If we add and subtract these equations, multiplied respectively by 1 and i, we obtain the following
where t is a real parameter de fining points of the curve. For a = 0, this will be
two equations:
along the hype rbola with foci at F 1 and F 2'
the type (2) hold simultaneously in the domain
P(C) -F(C.) = (l-~)
C-C.
CC.
or
(
F(C)=F(t.)+(C-C.) 1-
2ia e1)CC
•
In particular, from (16) with a= Tr/2 and 1(1 \ obtain the sharp inequality
-
e -2ia
=
1(2 \
,
'1= 1, 2,
a cut along the ellipse with foci at F 1 and F 2' For a= Tr/2, it will be a cut
Proof. Following the same reasoning as in the proof of Theorem 1, let us show that extremal functions exist, that each of them maps the domain \ (I onto a domain B without exterior points, and that each of them in
'1=1,2. p, where p> 1, we again
I(I > 1 satisfies
a differential equation of the form
C2F2 =
e2ia
(F-P
1
)
(P-P2 )
Q(q,
where Q e() is a rational function that is real and nonnegative on
1(\ =
When we solve this equation, we get the following dependence of w 1) Loewner [1919J.
>1
on (:
1. =
Fe()
137
§3. EXTREMA AND MAJORIZATIONS
IV. EXTREMAL QUESTIONS
136
for given (= (1' where 1(I lS>lL-and for F «() ranging over all the functions F«() = (+ ad( + '" € l. Obviously, this maximum coincides with the maximum
ei"log(yw-Ft which also holds on the citcle
yW-F 2 )= ~ yQ(C) ~C,
-
1(1 = 1.
of the quantity
f F= ffi (e gi " (F (C t )
From this we conclude, just as above,
among all the functions
Therefore, among the extremal functions of this last problem there are functions
yw-F2)=c+2lt,
-
without zeros in
where c = const and t is real.
image
the boundary of the form (19). This completes the proof of the theorem.
log
1"
B of the domain
following differential equation for
Theorem 3. 2 ) For functions F «() = ( + all ( + ..• €
where
Q«()
C
in
I(I
h-h-)
1 ( Km)
,c
I(I
l, we have the sharp
(m) ( 1 )' Km
2_ 2 E
-=
~1 Y (i-XS)dx(i
kllx l ) ,
E(k)=
~ -V I
1-- k x S dx . I-xs
=
If we divide (21) by
ffi
1(- (II,
(e i"
Since the function
is regular in part on
I(I
I(I
c
)? -
I(I
= 1. If we now
=
(21)
const.
C1C)
we get
C
.. / .
=cV
(22)
(C - C1 ) (l -C 1C)
/" .. IC (1" (C) - 1" (C »' V (C-C )(I-C C) 1
1
> 1 and, obviously, continuous on
1(' ?
1 and since its real
= 1 is given by formula (22), this function itself in
1(' > 1
is deter
mined by the Schwarz formula (the integral is taken in the counterclockwise direction): I" ... (CCF(C)- F(C 1 ))
1) Goluz:in [19471. 2) Theorems 3 and 4 were proved by the author (see Goluz:in [19431'11 ).
0 on
1
e
Inequality (20) defines the range of values for the quantity F«()/( for arbitrary
IF= ffi (e 2/" F (C))
(C-C J
C(1" (C) - 1" (C 1 ) \ (C - C1)(1- c1Q)
o
Proof. Let us look at the problem of maximizing the quantity
Q«() ?
1,
IC-C t I2 =
... (20' )
given ( in the domain \(\ > 1. Consequently, this range is a disk.
Q (C),
function F«(). We need only consider the case (I? 1. Then, we have, on ' "
J
=
Resting on this circumstance, we can construct an explicit expression for the
(20)
1
1"1
ffi (e i " yF (q - F (C t )) = c,
I 9
Si"
= 1 the equation
> 1. Here we us e the notation K(k)=
F «() does not have exterior
=
I(I > 1:
is a rational function such that
,~\
E
w
solve this equation, we conclude that the extremal function F«() satisfies on
use of elementary functions.
1 -2 + I - JTr
F «() in
C2F'2 /
radius nor the center of that disk can be expressed in terms of (I and (2 by the
F(C)
F «() denote one of them.
1(' > 1 under the mapping
(C 1) - p (C s) C1 - Cs
has, for arbitrary (I and (2, a disk for its range of values. But neither the
inequality
Let
points. If we then apply the varied function (26) of the same section, we get the
We mention without proof!) that the quantity
= F
I" > 1.
By applying the varied function (25) of §3 of Chapter 111, let us show that the
When we solve this equation for w, we obtain a parametric representation of
W
F «() € l. (Here a o is the constant term in the La urent €l.
expansion of F«().) But I~ has equal values for nonconstant functions F«()
that all boundary points of the domain B satisfy the equation
ei"log(yw-Ft
ao))
-
e
V
_
(C - C1 )(1 - C1 C) -
-
c 21ti
\
~
IC'I~I
-. (
V
c'
-C'+C dC'
(C' - C1 ) (1- C1C') C' - C"
+ let·
(23)
----~-----
13B
'~--~~~-~~""""""--~~-, ~.~---~-~;;~~:-- ~~'l'-;;,;_i:.;;~":""·'·1111~"l111j~!~!d=~J!'~~..:o".",",!!!!~,J'&~'!!!'1!"'" -~_,_ ,-,-::±~"~~;!,,.,,~-'02?;,!,~
IV. EXTREMAL QUESTIONS
§3. EXTREMA I;NO MAJORIZATIONS
From this we obtain FC(). To determine the real constants c and c 1 and also
If we make the substitution-x.-=1Cl(' in these two integrals and use the
notation (20') for complete elliptic integrals, we obtain
F((I) - a o, we use the normalization of FC(). Specifically, if we understand the radical appearing in (23) as referring to that branch that approaches 1 as ( ' --> ""
and equate the constant terms and coefficients of II ( in the expansions
F(Ct) -
cxo=Ct
of the two sides of (23) about (= "", we have ei~
~
c
--i
- - 21t
dC'
y~' (~ I _~') (1_1''-1'1") + I Ct, I t' 1=1
eia. ( 1 2i~ Ct+c;+CXO-F(Cl»)=-~ ~ 'I=l 1t
IC
00
"/""1'"
"'t\.11
"',..,\.
1/ (I
C') (1 - CtC') that branch that is positive on the right-hand side of the cut (0, 1/(1). Since the integrands in (24) and (25) are regular in the cut planes, we can deform the paths
t}
+lct,
,~
eia.
I
1
2i~ ~ Ct + ~ +
CXo - F
\
1
F (Ct) - CXo = Cl
+y + t
. 2 (e-ua. -
J yc'(Ct-C')(r----:.c1C') 1) -.,.°1------ t
.~
dC'
~
yC' (C 1-C')(1-C IO
> I,we have the sharp inequality
K(_I)
(1- E(I~I)\)'
41CII '"=ICI2-11\,
I CI
K(_I)'
(29)
I CI
Tnequality (29) detennines the range o[values o[the function CF"(C)IF'(C) for arbitrary given ( in '(I > 1. Consequently, this range is a disk. Proof. When the function F C'::) belongs to the class 1\'0
~'dC'
ICj2-1
(27)
F tr)=
I
c;
ICj2-1
F(C)
where the radica Is are understood to mean their positive values. By equating the.
1'1
~ in
~F"(C)+4ICII-2_ 41CI3 E(rh) , ;: :
~:
real (and imaginary), parts of the two sides of (26), we determine c and c l ' We then obtain from (27) an expression for FC(I) - a o :
Theorem 4. For functions F (Q €
(26)
1
C1 ) 2c \ C'dC' (C t) =;- ~ JI C' (C _ C') (I _ ~IC')-' I
(28)
range of values of F C()I( is proved just as in Theorem 1. This completes the proof of the theorem.
1
d~'
(1- :~~~\)
for (> 1. By virtue of the arbitrariness of a in (28), we obtain inequality (20).
As a result, we obtain
c \
~ and arbi
That this inequality also holds for complex ( in the domain I(I > 1 follows from examination of the function e-ieF(e ie (. The remaining conclusion regarding the
1/(1 to 0 along the upper side of the cut and then from 0 to II (I along the lower side of the cut (0, 1/(1)'
~=~ ~ YC'(C t -C')(1-C IC')
'{"'(F~)+C-+-
.,J,
of integration and, in particular, we can constrict them to the path extending from
i
K(c;)
\
From this it follows that for all the functions F (() = (+ all( + ... € trary given a, we have
and
YC' (Ct -
~
(e-~ia. -1/1- E(c;)).
IF~rn (e"'((c: -CI)+
(25)
C'dC'
and if we take for
tj
+ 2C1
For the maximum that we are seeking, we have
(24)
Let us make cuts along the real axis of the (' ·plane from 0 to from (1 to
++ 1
~
139
(1-IC 1 II)F'(C1) +l-(l_lr F(I+C IC I.,t 2 C CI ) -
for every (I in
+
1'1 > 1
nc, )
12)
(F"(C I) F'(C t )
I,
so does the function
_2r)=r+"I+ I., C'" 1.,1
and conversely.
If we apply inequality (20) to F lC() at the point (= ~1' we obtain inequal
ity (29) for ('= (1' Conversely, from (29) we can obtain (20). To do this, note first that, if fez) € S, then, by applying (29) to the function F(O "" obtain in I z I < I the inequality
II[(II (),
we
140
IV. EXTREMAL QUESTIONS
zj" (z) _ 2 z/' (z) I'(z) I(z)
I-lzl!
I-lzP'K(lzi) ~ 4 ( E(I z I») ~ I -I Z 12 I - K (i z I) •
g(C) = (where
I z I < 1)
I
I' (z) (1-1 z I") = C
z
Finally, if F«()
=
Iz I < 1
in
g
CgZ
From this we get the conclusions on the range of values. This completes
the proof of the theorem.
the method. of variations led to complete
over all possible systems of points , 11, p' v = 1, ••. , n, on the circle
points (11, p' let us consider the extremal problem of Theorem 1 of proof of Theorem 1 (equation (5) with Yll
other extremal problems. However, final solution of the problem frequently de pends on the determination of a cer tain number of unknown constants.. We shall
= p,
11
11' =
1 for
11
~
II'
and YlI
11'
=0
for
= v') that the function Fp «() mustsati~fy in the domain 1(( > 1 ;he differen
Let us now apply the method of variations to the investigation of the extremum of the average diameter of the boundary of the image of the domain
§ 3. One of
tial equ.ation
now Stop to look at certain problems of this type. 1)
1(1
> 1 under
[CF'(C)]g!
1
(F(C)-F.,p)(F(C)-f~"p'
C" F'
.-=t-.'
I. 2)
~
over all systems of n points all' 11 = 1," " n, located on the boundary of the image of the domain 1(1 > 1 under the function w = F(O. (The product (1) is
'.p
•-=t-.'
integer ~ 2. Let P F denote the least upper bound of the product
I
-,p
=n(n-l)+2"", CF.,p-F.,.p)(C-C.,p)
Theorem 1. Let F «() denote a function belonging to I. Let n denote an
a.,
'(I
where p>l, and we denote by Pp the maximum of Pp,F overall F«()€I. By vinue of the normality of the class I, the maximum Pp obviously is
the extremal functions in that problem is obviously F p «()' It follows from the
solution of the extremal problems. This method can also be applied to various
n Ia. -*.,
(3)
up for the function Fp «(), attains its maximum Pp ' For the system of these
problems
functions of the class
I. we denote
attained. Let Fp «() denote one of the extremal functions. We denote by (II,P' for v = 1, .• " n, the points on the circle I(I =: p for which the quantity (3), set
§4. Application of the method of variations to other extremal
§ 3,
(2)
n IF(C•. p)-F(C." p)1 .*.,
(31)
(+ a l / ( + ..• E I and F«(),j c in 1(1) 1, then, byapply cl € S, we obtain inequality (20) in
In all the cases considered in
n(n-l)
\
the theorem, we define the following: for a given function F«() € by Pp,F the maximum of the product
ing (31) to the function fez) =: I/[F(I!z) -
I(I > 1.
.*.'
1(1 > 1.
IPlr\_n \;Pln_n
Proof. In addition to the quantities PF and P mentioned in the statement of
the inequality
+ + 1-lz\-2 KE (I(\zl) z I) l~2 (I-K(lz\) E (I z I) ) .
141
attained by some function w = F «() that maps the domain 1(/ > 1 onto the entire w-plane with analytic cuts and that satisfies the differential equation
C2 [F'(l:)}g.!
+... E S
at the point (=: - z, we obtain in
j(z)
(30)
then, by applying (30) to the function
. I (C+Z) I + zC -/(z)
dF VARIATIONS
taken over all pairs of disti1leLnumlf~rs 11, 11' taken from 1,"', n.) Let P denote the least upper bound of the quantity PF over all functions F «() € I. Then P is
+ 2 ~ 4-21z I" + _4_E(1 z i) I
= z + C2z2 + •. ; € S,
Second, if fez)
§4. METHOD
C., pF~, p - 2v=l=v' ! (Fv, p-F."p)(CC., p-l)
(1)
(4)
where FII,p = FP«lI, ). The right~hand member of this equation can be simplified further. Note that, if we replace one of the values (11, p with (11, peia. in the expression (3) written
1) See Goluzin [1946£, 1947, 195Id]. 2) Goluzin [1947, 1949c 1.
for Fp «() and for the values (11, p' where
11
= 1, ... , n,
we obtain a function of
142
IV. EXTREMAL QUESTIONS
a defined in the interval
-TT
< a< TT
143
§4. METH9D OF VARIATIONS
that attains its maximum at a= O. Therefore,
we would have Pp -
0, which is impossible since it is easy to find a positive
the derivative of the logarithm of this function vanishes at a= 0 and this leads
lower bound for Pp that does not depend on p (calculating, for example, Pp,F'
to the condition
where F «() == ().
m(t~' C.,P~~.P ..:;.. F., p F.,. P
)=0,
If, for the sample made, we take the limit in equation (5), written for F
Fp ('), as p -
I"
.'=1
where the summation is over all v'
= 1, .' •. , n excluding v' = v. Since this
> 1: [~F' (~)P
equation holds for all v = 1,"', n, we conclude that the numbers A.
!
F
C., pF~ 'II,
-F."p P ' p
1
I
It is now easy to show that
':::F" takes the form ,1, _ ,~,
..,
.:/=.,
=n(n-l)-2C
1'1
~ .*.'
(IC., p IS - I ) C., pF~ p -' (F.,p-F.,. p) (C-C.,p) (C'C.,p-l)
2) 'v,p' v
=
F«() €~;
1,"', n, converge to some 'v' v
4)
1'1 > 1, under the mapping
1
=
1,"', n;
in
1(1) 1
n
-_ C j [ n (n - 1)
+ C..:;..~
1
(C _c.cy
]2 dCy.
(7) .=1 If some C v 1= 0, then, in a neighborbood of the point <:~ the left-hand member of (7) is bounded whereas the right-hand member behaves like log (, - ' ) , which
We need also to show that the function F (,) that we have obtained is the
1/1,\2)1 F'(')! ~ 1.
£
>P
P1
>0
and fixed PI sufficiently close to 1 but
,F - £.
Furthermore, for a sequence p'
-->
1, p'
> 1,
F«() we have, from some point on PPloF - £ ~ PPloFpl 2£ ~ Ppl ,Fp' - 2£ >= Pp ' - 2f. Consequently, P> Ppl - 2£. On the other hand, for these same values of p', we have Pp I ~ P. This shows that Pp I --> P; that is, such that Fpl (,)
-->
=
F«() maps the domain
1'1> 1
onto the entire
w-plane with analytic cuts. The analyticity of the arcs constituting the boundary of the image follows from equation (2). We denote this image by B. That B does
w = F('));
converge to certain finite numrers. This last is
we have (1-
,dF
,,~
It remains to show that w
We note that the points
a ' for v = 1,' . " n, are all distinct because, otherwise, for the chosen p v
not have any exterior points follows from the fact, that, if it did, we could, with out changing PF , add on to B a two-dimensional segment and then for the func
possible, because, for example, from the second of inequalities (15) of §3 for
F(') € ~
.=1 0 for v = 1, ... , n: We write equation (6) in inte
the function F (,) exhibited above is extremal.
3) Fv,p' v = 1,"', n, converge to a v (points on the boundary of the image of the domain
V.*., ! ,..
greater than 1, we have P
p-l: to a function
=
do this, we note that, for given
Thus, we can choose a sequence of functions that has the following properties as
1'1> 1
v
extremal function in the problem mentioned in the formulation of the theorem. To
can be applied to the functions Fp (!;,)' treated as functions of the p,arameter p> 1.
p
C
(6)
is impossible. Thus equation (7) reduces to the form (1).
(5)
for w = Fp in > 1. Now, by virtue of the normality of the class ~, the condensation principle
1) F (') converge in
~ (C~·C.)2' c.=const.
gral form:
~
Then, it can easily be combined with the preceding term. As a result, equation (4)
«,)
n
=n(n-l)+C
~ C.,pF~, p ..:;.. (F.,p-F",p)(CC.,p-l) •
!
1_ .•.
n,
v= , ... ,
=
F(c:> in
! ,.. ,., , '=F"
.'=1 are all real. Therefore, the last term on the right side of equation (4) is equal to
[~F' (C)]\1
1, we obtain the following differential equation for
1
tion w
=
F* ('), with IF' (08) I < 1, mapping
we would have
IF: ("") I < 1,
I" > 1
onto the modified domain B*
by Lindelof's principle. But then the function w
>=
\
\ \
F* (~")/F: (De) €
ber of (8) is equal to 12V3, which exceeds 16.
2 would give PF a higher value than the function F (<:"), in
But if the fraction cannot be reduced, the function F (,) must have expan
contradiction to the extremality of F(tj. This completes the proof of the theorem. The problem considered in Theorem 1 with n
=
sions of the form
2 constitutes the problem of
2.
under the mapping w = F (,) €
the extremal function is the function F (,)
Its solution is well known: P = 4, and =
'+ + TJI'
const, where ITJl
same result can easily be obtained by solving equation (2) with n
= 2.
=
1. This
I"
F(C)=ay+c~~-~y)~+.•. , cll*O,
the extremum of the ordinary diameter of the boundary of the image of the domain
1'1 > 1
145
§4. METHOD OF VARIATIONS
IV. EXTREMAL QUESTIONS
144
in neighborhoods of the points
"
'y,
for
v=
1, 2, 3, on
the points a I' a2' a 3' These expansions show that a y
The com
endpoints on the boundary of the image of the domain
,
=
1 corresponding to
for v
Iz I > 1
=
1, 2, 3, are the
under the mapping
plete solution of this problem for arbitrary n is laborious. We note that the func
w = F(') and the point (al + a2 + a3)/3 is a multiple boundary point out of which
tion F(') = '(1 + TJIC)2In € 2, satisfies equation (2) for arbitrary n 2: 2 also, for this function a y '" 2 21n e' TTYiln for v = 1,'" , n. However, for the case n = 3,
issue the boundary arcs extending to a y for v
we can carry the solution through to the end.
to it, let us normalize the function in such a way that al + a2 + a3 = O. If this
Since the extremality of the function F (,) is not lost when we add a constant
Theorem 2. Let F(') denote a function belonging to 2. Then, for arbitrary points a I ' a2, a3 on the boundary of the image of w =
1'1 > 1
extremal function has an expansion
under the mapping
F(q=~+IXo +
F('),
I(al -
a~) (al - ~s) (all - as) I ~ 12 V3,
with equality holding for the function
(9)
l1Jj=1.
Q
q= -
(8) over the points aI, a2, and a3 mentioned in the theorem and all functions at the problem considered in Theorem 1 with n =
3. One of
the functions providing this extremum must, in accordance with Theorem I, satisfy the differential equation (2), which for n
~2[F'(qlll._._,
=
~~~
all)(al -
as)(a~ -
The fraction on the left side of (0) cannot be reduced further. To see this, (a2 + a3)/2
= al'
Then, since la2 - a31 S 4, we would have lal - a2 \ S 2, lal - a3 \ S 2, and, consequendy, P
S 16,
(12)
Now, let us find a bound for
for the left-hand member of (8).
= aI, that is, that
as) I~ ~ 14 f.a y + 3~· 4IX~ I ~ 4( 4S 1IXll s + 3~ I ~ I~)·
Q = 43 \ IXI IS
Here, a I ' a2, and a3 are boundary points providing the corresponding maximum
suppose, for example, that (al + a2 + a3)/3
6~.
so that we have
(10)
.,=1.
=-
of the discriminant of the equation x 3 + px + q = 0, which is equal to 4p 3 + 27q2,
I (al -
+
a1 + aE as ,,_,., ~,~,.,
ala~aS
- 4IX1>
With regard to the left-hand member of (8), its square is the absolute value
3 can be written in the form
F (C)
(11)
is a polynomial with zeros x= aI, a2' a3' Then,
p = ala2 + alaS + a2 aS=
Proof. Let us consider the extremum of the left-hand member of inequality
F(,) € 2. We arrive
... ,
_ _ a1aS+a1aa+aE aa _ a1aEaa al4 ,tx<.l6 •
Suppose now that x 3 + PX +
2
io}a,
at +
then, by substituting it into equation (10) (with a I + a2 + a3 = 0), we find that
(8)
-0 ao ,
F(t:)=ql+
1, 2, 3.
=
whereas for the function (9), which maps the disk
1'1
>1
onto the w-plane with three rectilinear cuts extending from the point w = 0 to the points 2213TJ1I3e2TTYil3 for 11 '" 1, 2, 3, the maximum of the right-hand mem
'.1l.
over all the functions F(')
=
'+
al,-l
+ 3~ I all I~
+...
in the class
(13)
2°
to which our ex
tremal function belongs. Let us denote by C' the image of the circle
p> 1, under the mapping
w =
1'1 =
p, for
F(O. From inequality (4) of §7, we obtain, by con
stricting the cUl£ve C to the point w
=
0,
> 0.
~R~d
c'
Proof. We may assume that all the CJ,J are finite. Consider the function
From this we get, just as in §7, for A> 0 and p> 1, 2"
:p ~ IF(pe
iO
)
ary of the image of the domain
lAde ~ o.
\\
":,
)
=
!
~
.
3
(n- 2 ) ICIlI~p
-2
I(a1 -
3)
( 11--
(v. v'
=
1, 2, 3, we have,
2
ag) (al -
as) (a~ - as) I ~ 3
V31 a1 a2 aal· lal a2 a31 :s 4, which is
This last inequality, together with (8), yields equivalent to inequality (4). We verify directly that equality holds for the function (15) and the points CJ,J = 2-2/37J-1I3e21TJ,Ji/3, v = 1, 2, 3.
3
~2P'
Without going into detail, we mention the following. By using Schiffer's studies [1938], one can show first that equation (0) holds in the domain
I c~ I~ + 31 Cs Ii ~ 3,
1'1 > 1
for an arbitrary extremal function of the type mentioned in Theorem 2 and then that equality holds in (8) and (14) only for the functions (9) and (5) respectively. 1)
that is,
Iad~ +3 I eli \~ ~
Let us look at the following problem on the extrema of the coefficients in
: •
31 a 2 12 S 4/3 - \ a I 12 • Consequently, Q~ .13 I al 18 3s . 4 - 3& Ial Ii = 3s . 4 - 431 al Ii (~~ ~ Ia1
+
the class S. Theorem 3. Among all the functions w
I)
:f'
10:. 1 I < I, so that the parenthetical expression on the right is positive, we Q:s 3 3 • 4. Therefore, in accordance with inequality (12), we have (8) with
=
~':= I c J,JzJ,J €
S, the maximum
11
11: l.c.1
(6)
.=1
2: 2 and arbitrary given complex numbers y J,J' for v = I, ... 0, is attained only by functions that map the disk I z I < I
for arbitrary given n
This completes the proof of the theorem.
" ' , n, where Yn
t
onto the entire w-plane with analytic cuts that extend to w
This theorem leads to a number of corollaries.- We present one of them that
= oe
without finite
multiple points and that form an angle less than 17/4 with the corresponding radial
deals with the class S.
direction at each point.
Corollary. Let f(z) denote a function belonging to S. Then, for an arbitrary
Proof. The existence of extremal functions is obvious. Suppose that f(z)
that lie on three rays issuing from w = 0 forming
triple of points c I ,c 2' q
under the function w = f(z), we have
1 ICl c~ cSI~4
with equality holding for the function
f
(
Z)
=
.•.
z ._ 91.
,
1I 'IJ I =1.
IS
one of these. The function f(z) also maximizes the quantity
equal angles with each other and that belong to the boundary of the image of the
Iz I < 1
f(z)
=
of the quantity
r,'
equality holding for the function (9) with suitably chosen ai' a2' and a3 in (8).
disk
Vi
I a. I~ + I a., I~ - 2 cos ~ "3 1a.a., I I a.l~ + I a., \2 + I a.a.-I ~ 3 Ia.a., I .
/,/
If we take only the first twO terms of the series, we get, by letting p -> 1,
Therefore,
and, in fact,
and, conseqyently,
3
cS=21Lz'
11==2
have
a., I~ =
1, 00
Since
I a. \
= I:s/» (1 I ~~ + ~8 +...), 3
= F (,)
under the function w
2, we hav~ inequality (8). On the other hand, for v.j,
C~=2ab
>
1'1 > 1
on three rays issuing from w = 0 at equal angles with each other. From Theorem
Let us apply this res~lt with A = 3. We set
For p
~o. Then, the points aJ,J = l/cJ,J' v = 1, 2, 3, lie on the bound
F(CJ = l/fO/') €
o
F(I:)S/9
147
§4. METHOD OF VARIATIONS
IV. EXTREMAL QUESTIONS
146
11
Jj=ffi(e
iO
1:1.C.)
(17)
.=1
(14)
1) (Theorems 1 apd 2 and th~ more general corollary of Theorem 2 have recently been proved by Reich and Schiffer [1964] with the additional assenion that the extremal func· tions are unique.)
(5)
II ',.";7.,.,.
•
~~~7--=;~~~"f,:~;~""~
:~;:
148
IV. EXTREMAL QUESTIONS
§4. METIIOD OF VARIATIONS
149
where () is a real number, out of all functions of the class S.
Iz I < 1
Let us suppose that the image B of the disk
fez)
under the mapping w
=
i9 ffi ( e h
bas an exterior point Wo. Let us form- the varied function f* (z) in accord
1. Let us denote by {¢(()In the coefficient of zn in the expansion of an arbitrary function ¢(z) that is regular at z = 0 in a neighborhood of the point z = 0, taking.\A 1 = h. We have
ance with formula (21) of §3 of Chapter III with m
~ ~
{ f 19 (fo ~ 'Iv /(e)-f(zo) v- e h zof~ - ~ IVCv - ei9h
(;)2}
v= I
v= I
+eiBh
{f*(e)}.=Cv+h{f(~-~2Wnt+0(h2)=cv+h! {f(~:+1}v + k=1
°
Ih I
and arbitrary arg h, this can be true only if
i9 V { f(W }
e ~ Iv /(e) - f(zo) v
0
v=1
n
n
11* =11
- (-h.)D ~ (ei9IVCV + ei9'1v {Zoe!' (e)} -ei9i. { !oeSf' (e) } ) = 0. zofo "" e-zo v 1-z e v
v-I
+ m(h ! IV! {f(~:+1}v) + o (h v=1 k=l
2).
l/wo
Since (18) holds for arbitrary z 0 in I z I < 1, we can by replacing z in equation (18), obtain a differential equation for fez):
and the coefficient
in that expression is equal to Yn ~ 0, this polynomial is, for rhe value
of Wo that we are considering and w~ arbitrarily close to Wo, -also nonzero for the corresponding function f* (z) with small
iO (Zf'
e
h 1 and suitable arg h, If· > If'
I
n
(Z»)I
fez)
V
and arbitrary z 0 in the disk
+
I z I < 1.
_ h
h
(fO)8 ZoTo
§ 3 of Chapter III
1. -
vI
e
19h V
~ Iv Ii(e) _ f(zo)
n
- ei9h
v=1
v
Let us show that the right-hand side of formula (19) is negative for r:&;,
~l~
)8 V
zof~ ~ IVcv
__ n
lj; , (0)
> 0 and that maps the disk
"y.
f
I zI < 1
univalently onto the disk
(19)
=
fez)
The condition of extremality of the function f(z) requires that the inequality
S If hold, that is, that
1.
. + we 1 (1 + h) (. 1) -----r;;;= ze +-----;;;zeiT + 2h,
we''P
'f
T
For h, we obtain
1 +ze if (z) - z - hz 1 -ze if
,_
C(
with
issuing from the point w = e- ¢, can be determined from the
1
e2
•
h=4" 1-e
v=1
v=1
Iz I =
Iw I < 1
i
+ °(h ), 2
~'(o)= l-h,
h>O.
Let us use this function to define a function in the class S:
1(.
U.
We note that a function w = lj;(z) that satisfies the conditions lj;(0) = 0 and radial cut of length equation
19h ( fo
2 (4)8 VIv {zo~f' (e)} + eioh (A)8 VIv{ !oe f'_(e)} ) + 0(1 hI2). ~ ~ e-~v ~ ~ 1-~v v=1
~l- ei0i. { z~: ;~
011
v
v- e
{Z;r'
sists of a finite number of analytic arcs and that equation (19) holds for w the circle I z I = 1.
Cv
n
L f(e)2}
v
It follows from this differential equation that the boundary of the domain B con
8
n ro (
o
v v
and consequently,
+ 1
}
= V~l (e 191 C + ei l
i,l;
1 - 1
z
v=1
with m = 1
(4)2 {zoe!' (e)} + h( fo, \8 { zoe f' (e)} + 0(1 h 12) zofo ezofo I 1- zoe Zo
f (e)8
n
We have
-{ f' (e)2} h fee) _ f(zo) v -
{
° with
~ 'Iv /(e)-f(z) v
which is impossible. Thus B has no exterior points. Let us now consider the varied function (24) of
(18)
o
'=1
0
Since the expression in parentheses is a polynomial in
{f. (E)}. = c v
v
n
(h 2),
and, consequently,
l/w3- 1
o
(-4-)1 V IV{ !oe1f' (e) } ) + 0 (I h 12 )= 0. zofo ~ 1-zoe v=1
~"
For small
" (-4)1 V Iv {ZoE!' m} zofo ~ e-z v v=1
___ n
=
v-I
of
)1
f(o/ (z»
f* (z) = lI(O) =f(z)
+ hf (z) -
+
, 1 zei'P hzf (z) 1 _ ,~I
+ °(h ). 2
=-----_._--..
~':--==::-~-----='=:-"';,.-""-
~~..!'~~;~
.,
§ 4.
IV. EXTREMAL QUESTIONS
150
n
1/*=1/+ hffi
(e
iO
n
! i.e. - e ~ '=.1 lO
j.
i
{~f' (~) ~~:::}J + O(h~).
(e
ffi
! i.e. + e 1:
iO
iO
• =1
j.
n
.'::l
10 '" j { I. (E)2 } e • I*(E)-I*(z). =Q
1 (20)
n
'=1
.=1
~ (e iO 1: i.e. + elO !
j.
tion fez) with its values for the functions
e-ia.f(e'o.z) €
proves that the right-hand side of (19) is negative for
\
,/ If we divide (23) by (19), then, for
Iz \ =
S with real
ct.
This
where
='t~o,
=
then we also have jR(1/Q(w»
I
> 0 at all finite points on the boundary of B. But > 0 and we see from equation (19) that ffi (( zl'
(Z»)I) < 0
(24)
I(z)
n
{
f(z)=w o 1-}-"7 }-"7 =woT
O. Since this was excluded, the contradiction that we have
w = 0 if extended in the opposite direction, then the real part of the quantity '0 ~
i.e.
1, we have
This last equation shows that all boundary points lie on the straight line obtained shows that jR(Q(w»
onto the w-plane with rectilinear cuts that would include the point
Iz I =
(23)
T is a real number.
passing through w
1.
\1 (Zl' )1-Q. (j)=R* (z). f
Wo ( wo-I)
Wo )2 ( wo-/(z)
Let us now show that, if our extremal function w = f(z) does not map the
I z' < 1
"'-.\
{~f' (e)}. ).
But this quantity is equal to zero, as is shown by comparing its value for the func
disk
.
obtam
.=1
n
if (z»-Q (wo)=Q (/(z».
TherefO,fe, if we replace f. (z) in (22) with its expression in terms of fez), we
{~f' (~) : +:}J .s;;; o.
The difference between the right-hand side of (19) and the left-hand side of (20) is equal to
I(W I(E)-wo
and, consequently,
,=1
'Z 1 =
151
But
I"" (;)2 _ 1(;)2 I. (E)-I.(z)- I(E)-/(z) -
.
Consequently, by virtue of.the extremality of fez), we have on n n
I z I = 1.
R. (z) must again be negative on
This
For this function we have
METHOD OF VARIATIONS
(E)I
}
Q(w)=e' ~j. I(e)-w.'
at all points on
• =1
I z'
= 1 at which fez) is finite •
The additional conclusions on the boundary of the domain
which appears in equation (19), is positive at all finite boundary points of the
B follow from
equation (19) and inequality (24). We can see this as follows. If the boundary
domain B. Let Wo denote an arbitrary finite point on the boundary of B. The
B
consists only of radial rays, everything is proved. On the other hand, if this is
function
f*(z)= wof(z) "' , belongs to the class S and for this 1f • This shows that jR (Q (wo ~ O. If
=
»
I(Z)I
fez) - I(z)-Wg
not the case, then (21)
arcs that extend out to
n
iO
e
(Z/~)2 ~
1*
{
.=1
it follows from
B consists of analytic
Furthermore, it follows from inequality (24) that, for a
Iarg
z!' (z) I(z)
1t
+2
I< 4
1t
on the entire boundary of the domain. But the quantity on the left-hand side of
the equation corresponding to (19):
1* (;)2 } _ '- i. 1* (E)- 1* (z) • - R* (z);
oe.
suitable value of the argument, we have
This requires that Q(wo) be equal to zero.
f. (z)
~ 0 at all finite boundary points wand
the boundary. Consequently, the boundary of the domain
function we have 1f - ffi (Q (wo»' jR (Q (wo» were equal to zero, the function
(21) would also be extremal both for the quantity (17) and for the quantity (16). Let uS write for the function
Q(w)
equation (19) that none of these points can be a multiple or an angular point of
this inequality measures the angle between the tangent to the boundary of B at (22)
f(z) and the ray passing through the points w = 0 and w = f(z). Consequently, this angle must be less than 11/4. This completes the proof of the
the point w
=
---=--=--------
~o~~oo=~~"~~~=~:;;;;
-- -- "
1 P \
IV. EXTREMAL QUESTIONS
152
J- ,: --
_::::-:---;;-:-:'--=-~-----~-'---"~--'----------:-:-'';;:----:=---,_
:,\, ~\
-
'~.:
153
§4. METHOD OF VARIA TIONS
exist extremal functions such that c n > 0 and they exhaust all extremal functions in the sense of the characteristic of the images B.
Let us now apply the method of variations to the investigation of a particular
Suppose that fez)
extremal problem for the class Sa of functions
f(z) =CtZ
+ c~z~ +..., Iz I < 1
that are regular and univalent in the disk finite values a 1> ., • , am ,.where m
~
C
1. Let us look at the problem of maximizing
Theorem 4. A function w = fez)
C1 z + C2 z2 +... € Sa for which ICn I, where n is
=
a given integer ~ 1, attains its maximum maps the disk
I z 1< 1
onto the entire
w-plane with cuts along a finite number of analytic arcs and satisfies in the disk
n
> O.
R (f(z»
P (f(z»
z2f' (Z)2 = n
+ ~ (n _ ~
'I)
(C n _. z-'
• = I
Cn
+c~_.Cn z'),
(25)
~k=1 c k zk is one of the extremal functions such that
~
§ 3 of Chapter III,
N~~e can
easily obtain a differential equation for the function fez). Since
§ 3 of
the function\ (29) of distinct
zl,j···,
Chapter III is a function in the class Sa for arbitrary
ting M = 1/ in that formula
{f* (~)}n = Cn
I z 1< 1
in the disk
Zm
I
+ h f (~)
m-3
R (w) =
P(w)= Il(w-ak)'
~
and sufficiently small A, we have, set
til
II (( (e) -
ak) )
m
k;1
II
-
(((e)-f(Zk»
k=1
h
!
A~(Zk), {ej' (e) }
k= IZk! (Zk)
11
m
e-
\Zk
11
_
+ Ii ~~ Akf(Zk) {eSf' (e) } + 0 (\ h I~) Zkf'(Zk)2 1- Zke
where m
when we replace w with f(z)
and take WI = W2 = .•• = w n = 0, we can show, following the same reasoning as in the proof of Theorem 3, that the domain B does not have exterior points.
the differential equation n--I
=
Let B denote the' corresponding image. By using the varied function
obtained from equation (27) of
and that do not assume m given
for given n ~ 1 with respect to functions of the class Sa (see Goluzin [1946f]).
Iz I < I
,,~,:
f,?,,;/
theorem.
Ic n I
-
k bkw ,
n
k=1
let> (01 n denotes the coefficient of z n in the expansion of the function et>(z) about z=O. But since ~({(~)ln)~cn, we have the b k bidng constants. In equation (25), the right-hand member is real and non negative on the entire cir cle
Iz i =
where
k=-n-I
k=1
1, and the boundary cuts referred to are with
a suitable parametric representation w
=
wet) (t real), the integral curves of the
differential equation R(w) pew)
(dW)' + 1 =0. dt
(26)
m
I
II
rn(, h f (~) I m
(f(e)-ak) )
II
m
~k = I c k zk € Sa, we have the well-known inequality \c ~ I ~ en for the function f(z)/ c 1 = I. k= I C~ z k
I c n I has
and consequently,
1z 1/(1 + I z 1)2
a finite upper bound bec au se , if f(z)
I Cn I ~
en
I c 11.
(((e)- f(Zk» n
I c 11 < «1 + r)2/r) I a I.
Iz I =
r at which If(z)\
It follows from this that
I c II
fer
(e) } (Z)I C
k .. - Zk n
m
+Ii~ Ak/(Zk) {e2f'~e)} )+O(lh\~)=O. 1- Zke n
.'- zkf'(Zk)1
k=1
=
If we apply the familiar inequality If(z)/d ~
to a point z on the circle
~ z. P k=1 RJ
k=1
Proof. The existence of extremal functions is obvious. Specifically, the quantity
h ~ Akf (Z1l)
_
k=
< Iall,
we obtain
~ 4 I all as r
-->
1. Conse
quently, I Cn I:os:;; 4en I all· By virtue of the normality of the class Sa' this upper bound is attained.
If fez) is an extremal function, the function fee eLi z) for arbitrary real a, which has the same image B, will also be an extremal function. Therefore, there
If we replace the next to the last term with its conjugate (which leaves the real part unchanged), we obtain m
I
II
rn (h [ f (~) k = I
(f (e) - ak) )
m
II
m
_
(f(e)--f(Zk» n
~ Ak!(Zk) ({ ef' (e) }
.'- zkf k= I
e-
(Zk)1
Zk n
k=1
_ {e1-f' Zke ~e)} )]) + 0(1 h I~) = n 9
Because of the arbitrariness of arg h, this leads to the condition
O.
-
154
IV. EXTREMAL QUESTIONS m
(
I
II (f(E) -ak)
(~) k= 1
1
_
m
II
(f(E)-/(Zk» n
m
'\' Ak! (Zk)
I' 1- zkE
(fEr (E)} _ { E ~E)} ) = O. E- Zk n
(27)
n
= znc~
R (/ (j) Z2£' (Z)2 P (f ~;z» J
1
~ zkl (Zk)1 k= I
1
+... + nCn + (n -
where
m-3
k=1
Since the first term on the left does not vanish identically for all values of •• "
we can choose.
Z m,
let us change
Z m'
Z 2> •• "
Z b •• "
Z 1,
Zm
At =
Q (f(z»
considering it a variable z. Then, the A k take
'- ~__..
Bk .1:/_
(k=2, ... , m),
\
II (W-/(Zk))'
ak)
II
(f(e) - I(Zk»
1 n
~ (IEI'E- (E) }
P (f (z» Q (f(z» zl' (Z)ll
n
I
II
(f (E) -
I (Zk»
1= !
k=-n
"\' +k=2 ~
Ck .1:/_\
.I:
_
/_
\
-
Bkf(z)k,
V
~
so that we finally obtain
o/(z) where 0/(0)
k=2
=
0 and
wi
+ ze i9 + o (h
1
hz,
_nM
h
(nc n + 2 ~
2
).
+ ze_"Ini6 I' (z) + 0 (h it (z)
=
)
~k=1 c~l) zk, we have
kCke(n-kli6)
k=1
2
+ 0 (h
2 ).
From the extremality of the function fez) we have )R(c~l) ~ c n
I (z) - I (Zk)
,
which,
because of the smallness of h, leads to the condition
!
Dkl(zl,
k=O
~ [Z~~l + ~~~1
1
n-l
C n
.
= - . ~ ...
= -
1
obviously belongs to the class Sa' By setting
C"
=
I < 1 with a e- i e. For this function, we have the
0,
n-l
m-2
Ell (E)} _ {E 2 I' ~)}
{ E-z n 1- zE Il
=
It (z)=/(IjJ(z))=/(z) -
m
k=1
we have
but also on the circle
Iz I < 1
C~) =
-1
n
<1
univalently onto the disk
w=ljJ(z)=z-hz
or, in view of the fact that
(f(E)-ak)
Iz I < 1
0/ I (0) > 0, that maps the disk
small radial cut issuing from the point w
The function
Ell/' (E)})
1
I
=/ 1 except at a finite number of points. Consequently, the boundary of the
Z n
- { 1- zE
II
Z 1
expression in
m
f(~) ~ =
z
simpler form. To do this, let us consider a function w
k=1
m
1
The differential equation for the boundary of the domain B can be put in a
k=2
form
I
conclude, therefore, from the analytic theory of differential equa
domain B consists of a finite number of analytic arcs.
and the Bk are constants. Therefore, condition (27) can now be written in the
IT (f(E) -
a~~ b Ie' We
tions,,!that fez) is regular not only in the disk
m
Q(w)=
k= 1
~-
Admittedly, this differential equation contains the unknown constants
/
m
11 (W-ak)'
Thus, the function fez) satisfies the differential equation (28) in the disk
Izl:e 1. I
(I(~)
(28)
k=-n-l
Ck .1:/_\
where
P(w)=
+...+ CtZn-t,
such that this term is nonzero. Now, fixing
Ak =
,
1) Cn_tZ
k bkw .
~
R (w) =
Z b .••
the form p (f (z»
155
§4. MElHOD OF VARIATIONS
nC n
+ 2ffi ( ~
kCkei(n -k)
6) ~ O.
k=1
If we now set e iB = z, we obtain an inequality expressing the nonnegativeness of
the right-hand member of equation (28). Thus, on the circle
+ ... + nc n + (n-l) Cn_tZ +... + CtZn-t] ,
I< (f(z» P (f (z»
Iz I =
1,
z2f' (Z)2 :::>- O. ::0
It follows that, with a suitable parametric representation w = w (t), where t is
156
IV. EXTREMAL QUESTIONS
real, of the boundary of the domain B, we have for this boundary the differential equation
R (w)
pew)
where
" ~tl
k=l
ie
Let us stop for the important special case of n bound for
I C 1 I = I[ , (0) I.
=
bec~~se ['(0)
i~
This quantity represents the radius of the disk onto
= cp(w) normalized by the conditions cp(O)
= 0 and cp '(0)
= 1.
It is called
above concerning the bounds for the coefficients with n = 1 thus apply to the 00
and a given
finite system of points al"", am' The corresponding extremal function belong
1
where R (w)
=
I
;;01bk wk.
P(j(z»
If z
=
(Z»)I =
J(z)
rp (w)
This function is distinct from [(z)
IC I I
!c n I for
does not necessarily increase when the domain B is
/I
1 ,
over all possible functions
is the inverse function, we obtain the
h (z),
TI I/k (D) I k=l
for k
=
1, 2; ... , n?
In the case n = 2, the answer to this question was given by Lavrent 'ev [1934] (on the basis of his variational method). It consisted in the sharp ine quality If~ (O)/~ (0) I~ I a1- a~ \2,
R1 (w) rr -w!P (w)
= ...
in B. Consequently,
in which equality holds only for the functions [1 (z) and [2(Z) that map the disk
In (jl (W)=
r , / wabo(w-a +b w+ '" +b (w-a!) ... (w-a
J Y
m_1w m-
1
1)
1
m)
dw+ bm•
(29) '~l("
1z I < 1 onto the halfplanes about the common boundary of which the points a 1 and a2 are symmetric.
This yields an explicit analytical expression for the function cp(w). The con stants B 0, B 1, •. "
C
is also easy to show that Lindelof's principle does not apply to
J=
differential·equation· 'P'.. (W). If (!p)
1 n I.
[/(0) ~ (0).
ing to the class Sa satisfies the differential equation obtained from (25) with n = 1:
Rdj(z» (zl'
> 1, the function
In conclusion, let us examine the following problem on disjoint domains: Let aI, a2 .' .• , an, where n ~ 2, denote n distinct finite points in the w-plane. Functions w = h (z), k = 1, 2, ... , n, that are regular in the disk I z I < 1 map that disk univalently onto disjoint domains B k including the points ak, for k = 1, 2, .. " n, respectively in such a way that h (0) = ak for k = 1, 2, ... ,n, The question arises: can we speak of the maximum of the product
the conformal radius of the domain B (d. Chapter II, §2). The results obtained extremum of the conformal radius for domains B not containing
= e- 21T (n-l)ilm
n > l/, that is, that broadened.
1; that is, let us find a
which the domain B containing the point z = 0 is mapped univalently by the func tion z
for n
> 1 and nil (mod m). If fez) is one of the normalized extremal func
~I>0 a normalized extremal function for
n = I C n I e- . Let us apply the above reasoning to [( e i e/ n z). If we replace z with e -iel n z in the formulas obtained, we obtain the assertion of the theorem (!c n I is included in the coefficients bk ). C
C
C
tions, then the function [I(z) = [(ze21Tilm) e-21Tilm belongs to the class Sa and is
~-_l
Consider the function fez) with
I nI
> o.
n > 0 does not always have a unique extremum. Let us show this with an example. Suppose that ak = ae21Tkilm (k = 1, 2,,·, , m), where n
bkwk.
R(w)= ~
157
uniqueness of the extremal function fez) normalized by the condition (0)
fez) normalized by the condition
m-3
TI (W-ak),
§4. METHOD OF VARIATIONS
We note that, in the case of the extremum of
(dW)! dt + 1 =0,
m
P(w)=
,
Bm remain undetermined.
A formula of the form (29) was proved earlier by Lavrent'ev [1934] by starting with the principles of Lindelof and Montel. l ) These principles lead to the
The existence of extremal functions is easily proved. Specifically, since the function gk (z)
=
(h (z)
. ~ot assume in the disk
- ak )/[~(O),
Iz I < 1
belongs to S for k
the values (ak 1
follows from Koebe's theorem that (ak 1
41
- ak)
a k 1 - akl . This proves that the quantity
1, 2, ... , n, and does
(0), where k 1 ~ k, it that is,
If; (0)\ s
J has a bound independent of the form
h (z) for k = 1, 2, ... ,n. From this and the normality of the family of functions h (z) that we are considering, we conclude that the least
of the functions 1) Grotzsch [1930] studied the extremum of the conformal radius for the class Sa on the basis of other considerations. He also pointed out the analytic form of the extremal function in a certain particular case.
- ak)/ [~
[k (0) ~~;
=
upper bound J0 of the quantity J is attained.
.",=:€~=-""
----------
---~~-
,-
-
.-~.
,
-"'------"-----------
o
,
158
---
~=-----=~~,,~~~~~~~-;;;:,"==~~::~:~'!!::::::::~~_:::~:::~~~::-=--"---~~~~~
=
!k (z),
for k
=
;'
1, 2,"', n, denote a system of extremal func
LindelOf's principle ( p. 30), the domains Bk are such that the union of their
:8: (Ik (Zk) :8: (Ik (zk) _
zlk (z)
closures U k=lBk coincides with the entire w-plane.
-
•
Zk (I;' (Zk W
h
Let us prove certain variational formulas that we shall need. Let n denote an integer greater than 1. For fixed points ak, where k
+Ii
II
II (w-a.) .--:...=_11....:.1
2
z , 1k _.,(z)
~
~
""
1 ~ko~n,
_
II (w-w.) '1= k o
where h is a complex number and the w lJ ' for v
1,2,···, n, v.j k o are distinct points. This function can be represented in the form =
W*=W+h(W+AO+ ~
w:...·w.\,.
From this we easily see that, for sufficiently small
I
I h I,
the function w* is uni
for v = 1, 2,' .. , n, v.j k o are removed. Therefore, if w lJ = f)z lJ)' for v
disk
+ h :8:
(lk (z) _
=
I. (z.))
(31)
and f'ic(O)
= ak,
k
=
1, 2, " ' i n.
h (z)
=
constructed in accordance with formula (3), are also
II
J*= Dill:' (O)I=Jo 1 ~ko~n,
for k
11 +m fh (~----:'-.!;----"-k~_(ak_--_a_.)_
-
(30)
II (ak -I.(z.»
k=1
.=1
'1=k II
is, for k = k o, regular and univalent in the disk
§ 3 of
+ 0 (I h \2),
(Ik (Zk) - I. (z.))
easily obtain the equation
'1= k o
=
1_
1 -zk z
I h I onto disjoint domains
h o (z)
.=1
(30) has a simple pole z
)
admissible functions of the extremal problem that we are considering. Now, we
(Ik (z) - a.)
.=1
Ik (z)
for small
with the function
II
n (z) =
Iz I < 1
then the function
II
(zk) -a.)
1, 2, ... , n; k.j k o , let us now construct the varied functions (31) with zn ~ 0 for v = 1, 2,"', n, v.j k o , which, together
valent on the entire plane from which sufficiently small neighborhoods of all points
lJ
(lk
which is regular and univalent in the disk I z I < 1 for small values of I h \: also f'k (0) = ak' The functions w = f'k(z), k = 1, 2, .. " n, thus defined now map the
From the functions
)
.*ko
I z I < I,
:8: ( :8: .=1
.-1
Equations (30) and (31) are the desired variational formulas.
.=1
1, 2, ..• , n, where
I. (z.))
__
Z-Zk
• =tk, k o
.=1
W lJ'
.
'1=k, ko
1, 2,"', n, consider
=
a.)
I
.=1
the function
+h
159
§4. METHOD OF V ARIA TIONS
tions and let Bk denote the corresponding extremal domains. On the basis of
w* = 'w
...
'-""'-~~ _o~.".,.._...,.,""':::~c':."..~:=_::::===-::=
'",;,
~;_
IV. EXTREMAL QUESTIONS
Now, let w
~~.;~"''''--
--~;:;~~_=_:c
- --------- -
--------_.:--------
--,
I z I < 1.
+
For k.j k o , the function zk in the disk I z I < 1. Consequently, the theorem in
Chapter III can be applied to it. As a result, we obtain the function
:8: (lk (z) - a.) n (z) = Ik (z) + h :8: (lk (z) _ I. (z.)) .=1
·4:.ko
IT
.=1 (lk (Zk) -
(Zk»2
k=1
k=tko
n.
a'))J --
'~I (Ill (ZII) - I. (z.))
+ O(lhI2)} .
.*11, ko By virtue of the arbitrariness of arg h and the extremality of the functions h (z) for k = 1, 2, .. " n, this last relation can hold only if the expression in the large square brackets is equal to zero, that is only if II
II
II
.=1
L z: (11
L :*~ k=
I
II
.=1
"fko
(ak~a.)
(ak
-t. (z.))
II
II
II
+~
1
zZ (Ik(z k»'"
k=
I k =/=k
0
(lk (Zk) -
a.)
=0.
.-1
IT
.=1
'1=k, ko
(lk (Zk) _ I. (z.))
(32)
160
IV. EXTREMAL QUESTIONS
§4. METIlOD OF VARIATIONS
Consequently, this condition must be satisfied for an arbitrary choice of the points Zk,
for k
=
It follows from (32) that the circle
I I = 1. Z
I Z I < 1 and for arbitrary k o = 1, 2"", n. functions !k (z) are piecewise-analytic on the
1, 2,"', n in the disk
To see this, let us fix all the pointsz k> k = 1, 2, .. " n; k ~ k o ,
with the exception of one of them
Z
kI = Z
tion (32) becomes a differential equation for z~ (fk J (Z»i
< n).
(where 1.:s k I
for k = 1, 2, 3; that is, the functions w
!k (z)
for k
=
3
II
(ak- w)
11=1
zsw,a
We see that condi
_
(aI-as) (aj-aa)
-
a1 -w
j) (aa-a 8) + (aa-aaaW
(aI-as) +c + (al-al) al-w '
(z», 1
Of cdurse, the constant C may depend on ai, a2 and a3' However, we are not
theory of differential equations. Let us look at some particular cases. The case n
2 can be carried through
=
able !~o determine its value.
t. 0
and C = O.
to its completion without difficulty and we shall not stop to do this.
Let us consider separately the two cases C
Now let us look at the case n = 3. In this case, equation (32) with 'k o = 1 yields (a 1 - as) (a 1 - as) (as - a1 ) (as - as) (a 1 - Is (zs» (a 1 - la (z,» (as - Is (zs» (a 2 - la (za»
In the first case, equation (35) reduces to the form
l"Ptw) w'
+
3
II
3
(as - a1) (aa-as) (aa -Is (Z2» (aa - la (za»
+
1 I. (zs) -/a (zs)
If we multiply through by the difference [2 (Z2) -
(fs (zs) - a.)
z:
3
=
.=II
from ai, a2, and a3' For extremal domains B k , the union U~=IBk coincides
(fa (Z8) - a.)
with the entire w-plane. Therefore, the point w = De lies on the boundary of cer
I
zl (f~ (Za))lJ
_
0
tain of these domains Bk' let us say, for example, on the boundary of B I ' Let us
-.
denote the point on the circle
and decompose the first
A(/a (za»,
fdz) by z
as) (a 1 - aa) a1 - I (z)
+ (as -
=
1 corresponding to it under the mapping w =
I~
~ /1
2
z
=
z., we have
(A<+ 11'" B+ )_ -z'
A#- 0,
1)
Consequently, a1 ) (aa- aa) as - I (z)
~ (AI + _ 3
/12
II (f (z) - a + (aa-a 1) (aa-a s) + .=1 aa - I (z) --z--=a'-'(f"""""'('-'z)""')S~v)
1
Zl
Zi)"'
Zl
A'
#- o.
co W
=/1 (z) =
~ b. (z -'- ZI)",
b_~
#- O.
(37)
.=-2
(34)
A (fi (Zi» = A (fa (za».
Since equations (33) and (34) hold for arbitrary Z I' z2, and z 3 in the disk
it follows that in the disk \ z I < 1 we have A Uk (z»
00
~ c.(z £. r' + ...)=1j log !-=1j Z-Zj + .=2
B
This means that, in a neighborhood of z = z I,
Analogously, for k o = 2, we obtain from equation (32)
I z I < 1,
Izl
= Zl' Then, in a neighborhood of
(33)
where A is an operator defined by (a 1 -
(36)
(Z2»S
(f~
three fractions into partial fractions, we arrive at the form
A (j~(zi)
1j=+1,
where p (w) is a polynomial of exactly third degree, and its zeros are distinct
.=1
h (Z3)
(a1l- w)
1)
=z'
11=1
II
+ la (Za) - I la (Zs)
A (/(z»
(35)
C~~onst.
where R (w) is a rational function. Our assertion then follows from the analytic
+
1, 2, 3 are each a solution
of the differential equation
!k I (z):
= R (fk
=
161
= C = const
for
If the point w
=
De
were also a boundary point of some other domain Bk (k
the same would be true about the corresponding function
DO
1),
But this is impos
k = 1, 2, 3, are disjoint. Thus, in the present case, lies on the boundary of only one of the domains Bk' The other
sible since the domains Bk the point w =
fk (z).
t.
,
"~
''iff IV. EXTREMAL QUESTIONS
162
two domains must be bounded.
This is easy to verify directly by substituting the expressions (42) for wand the
Let us turn now to the case C = O. Let us suppose first that the points ak are vertices of an equilateral triangle, that is, that ak f =
e 217i / 3 • Then, equation (35) (with C
3 yasww'
=
k
= (Mk,
where
"IJ=-t-l.
f =
ik (z):
the disk ~-with
Ck=const,
Iz I < 1
h (0) = ak,
to be regular in the disk
1z \ < 1
and for t.he union =
e :J.k for k
1+ e~kz){ lak '
I -e
z
=
0, and with bisector
Jo
=
(64/27) I a 1 3 or,
what amounts to the same thing, ~.r- I (al 81 r 3
a~) (at -
as) (a2 - as) \.
ak, that are regular in the disk
=
(40)
1,2,3, that satisfy the
I z I < 1,
and that map this disk
J 0 can be calculated in the present case from formula (40). Consequently, again in this case we have inequality (41). The case C = 0 does not always hold. For this case to hold, it is necessary
64 V
3
I(at -
for k
=
a2)(at -- as)(a2 - as)\.
(41)
aak+ b ak= cak+ d '
W·
Z2
2
1=
k=l, 2, 3,
(42)
~-w
under the
Iz I < 1
is
(II (0) - f2 (0»
univalent in the disk
Iz I < 1
(0)
(ll\~)l_ f. (0» h (z),
l)
(a:-a 2)
~-w
(12 (0) - fa (0»
for k
=
1, 2, 3, that are regular and
and that map this disk onto disjoint domains. Since
a constant, we can confine ourselves to functions
+ (a2-~1)(a:-aB) + (as-_a ~-w
IT f k
I does not change when we replace the functions fk (z) with tion
(al-~)(~~-aB)
00
quantity
over all possible systems of functions
then ~ as a function of z satisfies the same ,differential equation
(ak- W)
=
ensured for all the functions fdz) defined by formulas (43).
II
same fractional-linear transformation, that is, if we set
cw+d'
corresponding to the point w
1, 2, 3, because it is only then that regularity in the disk
1, 2, 3, he arbitrary. Equation (35) with C = 0 has the
w= a~+b
= -d/ e
transformation (42) lie on the boundary of one of the corresponding domains "" Bk ,
property that if we subject the unknown function w and the constants ak to the
rr
k= 1 --=,-----,o---=k.:..==-=-I_",----;::,----:;;- a2 ) (a l --;- as) (a 2 - as) = (a l - a2) (Ur - as) (a2 - a.) ,
that C ,J. O. To show this, let us examine the question of the maximum of the
3
81
3
II Ik(O)
However, inequality (41) always holds, that is, even when the ak are such
univa:Iently onto disjoint domains, we obtain the sharp inequality
k=1
-
and sufficient that the point ;l;
64
Thus, for an arbitrary system of functions h(z), for k
a
is a boundary point of at least
3
passing through the point ak' Calculation then shows that
=
oe
II f k (0)
(39)
k= I, 2, 3.
rays with an angle of 2rr/3 between them with vertex at w
k=1
ik (z),
=
+ +
important for what follows) that the point w =
From this we conclude that each of the domains Bk is the domain between two
Now, let ak, k
of the form (39) that map
aJk (z) lJ k=l, 2,3, (43) clk (z) d ' corresponding to them will possess the property (obvious but
I k (z)
(a l
Ik (0) ~
"V
two of the three domains B k. Since we can verify directly that
Ik(z) =ae k ('
'n
""
that satisfy equation (35) with C = 0 are then of the form
and the domains B k =
1,2, 3, so that we finally obtain
ik (0) =
f k (0) = a k,
onto the three angles me.ntioned above. The functions w
\.
k=l, 2, 3.
to coincide with the entire w-plane, we must have e k
condition
"""'
"
f k (z)=ae k (11+ ckz CkZ)a '
Jo
are mapped into the points ~k = (Mk,
= 1, 2, 3,
e 2TTi/ 3. Then, the equation for ~ reduces to the form (38). Consequently,
its solutions are functions w = /k(z), with
(38)
2
Uk = 1 Bk
such a way that the points ak, k ""
'IJ
When we solve it, we obtain explicit expressions for the
h (z)
ak into equation (35). Using this property, we choose the transformation (42) in
1, 2, 3, where
=
0) reduce s to the form
z'
wB-aB
For the functions
163
§4. METHOD OF VARIATIONS
Ih (0) I < 1.
h (z)
eh (z),
where
that satisfy the condi
Now, just as above, let us show that
I/~ (0) I ~
e is
41/t (0) -12 (0) I, \/~ (0) I ~ 4 1/2 (0) -Is (0), l/~ (0) \ ~ 4 1/3 (0) - II (0) \.
:,--- 164
IV. EXTREMAL QUESTIONS
From this we con<;lude that the family of functions fk (z) that we are considering
This fact, in conjunction with the fact that the maximum of the quantity I is
is normal and that 1 :583. Consequently, the least upper bound of the quantity 1 is finite and is attained by some system of functions Let us set
nCO)
= ak, for k
fi. (z),
for k
=
attained by the functions
1, 2, 3.
fi. * (z),
leads us to the conclusion that this maximum is
equal to 64/8113. But then, for an arbitrary system of functions fk (z), k = 1,2,3,
= 1, 2, 3,and let us look at the question of the
that are regular in the disk
Iz I < 1
and that map this disk univalently onto dis
joint domains, we have l:s 64/8113, which is equivalent to inequality (41). We
maximum of the quantity
Dote, finally, that, since I
J= IJ]/k(O) I over all systems of functions
fk (z),
f~ (0), that are regular in the disk
for k
c'-c
.. =
I z I < 1,
1, 2, 3, that satisfy the condition and that map that disk univalently
onto disjoint domains. According to what was said above, this maximum is attained by the functions f'k(z). Therefore, on the basis of the exposition above, we con clude that the images Bk of the disk
Iz I < 1
under the mappings w
=
f'k(z) are
is invariant under an arbitrary fractional-linear trans-
formation of the functions fk (z) (but the same transformation for all three values
~fk), inequality (41) in th~, disk I z I < 1.
holds also when one of the functions
functions fk(z) for k
exists a point a that lies on the boundary of at least two of the domains Bk •
=
J=
These functions also map the disk
k=l, 2, 3. It (z) - a ' I z I < 1 onto disjoint domains,
disk \ z I
<1
Illtl f~ (0)1 =
fk (z), k = 1, 2, 3, each of which maps the
the center 0 f the triangle and with bisector passing through the point ak' On the and the quan
other hand, if the points a I ' a2 and a3 are not all situated in such a position,
1 is invariant under a single fractional linear transformation performed on all the
of which is bounded in the disk
=
1, 2, 3. This means that the functions f'k* (z), for k
=
1, 2, 3, are extremal under search for the maximum of the quantity I. If we set
f'k* (0) = 1/(ak - a) = a~, we conclude that the functions f'k*(z) are extremal also for the maximum of the quantity
f=)JJ1/k(O)/ with respect to all of the functions
fk (0) =
fk (z),
k
=
of the three domains onto which the disk
=
00
Theorem 6. For functions
I z I < 1. h (z), k = 1,
fk (Z),
for k
=
2, 3, that map the disk
1, 2, 3, one
Iz I < 1
univalently onto dis joint domains, we have the sharp inequality
U I,,-; Ii (0)
o. :'; •
I(/, (0) - I. (0)) if, (0) - /,(0»)(/.
(0) -
I, (0» I.
them by a~y fractional-linear trans formation (the same trans formation for k
=
1,
2, 3).
lies on the boundary of at least two
Iz I < 1
is mapped by w
=
§S. Limits of convexity and starlikeness
f'k* (z), it
follows in accordance with the analysis made above that the functions w
=
fZ*(z)
are solutions of a differential equation of the type (35) with C = O. But then, the
The degree of distortion of a conformal mapping can be indicated in another manner, one that is easier to grasp, by means of what are called the limits of
maximum J ~ of the quantity J is calculated, in accordance with (40), from the
convexity, the limits of starlikeness, etc. These tell us which of the disks
formula
I z I < r :s 1
].'0=
(44)
Equality holds in (44) only for functions (39) and for the functions obtained from
1, 2, 3, that satisfy the condition
a; , that are regular in the disk I z I < 1, and that map this disk univalently
onto disjoint domains. Since the point w
ab that are
onto the domain between two rays at an angle 211/3 with vertex at
then the same maximum is attained by the functions
for k
=
and that map that disk univalently onto disjoint do
tity 1 has the same value for them as for the functions {*(z) for k = 1, 2, 3 since
fk (z)
out of all possible systems of
1, 2,3 that satisfy the condition fk(O)
I z I < 1,
mains is attained by the functions w
w=t:*(z)
is meromorphic
Theorem S. Ifpoints aI, a2, and a3 are vertices of an equilateral triangle, then the maximum of the quantity regular in the disk
Let us define new functions
fk (z)
We formulate the results of this section in two theorems:
such that the union U ~=IBk coincides with the entire w-plane. Therefore, there
functions
165
§5. LIMITS OF CONVEXllY AND STARLIKENESS
64 • ') ( al-~ , ') 1. ,r I ( a1, --a j') ( aI-as 81 v 3
are mapped by an arbitrary function of the class S into domains of a
166
IV. EXTREMAL QUESTIONS
specified type, for, example, convex domains, starlike domains, etc.
of
§ 3,
that R e is a root of the equation
Limits of convexity. We define the limit of convexity for the class S as the least upper bound of the radii r of the disks
Iz I
Eq)_! _1 K(p) 8+ 8p2=0.
that are mapped by every
function of the class S onto a convex domain, that is, onto a domain such that the tangent at the boundary points of the domain always turns in the same direction
Calculations show that
as we move around the boundary. The limit of convexity proves to be nonzero. We denote it by Re . To deter~ine its value, we note that if we take a point z = re i ¢ on the circle I z I = r, then the tangent to I z I = r issuing from that point forms an angle 17/2 +
¢ with the real axis. Consequently, the corresponding tangent to the
image of the circle
¢ + arg
Iz I =
r under a function w =
r' (z)
with the real axis. It follows that domain if and only if
fez) € S forms an angle 17/2 +
Iz IS r
is mapped into a convex
I
z1
=
r or, what amounts ro \J\
R";
to the same thing, if and only
if
as the least-Upper bound R s of radii r of disks arbitrary w
=
function~,=
0, that is, onto
Z
I
To determine R s
,
1
z
aarg f
=
z!" (z) -+ 1 f' (z)
I
+ 1
2
I
Z
2
=
f
1
,
Z2
1'1
I - Z2
( 1)
the greatest lower bound of radii of circles
= p, where p > 1, that are mapped by an arbitrary function
w
= F (<:1 € I
onto convex curves, we can show by the same reasoning, but using only Theorem 4
1) R. Nevanlinna [1919-1920].
Iz I =
r, or, what amounts to the same
§ 1,
which gives a sharp inequality for
Rs
" l-e- 2 ---,,- = l+e- 2
1t
tanh
4 =
0.65 ....
(2)
Limits of generalized starlikeness. It is also possible to show what analytic
---=----:,-------';;-----
R e =2 - V3=0.26 ....
R";
(Zf' (z) ) ~ 0 f (z)
I arg (z( (z)/f(z»! for arbitrary z in I z I < 1, it now follows that Rs is a solu tion of the equation log [(1 + r)/(1 - r)] = 17/2; that is, 1)
•
I
Analogously, if we denote by
ffi
is mapped onto a starlike domain
z!' (z) \ 1t \ arg fez) ~ 2' By virtue of inequality (12) of
z 1 S r, where r > 2 - J3 are not mapped by every (z) € S OntO convex domains. Therefore 1)
This means that the disks function w
+
(z) =
Sr
thing,
I-Iz
1- 4z
1z I
re i¢, everywhere on the circle
I
we have
that are mapped by an
possessing the property that arg w always increases
we note that the disk
S r is indeed mapped onto a convex domain if r 2 - 4r + 1 ~ 0, that is, if r S 2 -.;3. Since this is true for an arbitrary function w = fez) € S, we see that Re ~ 2 -.;3. On the other hand, for the function f (z) = z/(I - z)2 € S, It follows that
Sr
fez) of the class S onto domains that are starlike about
'~omains
on that circle. But inequality (6) of §4, Chapter II yields, when we replace the
1-41
1z I
as a point w move~ around the boundary of such a domain in the positive direction.
where z
(Zl" (Z») + 1 ~ 0 f' (z) ~
(ZI" (Z») + 1 ~ !' (z)
1.78···.
a~
absolute value on the left with the real part taken with minus sign
ffi
=
The limit of starlikeness. We define the limit of starlikeness for the class S
if and only if
:tp (; + cp + argf' (z)) ~ 0 on the circle
167
§5. LIMITS OF CONVEXITY AND STARLIKENESS
z1
Sr
r exceeds
Rs
properties are possessed by mappings of disks of the form functions w
=
fez) that belong to the class S when
I
by means of ' 2)
To do this,
we introduce the concept of generalized starlikeness. We shall say that a closed domain is of the type D n (a) if every point of it can be connected with a point a by a broken line that lies entirely inside it and that consists of no more than n rectangular segments. Thus, a domain of the type D1(a) is a domain that is starlike about a point a. 1) Grunsky [1934]. 2) Goluzin [19351.
._-
----
--
...
--
_.
_;
-_._-_._---------.
-
.....
-_...........
---:
·1 ... · ··.'. ~J
Now, let us d~note by Dn (where n
=
I, 2",,) the least upper bound of all
p such that an arbitrary function of the class 5 maps the disk
Di
,;:;;
Iz \ < I
onto spiral-shaped domains. Furthermore, D1
Rs
=
Di = tanh 4' The exact values of D" for
a..
possible lower bounds for
n>
(3)
is mapped by an arbitrary [unction w
=
_
I+EC
g () ~ -
lel
f' (E) (I - EE)
I z I :s 2d n /(1
S maps the closed disk (7)
C+E I
+ec'
:s d"
I"z)1
~+b~"'+ ",
+ I e1
l z I'" -
2~ (~z)
I z 1 (1
- d~ Ie 1 ) 2
-
,;:;;
d~ - 2d~ffi (fz)
< Dn ,
2
0 by a broken line that lies entirely
(5)
+ d~ I ez
,;:;
by taking the limit as d"
-->
D", we see that
2D,,/(1 + D~) ~ D 2 " and hence that
2D"
'+D."
2m (Ez)( 1- d~';:;; d~ -
e(I - d~)) IE 1 (I - d~) z I _ d~ I HI +, ... ,,,.
=
exists an a of moduius unity such that the image of the disk (7) includes
this is true for arbitrary dn
onto a domain of the type Drr([(~)).
2
then the disks (7)cover
n rectilinear segments. We conclude from this that 2d,,/(l + d ~) :::: D 2 ". Since
...
Ir\~d '0 -.;: '"
eI'" -;~d"'ll ; ; : I. - fz I'"'
-
lal = I,
in it and that consists of no more than 2n rectilinear segments because there
I-DI ",;:;;
Iz
2m
I and
point [( ad,,) by broken lines lying entirely in it and consisting of not more than
In~
'+ I+D~ 2D."
(1-_D,,)2
I+D" ,
which proves the first of inequalities (6). To prove the second of inequalities (6), we set ~ = ar in (4), where
d n < r< I, and n >0. Since I EI (I - d~) -t- d" (I -I Ell) _ I EI ± d" 7 1- d~ I Ell 1- d~ i E12 - 1"'-_--I--d-=-,,--'I--=E ,
From (5), we get in succession
Izl -
Ia \ =
both these points, with the result that it is possible to connect them with the
Consequently, this function maps the closed disk
Z=
'(
in (4), where
+ d~). An arbitrary point in the image of this
disk can be connected with the point [(0)
w = [(z) maps the closed disk
9.
~ = ad"
n ~ O. We then see that every function w = [(z) €
< D", into a region of the type Dn (0). Therefore, the function
2
(6)
for arbitrary' m ~ I and n ~ 1.
completely the disk
[(z) € S into a domain o[ the type D,,([(~)).
I( C+3 ) -/(;)
e>/(I -
'-DI"+m~(I-D,,),,,. I-Dm I + Ds,,+m ~ I +D" I +Dm
Iz - I ~d~ I,;:;; I ~d~' d,,< D"
Proof. If [(z) belongs to the class 5, so does the function
I(z -
(I - I E)1)2 IE IS)2 ,
(I - d~
. . ., .
its boundary. If we let a move around the circle
I
(n= 1, 2, ...)
that is, the disk
,
onto a domain of the form D,,([( ad n )). The disk (7) includes the point z = 0 on
I
I ~I < 1..
I-D s",,;;;;(I-D,,)2 I +D" '
\I I +DI"
I are unknown. We shall give the greatest
E(1- d~) d" (I-I EIS) Z-I_d~IEI!';:;; I _ A S I C I ! '
where dn
\
We use the following
Lemma. The closed disk
:s d",
d~
d~1
-
oR
To dlrive the first inequality, we set ~
for arbitrary ~ in
+"
"
and, consequently, we have from (2)
I(,
IE 12 (I
We shall now use this lemma to prove the inequalities
.....•
D", ,;:;; D3 ,;:;; • • •
for ar!:>itrary n because there exist functions of the class 5
that map the open disk
E[2 ,,;;;; I - d~ I EII
from which (4) follows immediately.
Obviously, from the definition itself, we have
d~ -I
I z I :s p into a
region of the type Dn (0).
Furthermore, D"
169
§5. LIMITS OF CONVEXITY AND STARLIKENESS
IV. EXTREMAL QUESTIONS
168
Ial
= I,
'
2 1,
we conclude on the basis of geometrical considerations that the disk (4) is con
2 ,
tained in the annulus
Ie 1
IEI-d" I I IEI+d" l-dnl;I';:;; Z ';:;;'+dnl;j'
(8)
~
;..-- !!ii._
EQ~-
e
au:
~ "':'.7'£-
--- ==
_::i"a ::_, ' ..iEE__
:-
.._-:::~__
~
._._,. __.
':'-'-~E ~':.i~:'~:_
!!'"~.'~.
-:;:~-. -~-':~;:?~~~,~~~,,-~':,:.. _;,"::o·>__ n",g~~~~~:r~~~~i§~~~i~~~-:C::-iQ~7~"~~::~.:.!%:=&..:..:_~~.~~::':::'.~._~~
IV. EXTREMAL QUESTIONS
170
and is tangent to both its boundary circles. If we now let a move around the
circle
Ia I =
question is given by the theorem on p. 174. The proof of that theorem is based on
1, the disks ( 4) completely cover this annulus. Let us now choose
two lemmas, which we shall now prove.
~ in such a way that (j ~I - dn )/(1 - dn I~I ) = Dm , where m> O. This yields I~I = (Om + dm)/(1 + dmD m) < 1. Then, by the same reasoning used to derive the first of inequalities (6), we see that the image of the disk is of the type D2ntm (0) a~d consequently that This yields
(I
Lemma 1 (Grotzsch's principle).l) Suppose that we have a set of n disjoint
simplx connected domains S k, for k
I z I s (I ~I
+ dn )/(1 + d n I~I) ~I + d n )/(1 + d n I ~p .s D2ntm .
annulus-;<:l,\ I < R, where r
Sk can be
I z I = r and I z I = R. (The
regar~ed as strips extending from the inner to the outer circle.) If
these domains are mapped onto rectangles in the '-plane with sides equal re spectively to ak and bk in such a way that the arcs referred to are mapped into sides of lengths a k, then
Dn and divide the inequality by the equality, we
n
,., ak~~. ~ bk ~ R'
obtain the second of inequalities (6).
k=l
From these inequalities with n = 1, we have
I-D m
~
1 + Dm ~
Ii;
If
('-Dt)21-Dm_g D1 I D m_ g
+
,
m
> 2. (9)
•
m> 0
I-D m _"m '+D m ~e 2""' Inlt
>I-
log z
along an arc belonging to the
=
log R (see Figure
the line u
=
log r to the line u
9). Let z
=
gk (0 denote the functions inverse to
o < Tf < ak
"m
-
2e
2.
=
> 0.98.
h (,)
map
R k onto the strips
S; . The
Figure 8
images of the sides of length ak under this mapping lie on the straight lines u
=
log rand u = log
R.
We denote by Ik the area of the strip S~ . Then, we obviously have
In §4 of Chapter II, we showed that the image of the disk arbitrary function of the class S completely covers the disk
I
z 1 < I under an
I wi < 1/4.
r
~ Ik~2rdog~.
We now
k=l
pose the question of determining the length of the longest of the segments with 0 that lie on n arbitrary rays issuing from w
Rk : 0 < ~ < b k ,
(see Figure 10). Then the functions t =
log g k(') =
§6. Covering of segments and areas
=
I
u + iv (see Figure 8). This mapping maps the Strips S k into strips S~' that extend from
t =
the strips Sk onto the rectangles
This is the desired lower bound for Dm. In particular, D2 > 0.91 and D 3
ends at w
< I z I < R, ¢k <
completely filling the annulus.
those mentioned in the lemma. These functions map
that is,
"2
< ¢ktl
simply connected domain obtained onto the plane
tanh (77/4) we obtain from (9) for arbitrary
D m ~ tanh
arg z
Proof. Let us cut the annulus r < 1 z
m ('-D1)m
I-D '+D m ~ '+D 1 =
logy
boundary of one of the strips Sk and let us map the
F rom these we find by iteration that, for arbitrary m,
Noting that D1
( 1)
with equality holding only if the Sk are domains of the form r
('-D 1)2 '+D 1 '
'+
and R .j,.,., and that are bounded by Jordan
\
2
I-D g I+D g
1, 2, ... , n, that are contained in the
single poind, i1il common with each of the circles
! -D n+m ~ I-dn I-I ~I I + D g1Hm -= I +dn ' +1 eI' --,>
=0
=
curves each of ':which has a nondegenerate arc
I-D m_l+dn l-Iel I+D m -'-dn'+Ii;/'
If we now take the limit as d n
171
§6. COVERING OF SEGMENTS AND AREAS
=
0 at equal angles
from each other and that belong entirely to the same image. The answer to ihis 1) Grotzsch [192S].
.,..
._;i:z:::i
ettZ!!!:!!!!!!!!
.~.- --·---·------iz~~,
172
~~:: __ O~.
-=-:
._
";;'_~_,.:~
w.=~=:·;,..:"
_:-~::. ~~~~'·"~.-i .. iilHN~u
IV. EXTREMAL QUESTIONS
I, =
1,1 If. (1;) I' dt d~= ~
"..::~~~:?~~YW~~::~~~=~~~-
173
§6. COVERING OF SEGMENTS AND AREAS
On the other hand, a k b k
__
ak
these domains onto rectangles in the (-plane with sides equal respectively to a" and b k in such a way that the line segments referred to are mapped onto sides of
b
(~ 1M) I' d<) d~ ~ ~ ~, (~If; (C) 1d<)'d~,
length a", then n ~ all
A
(2)
Libk ~B'
the inequality being obtained on the basis of the Schwarz inequality. But the
k=1
integrals
b
r
k
~ Ifk (1:) Ide
~";/
I~
o
<.:
are the lengths of the images of the sides parallel to the sides of l,ength b,,; that is, they are the lengths of certain curves extending from the line u "" log r to the
fg"f~ (()dt?log(Rlr), and,consequently, I,,~
line u=logR. Therefore
(a" Ib,,) (log (RI r»2. If we substitute this into I ~ =11k ~ 211 log (RI r) and divide by (log (Rlr» 2, we obtain (1).
_E1j1w.IJ,JY holds only if the S" are rectangles entirely filling the rectangle of sides ".
A and ~\
\
to those mentioned in the lemma. These
tangles R,,:
0< t < b" , 0 < 'T/ < ak'
and that the images (referred to above) of
all the lines parallel to the sides of length b k be lines parallel to the real axis, that is, that the S k be of the form r < I z I < R, ¢" < arg z < ¢k+l. A simple check shows that in this case equality does indeed hold in (1).
Let I" denote the area of S". Then, on the
11
~ Ik~AB
k'>= 1
and, on the other,
ak
Ik =
bk
a
~ ~ Ifk(C)I~d~d1j;:=: ~ ;k (~lfk(C)!def dTJ ~ (8 Rk
0
~
0
dTj=Qk
bk
bk
B~.
in (2) only when the S" themselves are rectangles completely filling the rectangle of sides A and B is proved just as in Lemma 1.
-
Rtf
~og
I
r
Figure 9
~ogR
.. u
01
Z
strips asserted in Le'mmas 1 and 2 follows from
~
I
y
The possibility of conformal mapping of the
lj
01
2
Consequently, I~=1a"B2Ibk SAB, from which we obtain (2). That equality holds
L
Au
=
one hand, we have
For equality to hold in (1), it is obviously necessary that the strips S" com pletely fill the annulus r
h (() denote the functions inverse functions fk (() map the 5" onto rec
Proof\(analogous to the preceding). Let z
8"
'-
"
"
the fact that every simply connected region having more than a single boundary point can be mapped lair ~~
Figure 10
Lemma 2. Suppose that we have a set of n disjoint simply connected domains
S k> for k
= 1", " n, contained in a rectangle with sides oflengths A and B (see Figure 11). Suppose that the boundary of each of these domains is a closed
onto some rectangle in such a way that four given
-
-
W
the vertices of the rectangle. We prove this last
-
A
1'////'£;/;;;/;;1;
-..
x
accessible points on the boundary are mapped into U I
B Figure 11
assertion as follows: Let us first map our domain onto the upper halfplane and let uS denote by a I, a2, a 3, and a4 the points on the real axis into which the four given boundary points are mapped. (We can arrange for all the ak to be finite.) We then map the upper halfplane onto the rectangle by using the function
Jordan curve that contains a nondegenerate segment (that is, a segment not con sisting of a single point) of each side of length A. (The domains Sk can be regarded as strips extending from one side of length A to the other.) If we map
~
"
~
«
w=oV~-~~-~«-~~-~·
Then, the points a I, a2, a 3, and a4 are mapped into the vertices of the rectangle.
174
175
§6. COVERING OF SEGMENTS AND AREAS
IV. EXTREMAL QUESTIONS
fo (z)
As a consequence of Lemma 2, we obtain the result that the ratio of the sides of
strips Ski and Sk" whose images Lk' and Lk" under the function w
the rectangle is determined uniquely by the mapped domain and the points that are
will have a common boundary segment on the rays L k and will together constitute
mapped into the vertices.
=
Lz
Theorem. I) Let f(z) denote a member of S. Consider n rays issuing from the point w
=
0 forming equal angles each with the next. On each of these n rays
consider that point of intfrsection of the ray with the boundary C of the image of the disk
Iz I < 1
under the mapping w = fez) which is closest to the point w =
Then, at least one of these n points is at a distance no less than w
=
\Il1;;"
o.
from
o.
L1
I
Proof. Let us suppose first that the boundary C is an analytic curve and that the theorem is not valid. Then there exists a system of rays L k , k
=
1, "', n,
that issue from w
=
0 at equal angles such that the boundary points on them that
are closest to w
=
0 are all at a distance not exceeding
d < \11/4
from w
=
O.
c
The function
w=!o(z)=dl/Lf
z
2
az+ ... , a=dl/=f
< 1,
(1 +eiazn)n
for suitable a maps the disk
Iz I < 1
Figure 12
onto the plane with cuts in the form of n
halflines lying on the rays L k issuing from the points A k = e-(a+2kTT)iln for k = 1, " " n that include the point w = 0 on their extensions. Since a < 1, it follows that for r sufficiently small, the image function w
=
f (z)
S
contains in its interior the image So of a
where Q> 1, under the function w
=
I z I = r under circle I z I = Qr,
of the circle
the
fo (z). Let us now partition the annulus
Qr < I z I < 1 into a system of strips by drawing an atbitrary set of rays issuing from z
=
0 in such a way that the following three conditions are satisfied:
1)
The set includes all rays lk containing segments that the function w = fo(z) maps into segments OAk and all rays that are bisectors of the angles between adjacent lk (see Figure 12);
2) it includes ;111 rays that the function w
maps into curves tangent to C;
=
fo (z)
a strip L k ' U L k" which extends from So and returns to So. The entire w-plane exclusive of So and, consequently, the image of the annulus r < the function w
I) Lavrent'ev and Sepelev [1930, 1937]
under
Also, by hypothesis, the number of such subsets of the image in question that lie
u",
and
uk" of these subsets contained in the strips Lk 'U Lk" and adjacent to s. Let us denote by ak I , b k I and ak", b k " the sides of the rectangles into which they are mapped in accordance with Lemma 1 and let us denote by their pre images under the mapping w
fez) lying in r < I z
=
sk'
and
Sk "
I < 1.
Then, in accordance with Lemma 1, we have ~
3) if this set includes a certain ray, it also
constructed in this way can be partitioned into pairs of symmetric
I z I< 1
fez) is partitioned into subsets contained in the strips Lk'U Lk'"
in each strip Lk' UL k" will not be less than two. Consider the pairs
.l.J
includes the ray symmetric to it about the closest of the rays lk' The system of strips S k
=
(a
k,
bk ,
+a
k ,,)
bk "
Now, let us denote by A k' and Bk
2lt
~ -1- 1 .
0)
ogr
(A k" and B k ,,) the lengths of the sides of the rectangles into which the strip Sk' (S k "), and hence the strip L k I (Lk") is mapped (as described in Lemma 1). Obviously, we may assume that A k , = A k " and B k , = B k ". Then, the strip Lk' ULk" is mapped into the rec tangle R k
I
k"
with sides A k
I
I
and 2B k
I
such that corresponding to sides of
length A k I are the parts of the boundary Lk'
U Lk"
lying on So. These last
~~~~~~~~-
176
.........~~-.-.-..--------_·_-------
",~~~~---=~~~~:~~~-===:=:::~~~~c;;:,~..;;;_-=.:~~.,=-----,-
--=-=u~~~~~~~~=:-:-:-~--=--~-=--
mappings ok in Ric
I ,,,"
I
and Ok
II
177
§6. COVERING OF SEGMENlS AND AREAS
IV. EXTREMAL QUESTIONS
Theorem 2.1) Let f(z) denote a member of S. Then, in the image of the disk
are mapped into strips contained
and extending from one side of length 2B k
Iz I <1
I
under this function there are n rectilinear segments issuing from w
=
0
to the other. If we map these strips into rectangles as
at equal angles (2"/n) to each other the sum of the lengths of which can be made
indicated in Lemma 2 and if we take for the continua on
arbitrarily close to n.
the sides of length 2B Ie
I
segments on
I k UI I
Proof. Let us look at the image B of the disk
corre
k"
2BK ,
sponding to the segments that, in turn, correspond under the mapping w
=
we have bk , Ok'
+b
< I z I < I,
"~';;~/./X1//i
then, from Lemma 2
Ok" --=:
2B k , •
A k ,
---
AK ,
nected domain the inner boundary s of which will, for small r, lie outside the
$b,
disk where
and b are both positive, we obtain all' + 011 " b , b~2 II" ll
that have the following properties:
bll , bk " ='
1J II ,bll "
through an angle 2"k/n' for k = 1, .. " n; and 3) the fluctuations in
::>-: 2A II , _ bll,,::O--
BII ,
All' BII ,
+B
All" II ,,'
all"
! (A + Ak")
=0
1) the system contains all tangents to the
boundary C of the domain B:
2) it is mapped into itself by a rotation about w = 0
I w I on the
portion of C between two adjacent rays (though not on the rays themselves) is less than a given
f
> O. Since C is an analytic curve, such a system does exist.
There are nm such rays, where m is an integer. Let us denote them in order by l1, l2, ••• , lnm' Here, lmk+ q is obtained from
But, in accordance with Lemma 1, we have B::
(1 - or) = p, where or - .... 0 as r -- O. Let us partition the part of
2
-+ all'
Iw I < r
the star lying outside s by drawing a finite system of rays issuing from w
011,0 11"
all' all"
:>: __4 ::0-- b ll ,
the function w = f(z). We define the star of the domain B as the largest domain
stra~ht line segment containing an arbitrary point in it with w 0 is contained in it. ', Corresponding to the annulus r < \ z I < R in the w-plane is a doubly con
Figure 13
On the basis of this, when we twice apply the inequality (a + b)/2 > a
< 1, under
=0
0' k" &
where R
~()~tained in B that is starlike about w = 0, that is, that has the property that a
f(z) to the parts of the boundaries s k '
and Sic" contained in Qr
I z I < R,
lq
by a rotation through an angle
2"k/n (see Figure 14). Let us denote by p" (k = 1, 2"", nm) the greatest
~
B kIt = log
Jr .
distance between w
=
0 and points of the part gk ,?f the star lying between lk
and l"+1 and outside s (for k
= 1, 2,'" ,
nm, gnm+1
= g1)'
By means of the func
tion ~ = log (wi p), we transform this system of domains gle into a system of do
Consequently,
ak' 21' - +011") -~--
~( b
k,
bll "
log _1_ Qr
mains h k lying respectively in rectangles with sides of length log (Pk / p) and ak . lK"
When we compare this with (3), we obtain 2'lt' :>2l' . Q. ~ 1. --1-::0-I ' I.e. loglog r Qr But this contradicts the fact that Q > 1. This proves the theorem for the case in
lit
which C is an analytic curve. In the general case, we need only apply this result to the function f(pz)/ P €
S, where p < I, and then take the limit as p -- 1. By the same method we can Figure 14
also prove the following theorem on the covering of the segments.
1) Goluzin [1936a]. A generalization of this theorem was obtained by Bermant [1938, 1944 ].
----_._. -~--~~_....~". ._-""'~=---~--~--"""""'--...,--"~. ---~-~---~~---_."----~~-~------§6. COVERING OF SEGMENTS AND AREAS
179
IV. EXTREMAL QUESTIONS
178
Here, ~~:. ak = 217 where amk+ 1= al. On the horizontal sides -of each of these
n
1:
~-
rectangles is a boundary segment (see Figure 15).
n
Ck~ -~
1
k=1
11
n
IT
Ck'
Ck~O,
(5)
k=1
we obtain .S:~+iry
I
n-I
V
'k=O
\,
Qt/(
~vn-I 'II_log
~
n
log Pmk+l -P-
\
Pmk+l -P-
n-I
!
k=O
n!
log Pmk+1 _P_
k=O
\
\
\
Consequently, (4) yields
ooi
n-I
' .. §
n log R ~ max V log r
lDg.ftJ p
we have, in accordance with Lemma 2,
< (Rp)n r Therefore, by taking the limit as r
ak ~ --.!!:!:.
bk
P
n-1
When the domain h k is mapped onto a rectangle with sides ak and h such that the boundary segments mentioned above are mapped into the sides of length ,
Pmk+1
'k=O
or
Figure 15
bk
1
max 1
II
Pmk+I'
k,;""O
0, we obtain for some II in the closed
-->
interval 1:S [1 :S m
log Pk
n-I
P
Rn <
Therefore,
IT
(6)
Pmk+11
k=O
m n-I
nm
'Y
ak .... bk k=1
m
~ "\1 ~
!tmk+1 Pmk+1
~ '- I 1=1 k=O og - -
=~,
P
~ 1=1
n-l
~
CT.I
or, again using (5),
'k=O
log
n-l
Pmk+l
nR ~ ~
P
Pmk+I,'
k=O
But the system of domains gk is the image of a system of strips contained in the annulus r
< Iz I < R
under the mapping w = f(z) and the rectangles mentioned
above with sides ak and b k are the rectangles mentioned in Lemma 1. Conse quently, from what was said above, we obtain on the basis of Lemma 1 m
~:>-: ~
R =- ' 1=1 I og~ r
'Y
2"
___ :>-: min
R =I ogr
1
~ 7.
I P
p(
n-
1)
I-f. m+ 1
n
. ,., ,
'-i
1=1
If we now use twice the familiar inequality
CT. 1
n-I
~ Ak~ ~ I
n
~ - I = -2". mm 'n 1 olog Pmk+1 k=
P
(4)
P/l - E, PI2 - E,' " ,
Consequently,
k=1
m
.... I Pmk+1 k=Oog---
.. "
.
A2" ", An and hence are respectively greater than
P
Therefore, since ~T=1 al = 2TT/n, we have
n-I
~1'
n-I CT. I 1.- I Pmk+l k=O o g -
Let us now take an arbitrary system of n rays issuing from w = 0 at equal angles (2TT/n) with each other and intersecting respectively gIl' grn+/l"" " ' , g(n-l)m+ 11 • The parts of these rays lying inside the star are of lengths
Pmk+l.
-ns~n(R-e).
k=O
For R sufficiently close to 1 and
f
sufficiently small, this is arbitrarily close
to n. This completes the proof of Theorem 2. As a special case (with n = 2) of this theorem, we obtain the result that there exists an interval of length arbitrarily close point w
=
to
2 that passes through the
0 and that lies entirely in the image of the disk
1
z I < 1 under the
mapping w = fez) €
that is, by setting z = rei
S. Furthermore, when we rewrite inequality (6) in the form n-I
1
~
-2; ~
log Pmk+I, •
_~
2n n ;?: log R
k=O
and take the limit as n
-> 00,
it;
we obtain the following inequality: 1)
~
j.
log P dtJi ;?: log R,
where w = pei'!J and the integral is over the boundary of the star of the image of the
< R under the mapping w
=
~ log p dtJi \
0=3 (~ f¥2 dZ) = ~
fez).
I 2"1 r~ p'd tJi~'lrR,
~ log ~- d'f - ~ log, d (ljJ -
under the mapping w = f(z) and the area of
x + iy and using the Cauchy-Riemann conditions
~ and a small circle
= r r (au av _ au av) dx dy = ~ ~
Izl
ox oy
oy ax
\ \ IF' (z) I' dx dy, ~ ~
Izl
y:
Iw I = f.
From Green's formula, we have, just as above I
J=I -
log P dljJ=
t~ ~~~~,
(9)
I
with a double integral taken over the domain BE' Now, if we denote by SR the area of the domain B bounded by the curve ~ (that is, the area of the star) and
I w I < J-S R/1T =
not belonging to the domain B than in the part (of equal area) of
the domain B lying outside that disk, we conclude that
~". 1WJ'~ ·< J
1) Goluzin [19~7aI.
2) GoluzinlI937~]. 3) Bermant [1938a].
(8)
I
~ log p dljJ -
r du du ,,;;
~ UdV=.\ U:~dX+U~~dY
const on the radial seg
note that the integrand in the last integral in (9) is no less in the part of the disk
Z
Consequently, from Green's formula, we have, taking z
t/J =
Let us now consider the doubly connected domain BE bounded by the curve
P(z)=log!(Z)=U+1V, P(O)=l);
1
log, dljJ,
+ ~ log r d(jl ~ 41t log R.
A
Proor. Consider the star of the image of the disk I z I < R, where R < 1, under the mapping w = f( z) and let ~ denote its boundary and 1 the curve corre sponding to it in I z 1< 1. This last curve is a closed Jordan curve. The function
h
~ log Pd'f - ~
(7) takes the form
the preimage of this star is not less than 1T 2 •
I z I < 1.
'f) =
'f) d log,
ments lying on ~ and log R on the remaining arcs. On the basis of this, inequality
~ log P dljJ
Theorem 3'. If f(z} is a member of S, then the product of the area of the
is regular in
1
But this last integral is equal to 21T log R becaus e
Starting with other considerations, we can obtain the stronger result: 3 )
-
+ ~ (ljJ -
h
'I
under the mapping w = fez) is no less than 1T.
Iz I < 1
d'f
~logpd(jl=~ logrdljJ.
Theorem 3. 2 ) If f(z) is a member of S, then the area of the star of the image
star of the image of the disk
~
(7)
1
i.e.
to the area of the star. By letting R approach 1, we obtain
I z 1< 1
log
1
1
=
where the integral is over the boundary of the same star. This integral is equal
of the disk
A
1
-'Ori\the " other hand, if we integrate by parts, we obtain
On the other hand, if we raise (6) to the power 2fn and if we apply inequality = "", we obtain the inequality
+ ~ log r d'f ~ ~ log, dljJ + ~ log Pd'f.
h
(5) to the right side of the resulting inequality, then, when we take the limit as
n
~ d(ljJ-(jl)~O
log
or
1 21t
disk 1 z I
181
§6. COVERING OF SEGMENTS AND AREAS
IV. EXTREMAL QUESTIONS
180
Remembering that
r~ J
I .,
du I"Id~
=
d,!, ~ ~ jI§ -pdp -p-t
,r~":_IOg,).
,<,.,<. ~ 2. (lOg If
-if
J'" ~=,.=~~~==,~
__ ~~:.
182
.. _.
');;
•
-.=-
-'ii~ ~,,_.-..
§iL ~"__ # ..:.
----.
-5iii-
.~
~;;::;.
~'
:..... :..-~ ... -.
F5F5- ._._..
:eEE';i
..
E
,.......
..
--
=....
_.
183
§7. LEMMAS ON THE MEAN MODULUS
IV. EXTREMAL QUESTIONS
-:
~
know bounds for certain moduli. These bounds are provided in the following lemmas.
~
Lemma 1.1) Let f(z) denote a function belonging to S. For A> 0 and 0
log p dt = 21t log 5,
Iwl=<
T
< 1,
r
we obtain from (9)
r log pdt ~ ~
1t
logS R
I
It
I z I = f.
Thus, if we denote by
S R
r
s:.
k
-#r ~ /f(re
(11)
I9
+ 1t log -
'It
~ 41t log R,
Therefore, when we set f(z)
'iI , we obtain
= ReI
2"
~\ dr ~
approaches the area s of the image of the star, and we obtain
S· s~ 1t'a,
\/(reIO)IAdB=!r
Iz I =
where Cr is the image of the circle
Jr RAd,
w
As a consequence of this result and of the property of minimization of the
r
r under the mapping w
=
f(z)' Now,
then use the Cauchy-Riemann conditions, we obtain successively
We note that all these theorems except Theorem 1 remain valid for an arbi
I z I < 1. 1)
consider the Riemann surface onto which the disk
~ RAdiP -
The only difference is
that, in this case, instead of the univalent image of the disk
Iz I < 1
I z I < 1,
we need to
is mapped bijectively
~'" "
,',
Lemmas on the mean modulus. Bounds for the coefficients
J
I
·0
~~:
A
R diP =
=
by the function w = f(z) and, consequently, the star located on that Riemann sur face. All the above reasoning is valid word for word in this case.
u + iv, such that C encircles the point w = 0 but is itself encircled by C'.
Green's formula to the doubly connected domain B bounded by these curves and
only if f(z) == z.
TT
=
Denote the polar coordinates of points in the w-plane by Rand
area under a mapping (see §2 of Chapter II), we obtain a supplement to Theorem 3: the area of the star mentioned in that theorem is equal to
\
cl
~: du + R :
A cR
r ARA [0 logau R 01fJov _
JBJ
.
A
dv
alog R 01fJ] du dv
ov au
= ~ ~ AR A [ (OI~~Rr +(OI~Rr]d(1dv=A S~ B that is, we obtain the formula
Let us now investigate the influence of the univalence of a mapping on the values of the coefficients of the mapping functions. To do this, we need to 1) Goluzin [I937a] and Bennant [1947).
(2)
let C and C' denote twO closed analytic Jordan curves in the w-plane, where
which completes the proof of the theorem.
§7.
0
alog If (re I9 ) I_ aarg f (rei B) alog r 06
1t'aR4.
trary function f(z) = z + .. , that is regular in
k
\ 1/(relo)IAolog/!(reIO) I dB. 0
But, from the Cauchy-Riemann conditions,
As we let R approach 1, the area 5 R approaches the area S referred to in the S R
k
)I AdB= .~ Olf~:iO)IA dB=A
u
sR
SR' SR ~
star theorem and
( 1)
M :P)A dp,
I z I =P
From (9)-(11), we obtain
that is,
r
Proof. We have
} log dr.p:=:;; 1t log
.
~ A~
M (p)=max I/(z) \.
the area of the
domain bounded by the curve l, we have
S 1t log --8 It
lo
I/(re ) IA dB
'~re
Analogous reasoning is applicable to a doubly connected domain bounded by a curve l and a small circle
~.
21t
( 10)
•
<
1) Prawitz [1927-1928]. See also Goluzin [1949d).
B
RA-2dudv;
.. ~ ..
.....
__ -_~·~2b~~---_F-_~-;:o~~~-:_~~__ __: -.
-~ --~~~~~=-"""----=:.--==-=~~"':"'~==="L"_""""""""'''''''''''''_'''~'''~'''.IlI"m"_ __ =!!!=!!!!!!"",,~!!!!!!;;;;;;;;;;;;;;;;;;~;;;;;;;;;;;;;;;;;;~~,",,,~,",,,~~~fWJ~m~_iIll~~ry=n~"'~':"'''-!1i-i.~_~_~~~iij __ ~~ ..s.i~iirli!.i.lii-&d~~~~~'!;:~~~~~~~~~~~~~
184
IV. EXTREMAL QUESTIONS
§7. LEMMAS ON THE MEAN MODULUS
~ RA d - ~ R" d= A ~ ~ RA-<J dll dv. C'
C
.!. -
(3)
f; (z) = ZP-I f (zP)P
B
t
185
f' (zP),
so that From this formula with A> 0 (it obviously holds also with A < 0) we get the ine quality
~ R Ad ~ ~ RAd. C
(4) ~
C'
In particular, if C I is the circle \ w I = R 0, then (4) yields
~
"!"-t I-"!" .! f'(z)=zP fez) p f;(zp).
RAd ~ 21tR~.
Let us denote the left-hand side of ( 6) by I, replacing above expression. We obtain
(5)
l=r
C
If we apply inequality (5) to the case in which C is Cr and R0 = M(r) + (, where ( > 0, and then let ( approach 0, we obtain from (2) the inequality
b
l~r
~ If(reiO)IAdB~21tAMr)A,
2~ ~1j(reiDJI' -IF (reiD) IdB·
'\ <~ -V 2p(l+A)-2 \
r
where
e
i
2
(8)
0
2"
2
,
2~ ~ \f(reiO)12A+2-PdB~(2A+2_~)~ o p ~
. 2A+2-! M(p) Pdp. P
(9)
With regard to the second integral, we have from equation (7)
{
~~)IIP f~(p»2A+2-% d, l-r
~
p
p
:m
1
00
2~ ~ If; (zP) I' dB = n~o(nJl+ 1)'il1 c~) lir'lI.
(6) But, for 0
m
m
x (l - x) ~ max (x (1 - x» =
M (p)=max If(z) I,
(10)
< x < 1 and m > 0, O::::x~l
Izl=p
(m
+mml)m+l 1
p is a positive integer, and A + 1 - p-l > o.
1
.
1
~_
=m "(1 +~r+l ---= em •
Proof. Let us consider the function
from which we get
1
00
fp(z)=!Vf(zP)= ~c~)znp+\ n=O
m~--
-.-" ex m (1 - x) ,
(7)
O<XO.
If we now set m = (np + 1)/p and x = r, we obtain
which, as we showed in §4 of Chapter II, belongs to the class S.
p
np+ 1 ~ np+1 f!r-P-O-r)
From ( 7), we have Equation (10) yields 1) Goluzin L1949cl.
b
Let us find a bound for each of the integrals under the radical. Lemma 1 provides a bound for the first of these:
./
\
1
2
D
r
\
I
~ If(relO)IA+l-Plf~(zP)ldB,
1 iO 2A 2 If(re ) 1 + - p dB. 271: ~ r P II' (zP)l'il dB.
2~ ~
1
integration of which with respect to r from zero to r leads to inequality (1). This completes the proof of the lemma. Lemma 2. 1) Let fez) € S. For 0 < r < 1 2"
2n
2"
in it with the
from which we get
2"
:,
..!.. P -I
f' (z)
(1)
IV. EXTREMAL QUESTIONS
186 2"
I
2
By using this last inequality, we obtain from (16)
np+ I
00
2~ ~ IIp (Zp) 1 r P dO ~ e (1 ~ r) n~o(np + 1) I C~) Ii r-----P- • 2
(12) 1
21t But the last summation is equal to (1111) times the area of the image of the disk
I z I < r l /2p
under the mappi~g w
(maxI zl=r1l2plfp (z)1)2; fhat
fr, (z). Consequently, it does not exceed p i~>j,t__~~e~_.notexceed maxI zl=I,-lf(zW/p =- M(yr)2/ • =
Thus, it follows from (12) that
~I
2"
\
21t ~
o
187
§7. LEMMAS ON THE MEAN MODULUS
2"
~ II
,
iO
(re ) IdO
~
Y
~+I _1..
_1
P M2M'2 P r P M"r 2p I- r . (1 - r)2/1 I = (1 - r)'"
Here, M II depends only on a and p. Inequality (15) then follows for But (15) obviously holds also for 0
Yz < r < 1.
< r'S Yz when M1 is suitably chosen. This
completes the proof of the lemma. Now, we have some theorems on bounds for the coefficients of functions
If'p (ZlIP) \2 r21P dO ~ e (1 P-
r)
M
0/')2 IP•
(13)
belonging to the class 5.
Theorem 1. 1) For functions fez)
=
I';;' =1 cnz n €
5, we have
Using inequalities (9) and (13), we obtain inequality (6) from (8). This completes
lcnl<en,
the proof of the lemma.
Lemma 3. Let f(z) denote a member of 5 such that, in the disk
I z I < 1,
Mizi
Proof. Beginning with the integral representation of the coefficients
(14)
I/(z) I ~ (I-Iz I )a'
C n
=_1
21ti
2"
11
u
where M1 depends only on M and a.
I z I=p
I/(z)l~ (I~ P\a'
by applying Lemma 2 with A = 0, we obtain, for 0 2"
2~ ~ If' (refS) IdO ~ f i r
sible since a
j
> Yz).
p>
(l_p)2- p
1
1- r
O
But, from inequality (10) of §4 of Chapter 11 with 0 < P < 1, we have M(p) pl(I - p)2. Consequently,
< r < 1,
1.. ~+l"
M 2r P 2P
I ~ M (p) 1 Cn I ~ ----n r p- dp.
j
2a
P
1 \
P 2
. (I_P)(2- p
(l-r)P
1 in such a way that (2 - 2/p) a
>1
dp ,,; r I -
ff
)a
dp.
a
where M I depends only on a and p.
2)
o (l_p)2- p
~
M'r
(1- r)
dp
ICnl~rn ~ (I_p)ll=
(16)
1 rn-1(I-r)'
Here, r is an arbitrary number in the interval 0 have
< r < 1 . If we set r = 1 - lin, we
(which is pos
2-!
d( P
'S
r
l-!
We then have
r _ _p!-I-;-(--*-;;2-;-)_
(18)
r
M(p)= max
Let us now choose
n=2, 3, ... , O
dz,
zn+1
If we now apply Lemma 1 with A = 1, we obtain
Proof. Since, by (14),
o
j(z)
~
2" 1 \ Icn I ~ 21tr n ~ I/(re i6) IdO.
(15)
M1_\a ,
\
I z I=r
we have
where M and a are independent of r and where a> 1/2. Then, for 0 < r < 1,
2~ ~ If' (re i6) I d(t <
(17)
n=2, 3, ...
I cn I ~ 1_
nn 1\n
1=
(
n 1
1 )n-I + n -1 < en.
P
(2-~)a-I'
Thus, we have obtained inequality (17). This completes the proof of the theorem.
P
1) Littlewood [1925].
__
"0--
_-
__
-
- "'-_, _
__ _
=.
~_=---~
§7. LEMMAS ON THE MEAN MODULUS
IV. EXTR)MAL QUESTIONS
188
_~
Theorem 2. I) For an odlfrmction fez) = :£':=IC2n_1Zh-1 €
Let us take r = (2n - 1)/(2n + 1). Then, remembering that
5
1 1 1
\C n
l<2 4 32 e2
I
(19)
=3.39 ....
= l- .. /
1
Proof. We have
r
nc n
= ~1ti - \.) f'----zn (z) d z,
n=2, 3, ... ,
2(1-r)
r
2
=M
~ If' (re i8) \ d6. = 4,
we have
I ~ _1 .. 0 ... 1M (v;:-) III
< P < 1,
II
J!
M(p)p ~
{
l-r
III
(20)
dp
<2
(I - r)1/1
(l -
1
/t
z
(I - pl)·/1
)-1
< en,
31 /2 elll n,
2: nz",
(21)
2:
(22)
11=1
=
co
zllll-t,
(21) and (22) are extremal. In connection with this, mathematicians came to feel
~
r It - . r I I••
(1 -
r)·/1
-6
1
problems dealing with the functions that we have been considering, the functions r
V
1
3" (211)3+'" "
Theorems 1 and 2 with regard to order are sharp. However, in many extremal
( (I
r
+ r)1
dp pl)·/1
(1 )1 /8=
1 e r"--f(l-r) I
that these functions are also extremal in the problem of sharp bounds on the coef ficients, that is, that the inequality
"4
len I S n
provided the answer to the question
in the case of the entire class 5 and that the inequality
)1/8
- ' · - ' (11 - ' )
~
1
which belong to the class 5, show that the bounds on the coefficients given by
+
r
1
211 +"2 (211)1 +
co
(I-Z)I=
z
1
r1/4
111
1)( 1
2n
The functions
1 -Zl
P Iidp
Jo
1-
11=1
1
rn -. ~...!..111 e
=
r
\
(
that is, we have obtained inequality (19). This completes the proof of the theorem.
When we substitute this into (20), we obtain
rn -e . nICn\~...!..1I1
)
(I 1 1 I ) I 1 I 1 ) ( "3 {2n)2 + r; (211)4 +... "2 (211)2 + 4" (211)t + ...
nj CII I
•
p
t
1
2
we obtain
M (p) ~ I-pl'
II
(1-2~) 11-
... +
we have
r
2n
(I +..!) log (I +~) - n (I-~) log (I _~) ~ ~ ~ ~
= ne 1+
But if the odd function f(z) is a member of 5, then so is the function g(z) = fez '12)2. To see this, suppose that g (z 1) = g (z 2), where z 1 and Z2 are points in the disk I z I < 1. We can easily show then that z 1 = z2' H we now apply inequality 2 (10) of §4 of Chapter II to g (z1 we obtain If(z) \ ~ I z \/0 - I z 1 ) for I z \ < 1. Consequently, for 0
(I + -.!..-) II +2
1)111+1
1 1 1 I = ne'l [(- 1 + 21i1 )( 2n1 -:f(2n)l+ 3 (211)3 +
o
Consequently, from Lemma 2 with ,\ = 0 and p
n-=rnVe
+
(2n-I)ln-l =n
2"
n I Cn I ~ 21t;n-l
n IC
(2n
O
\zl=r
From this we get
189
\en I ~ 1
provided the
answer in the case of the subclass of odd functions. The first inequality with
,'1'3'1"
-1/1
1
r n- 2 (1 _ r)
1) The elCistence of an absolute constant bounding \ c n \ was proved by Littlewood and Paley [19321. For the numerical value of the constant, see Levin Lt9351.
arbitrary n for the entire class 5 has not at the present time been proved either true or false. This question constitutes a widely known problem today. Except fgr the inequality \C
31 S 3.
1
)
\e21
~ 2, we have only succeeded in proving so far that
In the general case, all that we have are various improvements in the
o 1) Loewner [1923]. (At the present time, it has been shown that 3 of the supplement).)
IC n I Se
(d. §2,
inequality, for example, the inequalities \c n \
~ %en
191
§S. THE RELATIVE GROWTH OF COEFFICIENTS
IV. EXTREMAL QUESTIONS
190
and
1c n I
~ (el2)n
1
+ 2. )
lenl
~cna, where a~o, holds for all n=/lk+v for k=O, 1,···. Then, the
ICn 1 ~ c ' nr:J., where c' depends only on c, a and /l, holds for all 2, 3, ., '. For a < 0, this assertion does not hold for all /l and v.
With regard to odd functions in the class S, it has now been proved that there
inequality
exist functions for which \c 2 n-l \ exceeds unity for given n ~ 3 (for
n
Ie 51
see
§9). Furthermore, there even exist functions of this kind with real coefficients. 2)
=
Proof. We define CD
§s.
The relative growth of coefficients of univalent functions
11'0," (z)=
!
Now, let us look at the question of the relative growth of the coefficients of Then,
univalent functions. As a preliminary, let us prove the
I'
Lemma. All the minors of the /l x /l Vandermonde determinant, where /l ~ 2, compose d
0
f
t he num bers 7J v
=
e 21Tvi! iJ. , where
cl'.n+.·zIW+,', ,,/'= 1, ... , p.•
,,=0
~
I(z)=
1, ... , /l are nonzero.
J/ =
:~~ =
II'-, .' (z).
Now, from the conditions of the theorem, we have in the disk
Proof. We set
1Jl' •.. , 1Jp.
A=
CD
IIp..• (z) I ~
2
Q
1Ji' ... , 1Jp.
!
(1)
1
Iz I < 1
CD
IC'ln+. I rp.,,+· ~ cr
,,=0
!
(!J.n+ "/)"rp."
,,=0 ro
CD
1Jp.1 ,
•• , ,
~cr!J." ...., )1 (n+ltrP."~crp'''(1+2'' V n"rP."), \.i.J
1Jp.P.
We denote by A •v' the cofactor of the element 7J~' of this determinant. In v
,,=0
where r
=
I z 1.
It then follows for a
accordance with the properties of determinants,
p . .o
!
1Jk A•0, •,,,,,:,,,,,
{ /1 for Il = 0
'0 = I
,,/',
for k -=i::. ,,/'.
k= 1' ... t
It'
r'
On the other hand, if a> 0, when we apply inequality (11) of §7 with m
1," . ,/l, and add
!J.A., .' = A1J., -'.
0, it follows that A v • v' the proof of the lemma.
f-
0 for v, v'
=
(p.nt ~
I, ... , /l. This completes
Consequently,
I"
Theorem 1. 3 ) Let f(z) denote a function belonging to S. Let /l and v
J/
~ /l. Suppose that the inequality
p.n(2a)" r 2"(1-r)"
J 1'-.'
denote a pair of natural numbers with 1 ~
=
/In/2a and x = r, we obtain =
the resulting equations, we get, the properties of the roots of unity,
f-
0 that
I ~ 1- crrl'- -_ clr IJ"p.. • () Z 1- r'
if we multiply these equations respectively by 7JI/, for k
Since ~
=
,,=1
2a. Cl
W
lJon
() ~ cr!J. (" + (12_ ar)" ki V r2) ~ "".>-_. Mlr Z I~ n=l
Thus, in the disk I z 1 < 1 we have
Mlr
l) Goluzin [194Sa1. Bazilevic [1951] and, independently, Milin (see Lebedev and Milin [1951}) have given a sharp bound for in rhe class S. The bound given in rhe text was found by Milin (see Lebedev and Milin (19491). For an improvement in the bounds for the coefficients of odd functions, see Alenicyn [19511. (In 195~, Gong Sheng (Kung Sheng = Kung Sun) showed that, for the coefficients of odd functions, ICn I < 2.55 (see §2 of the
lenl
I
supplement>. )
2) Schaeffer and sfencer [19431 and also Goluzin [1949c].
3) Goluzin [1948b .
f.(z)I~.oM2~0, with the same M2 • We set 7J v
I
= e
21TV
I
i/iJ., J/' =
(3)
1,"', /l. For fixed z in the
disk
1zl < 1, let us look at the values of If(7Jv' z)j, v'
=
that the greatest of these absolute values is obtained for v Theorem 7 of §2 ( with n
I
=
2) applied to If(7J v z)1 and If(7J v 1 we obtain the Il - 1 inequalities
v'
inequalities
1,"', Il· Suppose
=
II (7j" z) I ~ 1Mrr '
I
z)\' where v' -!
Vb
(4)
= 7j,fp., 1 (z) + 7j~11'-, 2(z) + ... + 7j~1p.. p.'
= 2 and a = 0, this theorem is a di
rect sharpening of Theorem 2 of §7 in the sense that it establishes the relative
,
1
Ilc nt ll-lc n llE;cn 4 Iogn, n=2, 3, ... ,
(5)
VI'
Now, let v. denote any of the numbers 1,"', Il distinct from v. Consider
(9)
where c is an absolute constant.
where v =
1,"', Il and v' ~Vl:
Proof. Here, we shall need a more definite form of sharpening of the dis tortion theorem than that provided by Theorem 7 of §2 (with n = 2). Specifically, ,
1,'7j~' =
{ 1 for s=V*, s =F v*, 0 for
"1:-'1
let fez) denote a member of 5 and suppose that the maximum value of If(z)1 on s~1
- , ... ,
(Jo,
(6)
s =F v.
the circle
I z I = r,
,
where 0
ing inequality (16) of
According to the lemma, this system has solutions. Let us denote these solutions I
n do not even approach zero for
odd n. This completes the proof of the theorem.
have
the system of Il - 1 linear equations with the Il - 1 unknowns Y v I
by Yv' . For ea c h v
Ilk +
Theorem 2.1) For functions fez) = I.~ =1 C n zn belonging to the class 5, we
+ 7j~1p..2(Z) + ... + 7j~'/p., p.(z) I E; 1Mrr , '/' =F
!
C
=
class 5.
Therefore, inequalities (4) can be written in the form
\ 7j.-!p., 1 (z)
are satisfied for arbitrary even Il and v for all n
boundedness of the even and odd coefficients of functions in the class 5. We now give bounds on the relative growth of successive coefficients of functions in the
It follows from (1) that
(7j,.z)
Ie n I ~ c n a.
We note that, in the simplest case of Il
=F vI'
193
THE RELATIVE GROWTH OF COEFFICIENTS
v with k = 0, 1, " ' , whereas the coefficients
Then, from
V I .
where M depends only on-Il'
I
§s.
.~
IV. EXTREMAL QUESTIONS
192
§ 2 to the
is attained at the point z l ' Then, by apply~
function F «() =
I
(7)
M'r IIp., '. (z) IE; (1- ~)1+"
(8) =
I z -- Zl II I
Thus, the assertion of the theorem is proved when a ~ O. That this assertion is
IE; 1_2r rll .
(10)
Let us now prove inequality (9). From
1,," , Il but
But now that we have proved inequality (3), it follows immediately on the . • baSIS of Lemma 3 of §6 that 1c n I ~ c ' n a., where c I IS the c I 0 f the theorem.
2
(z)
This is the sharpened form of the distortion theorem that we shall need. 00
v. ,ft v. But in accordance with (2), such an inequality holds for v. = v. In view of this, we obtain from (1) inequality (3) with M2 depending only on Il' a, and c.
<0
I
that is, we have
where M{ and a depend only on Il. Using (2), we obtain from (7) the inequality
not valid for all Il and v when a
I. and to the points
1 _,,;:::_2_ l - r2 1 I _1_ _L _ Izl-zl-----,::JE; f(Zl)-f(z) E; If(Zl)1 l-lf(z)I---=lf(z)/'
+ alp., , (z) I E; t~rr'
where M~ depends only on Il, a, and c. This is true for all v.
€
we have
obtain the inequality
IIp., '. (z)
1/f (1/()
1 1 ~I=-' ~~=-, jzl=lzII=r, Zl Z
we multiply inequality ( 5) by \Yv' \ and add, replac ing
the sum of absolute values on the left with the absolute value of the sum. We then
< r < 1,
(z-
ZI)/' (z)=
-
ZI
+ ~ [- (n+ 1) CntlZ +n£'nl zn I
n=1
we have
- (n and, consequently,
+ 1) CntlZ + nCn = I
2~i ~
(z - zl)/' (z)
z~~i
1:1=7
is shown by the example of the function
(22) of §7 for which c 2 n = 0 for n = 1, 2, .. '. Consequently, when a
< 0,
the
1)
mend.)
Goluzin [1946e]. (This result has been improved by Hayman (see §2 of the supple
--------
-------~~~~~~~--=~~~~~~~-~ ~
~
~
§s.
IV. EXTREMAL QUESTIONS
194
with \c r
l(n+l)lcn+l\r-n!cn\I~271lrn ~
I < c,
!(Z-ZI)f'(Z)ld8.
Let ,\ denote any number in the interval - 1
< ,\ < 0 and set 11 =
-
~
I
Finally, if we set r = n/(n +
0,
we obtain inequality (9) from (14) for n suf
theorem.
(z - ZI)1' (z) 1 d8
Theorem 2'. For odd functions f(z) = I:: =1 c 2n-l z2n-l belonging to the
= ;71 ~
zili/(z) 1fJ. .I!(z) \A\I' (z) \ d8
IZ -
class 5, we have
I zl=r
1 1-1'-
~2
~
2r2 )fL • __ I ( l-r2 271
Ilcn+ll-lcn_I'I~cn-4logn,
l/(Z) IAIf' (z) I d8.
I z I=r
C n+l
~
Proof. Let us represent f(z) in the form f(z)
I r -- n I Cn II
/1
1!
Yi
2
:0::::
.\11
r
2
~
r
-
where ,\ and p are connected by the condition ,\ + 1 - l/p
p,
( 11)
~(I_p)2'
1
+%yp
----:........0.'1-"2 1'0+-+r(1-r)
I"c
~
0
P
2
2A
p
Now, let us choose ,\ so that 4(,\ + 1 - l/p)
(l--p)
+( -%
1
2
- (n+ l)C n+IZl
r (1- r)
~+.!.V
I) dp.
r
=
-% + l/p.
(13)
r (1 r) T
=
log 1- r'
z
2711
~
(zg -
Zl) g (zg)
_2.. 2
dz
g' (zg)il&+l'
1-1-y"'T Let ,\ denote an arbitrary number belonging to -1 < ,\ < Y2 and set 11 = -,\ - Y2 so that ,\ + 11
= -
Y2. Then,
l(n+l)lc n+llr-(n-l)lc n
I
D
1) Cn"_l] zn+l,
\3 (Z2 - ZI )/' (Z) ~ ntl
111=Vr
P
Furthermore, if we set p = [log (1/(1 r))] and assume r to be such that p> 1,
C
+ ~ [- (n+ 1) Cn+lZl + (n -
I-Vr
(13) reduce s to the form
is
r with
1) Cn_l
= ~ ~l
(12)
P
4
< r < 1,
Iz I =
co
-Zl
+ (n -
4 A+I-
.. /l~gl+Yr .
+ PYP
0
from which we get
Then, the quantity (12) takes the form 2
I z I = r,
=
f(z) replaced by g(z). We now have
1
1; that is, let us set ,\
=
Jg(z2), where g(z)
n=2
so that the right-hand member of inequality (11) does not exceed
2
(15)
attained at z = z l ' Then, just as above, we have inequality (10) on
(Z2 - Zl)/'(Z) =
> O. But
M(Vr)~ vr 4 --=(I_Yr)2~(I-r)2"
M(p),;:;::-P-
.
=
[f(z~)]2 E: 5. Suppose that the maximum value of \g(z)! on
2
M(Yr)P~ M(p)2A+2- p d I-
n=2, 4, ... ,
where c is an absolute constant.
If we now apply Lemma 2 of §7, we obtain
+ 1) I
(14)
ficiently large and hence for all n = 2, 3, .. '. This completes the proof of the
Izl=r
r n I (n
I
C
n I Cnll ~rn+1 (I-r)"/t log I - r '
A, so that
,\ + 11 = O. From (10), we have 1 271
where c is an absolute constant. As a result, inequality (11) yields
I(n+ 1) ICn+l! r -
I z I =r
195
THE RELATIVE GROWTH OF COEFFICIENTS
~ I~
~
til
g Iz -zlllg(Z2)IIJ.'lg(zg)\AIg'(Z2)!dO
271r 2 I z I = Vr
~
!!_ 2 ~ I g(z') IA\ g' (z') I dO. 271r 2 (1 __ r)fL I z I = r
------~-
196
~-::::-~~=---,-----:--===;
IV. EXTREMAL QUESTIONS
§9. SHARP BOUNDS ON THE COEFFICIENlS
197
We get a bound for the last integral just as in the proof of Theorem 2. As a
(I-'- ('t) -I A('t) D(2e-' - tJo ('t) -[ A('t) I) ~ 0,
result, for r sufficiently close to I, we have
l(n+I)lcn+t1r-(n-1)lclI_t1I~ -+1 11 C
1
a lOg t
1
"
because the left-hand meml}er is, for 0
where c 1 is an absolute constant. For r = (n - O/(n +
0,
< r(
v, equal to (e -II _\'\(r)1)2 and, for
r> v, we have e -II ~ 11 (r) ~ I '\(r)l. From this we obtain
(1_ ,)4
,2
tJo' ('t) - A' ('t) ~ 2e--' (I-'- ('t) -I A('t) I).
this yields inequality
(15), proving the theorem.
(4)
Integrating equation (4) from 0 to "", we obtain in accordance with (3)
The inequalities given by Theorems 2 and 2 I are undoubtedly not sharp, but they are the best that have been obtained.
00
00
•
00
~ I A('t) I d't ~ ~ I-'- ('t)d't = ~ e-' d't + ~ e--~ d't o 0 0 •
= ('I +
1) e-',
§9. Sharp bounds on the coefficients As we stated above, in addition to a sharp bound for the second coefficient, we have a sharp bound for the third coefficient for functions in the class S. To illustrate the nontrivial nature of solving certain problems on the extrema of the coefficients, we now present a more general result. Its derivation is based on
Consequently we obtain inequality (2). From what was said above, we easily see that equality holds in (2) only for the function '\(r) = ±Il(r). This completes the proof of the lemma.
Theorem 1. 1) For functions 00
f(z) = ~ cn z ll
Lemma 1. Let '\(r) denote an arbitrary real function that is continuous for
0< r < "" except possibly at a finite number of points at which there are discon tinuities of the firs t kind. Suppose that I'\(r) I :5 e -r for r ~ O. Then, by setting
11_1
belonging to the class S and an arbitrary real number a in the interval 0 ~ a we have the sharp inequality
I ca -
00
~ Ai('t)d't=(~+ ;)e-i .,
'1>0,
(1)
we have
I~A('t)d'tIE:;;(~+])e-·.
(2)
o
Inequality (2) is s harp for each given v with equality holding only for the func tion '\(r) = ±Il(r), where Il(r) = e- II for 0 < r< v and Il(r) = e- r for r> v. [We note that a v satisfying condition (1) always exists because the left hand member does not exceed e- 2rdr = Yz, and the right-hand member decreases
JO'
from
Yz
In particular Ic 3-'
2
IXC: I ~ 2e- 1- II
+ 1.
(5)
:5 3.
Proof. It will be sufficient for us to consider a function fez) € 00
to 0 as v increases from 0 to
"".J
co
00
ca-lXc~=4(1-1X)(~ e-~k('t)d'tY -2~ e-i~ki('t)d't. (6) o 0 Since the function e i f3 f(e- i f3 z ) for arbitrary real f3 belongs to S' whenever fez) does, the maximum value of IC3 - a d I in the class S coinc ides with the maximum
ad). = e i8 (r),
00
v
00
~
tJoi ('t) d't =
~
ffi (ca-elC;) = 4 (I-IX) [( ~ e-~ cos a ('t) d't
00
e-i ' d't +
~ e-'~ d't =
('I +
~) e-"
o
00 00
= } A' ('t) d't.
(3)
Now, for an arbitrary function '\(r) of the type in question, we have for r ~ 0
§ I).
For such functions, formulas (23) and (24) of §2 of Chapter III are valid. On the
Now, we have from (6), by setting k(r)
given v is easily verified. Specifically,
S' (see
basis of these formulas, we have
of the quantity R(c3 -
Proof. That the function 11 (r) belongs to the type of functions ,\ (r) with
< I,
Y
00
- (~ e-~ sin a('t) d't )'] - 4 ~ e-i~ cosia ('t)d't + 1 ~ 1) Fekete and Szego [1933]. See also Bazilevi~ [1936].
~
__
~
~
.
_
=
~
~
o
~
_
___
~
-------, 198
IV. EXTREMAL QUESTIONS ex>
ex>
~ 4 (I - Il) (~ e-< cos 6 ('t) dt )2 o
-
4 ~ e-~< cos~ 6 (t) dt
fp (z) = ~ c~ + IZ"P+1
ex>
+ I,
where p > 1, we have (he sharp inequality
(7)
I
the form (1), we have from (2)
\
m(ca - IlC~) ~ 4 (I -
Il) (\I
r
= 4 (l -
+ I)g eII)
\I~ -
I-
• -
(I -
I
+ ~ )t,-g· + I 2(1) + -} - II1 e-'J' + 1.
4 ("
(8)
\I
In particular, Ic~2)1:s e- 2/3 + 2- 1
This maximum is attained at v
00.
This maximum is attained at v
=
al( 1 - a).
al(1 - a). Consequently, we obtain the inequality
=
2/1 IIc:) ~ 2e -I -/1
ffi (Ca -
I~-e 2 -
and, hence, inequality (5). Now, for equality to hold in (9), it is obviously neces'
( 13)
p
1.013 ....
=
=
f/f(zP),
where fez) = k~ =1 C n zn E: S. We have (p)
C2p
+ 1_ - IiI ( Ca _
p- I 2p
2)
(14)
C~ •
But, in accordance with Theorem 1, we have the sharp inequality
I
(9)
sary and sufficient that k (r) be such that
I
p+I+-.
p
I
+ 1,
p-I
2-'
Proof. The function fp (z) can be represented in the form fp (z)
It remains for us to find the maximum of the right-hand member of inequality (8) for v in the interval 0
(P)
C2p+1
e -r cos 0 (r). Since IA(r) I.:::: e -r, by representing the last integral in
g
E S,
,.=0
0
ex>
=
ex>
+ I'
= 4 (I - Il) (~ A(t) dt)g - 4 ~ A~ (t) dt o 0 where ,\ (r)
199
§9. SHARP BOUNDS ON THE COEFFICIENTS
Ca - p 2p
C~ I ~ 2e
-2 P - 1 p+ 1
+ 1.
This together with (14) yields (13). For certain special subclasses of functions of the class S, sharp bounds for
00
~ e-' sin 6(t)dt=O', o
and that A(r)
=
(10)
e -r cos O(r) coincide, in accordance with the lemma, with the
function fl (r) (with v
k('t)=
=
{
al( 1 - a», that is, that
1"
2 e-('-')+IVI -e(.-.)
for 't~\I, for
(11)
t>".
the coefficients c n have been obtained. For this, we need Lemma 2. Suppose that a function fez) = 1+ alz + a2 z2 + ... is regular in the disk z I < 1 and satisfies in that disk the condition 'R(f(z» ~ O. Then, Ian l:s 2 for arbitrary n = 1, 2,···. J
Proof. The condition 'iR(f(z» ~ 0 is equiv;'llent to the inequality 11+ f(z)1 ~
11 - f(z)J.
It follows that the function
It is easy to show that we can satisfy condition (10) by choosing the sign ± properly in (11). Specifically, if we take the plus sign in (11) for 0 and the minus sign for
f3 < r < v,
then, by virtue of ( 11), condition (10) can be
.
written in the form ~
~e-'V I-e o For
f3
=
g(.-.)
g is regular in the disk
dt-~e-
in the interval 0
Z
Iz I < I
j(z)-I
aj
=j(Z)+1=2 Z
+'"
(12)
< f3 < v
f3
=
m(~
v, it is positive
such that (12) is satis
Ig(z)1 :s 1 in that disk. Ia 11 ::: 2. If we now denote
and satisfies the condition
by TJk, for k= 1"", n the nth roots of unity, we have
~
f3
()
It then follows on the basis of the Schwarz lemma that
0, the left-hand member of (12) is negative and, for
Consequently, there exists a
< r < f3 < v
n
1
! f ("Ilkzn») ~
O.
k=1
Consequently, the function
fied for the corresponding function k(r) in (9) and equality holds in (9). This
n
'\'I
I
n ""-
completes the proof of the theorem.
1
f("Ilk Zn ) = I +anz+
k=1
Corollary 1. For functions
"
~. ~ ~
~
1...
_L
, I , __ L ~ __ J. _ _
,~
L ~ _ ... a
Ian I :s 2
also satisfies the cQnditions of the lemma. Therefore,
Theorem 3. 1 ) For odd functions
for all n = 1,
2, .. '. This completes the proof of the lemma. Theorem 2. I) For functions fez)
=
00
I';:' =1 C n zn € S
f (z) =
with real coefficients, we
Proof. By virtue of th~ univalence of fez) in Z1
and z 2 in
1
I z I < 1,
ICnl~l+cn-4Iogn, n=2, 4, ....
we have, for arbitrary
Iz I < 1
=1+ "'\1
c/~-4
*0.
Proof. Since inequalities (16) are applicable both to the function fez) and to
Zl-ZI
11=1
the function
In particular, for z 1 = re i ¢ and Z2 = re- i ¢, where 0
< r < I,
+
this shows that the
j(iz)=
expression
+~
"-
C rll-1 si~ kIf k sm 'f
we have
is real and nonzero, that is, that it retains sign for 0
<
<1
r
and arbitrary values
Consequently,
+
+ (csr\! - 1) cos 2rp +." + (cllr\! - cll_\!) r
2 sin\! rpcII (r, rp) = 1 c\!r cos rp (Ci r \!- C\I) r cos 3rp
+
alsQ retains sign. Since this expression is positive at r
ll 8 =
cos (k -
I) rp
+...
...
- 611_\!) r II-Sz 11-1
n=3, 5, ....
(19)
Adding (IS) and (19), we obtain (17). In connection with Theorem 3, we state without proof 2 ) that there exists a
+ .. "
j>
function fez) satisfying the conditions of this theorem for which
and satisfies the condition ~(F(z)) ~ O. In accord
arbitrary given n
~
\c 2 n +1 I with
2 is greater than unity.
In §S, we gave definitions of convex and starlike domains for the case of
Ic\!I~2, IC8rll-II~2, ICir\!-c\!I~2, ... ,
ICnrll-clI_\!1~2,
domains with analytic boundaries since we were then speaking of the images of disks
or, taking the limit as r --- 1,
IC\lI~2, Ics-ll~2, ICi-C\!I~2, ... , ICII-clI-\!I~2.
Ic n I :s n
(IS)
1
ance with the lemma,we have
It follows by induction that
n= 3, 5.....
Ilclll-lcn_\!II~cn-4Iogn,
0, the sign is plus.
s
+ (cnr\!
3, 5, ...
theorem,
~
F (z) = 1 + c\lrz + (csr\! - 1) z\l + (cir\l - c\!) rz
I Z 1< 1
ES
On the other hand, Theorem 2' of §S can be applied to fez). According to this
On the basis of this, the function
is regular in the disk
(_1)n-l C\In_lZIlIl-l
I clI-1 + I clI_III ~ 2,
cp. Consequently, the function
!
ICn -t- C11_\1 I ~ 2, n =
1l~2-
of
00
11=1
00
cII (r, rp) = 1
(17)
where c is an absolute cons tanto
00
!(Zl)-!(Zg) Zl-ZJ
ES
with real coefficients,
(15)
n=I, 2, ....
~ ClllI _t Z \lIl-1 II~I
have the sharp inequalities ICIII~n,
201
§9. SHARP BOUNDS ON THE COEFFICIENTS
IV. EXTREMAL QUESTIONS
200
Iz I < r
with r
<1
uilder functions of the class S. We shall now give defi
nitions of convex and starlike domains that do not involve properties of the bound
(16)
ary of a domain.
for arbitrary n. This completes the proof of
A one-sheeted domain is said to be convex if the segment connecting any two
the theorem.
points in it is entirely contained in that domain.
1) Goluzin[1949cl. 2) This was mentioned on page 190 with bibliographic references .
1) Dieudonne [19311; Rogosinski u9321.
.' ....
,
Proof. If we set
A one-sheeted domain is said to be starlike about a point if the segment con
z!'(z) = I
necting an arbitrary point in the domain with the point· in question is contained in that domain. Let us now look at functions [(z)
Iz I < 1
z + C2 z2 + ... €
=
onto domains that are starlike about
tion and let B denote the image of the disk \ z \
<1
under the mapping w
is also starlike about w
I z \ < r,
=
fez).
where
j
Br • This shows that the domain Br is starlike about w = O. A consequence of this is that, if a function w = [(z) maps the disk \ z\ < 1
that is, tw I €
Cn
§5,
in the disk \ z \
< 1.
(z/' (z)) If(z)
> 0,
(20)
In accordance with the maximum principle of harmonic func
tions, equality cannot hold here.
For n = 2, it follows tbat
=
~
ICnl~n,
n=2,
I)
3, ... ,
n
1
n-l
Ick Is k
for k
< n.
We
~ 2 1+~+ ... +(n-l) . n-l =n,
so that inequality (21) holds for all n. Inequality (22) is proved in an analogous Consider the functions [(z)
, ~
=
z + C2 z2 + ..• €
S that map the disk
Iz I < 1
onto convex domains. Let [(z) denote such a function and let B denote the image of tbe disk.\ z
I<1
under tbe mapping w = fez). From the definition given above
of a convex domain, for two points WI and W2 belonging to B, all points tWI + (l - t)w2, for 0
< t < 1,
that is, all points of the line segment connecting wI and
w2, also belong to B. Let us sbow that, in this case, the image B r of the disk I z 1< r, for 0 < r < 1, is al~o a convex domain. If WI = ((Zl) and W2 = [(Z2), wbere Iz II Iz 2 I < r, we easily see that the points ¢ (z) = t[(z 1 z/z 2) + (l - e) fez) for I z I < 1 and 0 < t < 1 all belong to B. Therefore, the function tjJ(z) = [-I (¢(z)) is regular in I z 1< 1, tjJ(O) = 0, and \tjJ(z)1 1 in the disk I z I < 1. From the Schwarz lemma, we have ItjJ(z)1 :5 I z I. For z = z2, this yields 1[-1 (twI + (1 - t)w2)1 Sr. But tWI + (1 - e)W2 = [(zo), where Izo I < 1. There fore, Jzo 1:5 r, so that tWI + 0- t )W2 € Br • This proves that the domain B,. is
s
s
.
(21)
)
and under the additional assumption that [(z) is an odd [unction, we have the
Let us assume that
cll+ .. ·+1 cn-II
manner.
::£':=1 cnz n € S that map the disk Izi < 1
onto starlike domains, we have the sharp inequalities
IC21s 2. 1c
!~
We now have
Theorem 4. For [unctions [(z)
(11
n-1
""""='
tben conclude tbat
in
this is equivalent to satisfying the inequality j(
n = 1, 2,' . '. On the other
=21 +1
a specified direction, arg [(z) changes in the same direction, and, in accordance with
s 2 for
'\ I,,;::: I an I I+ I an_III· I C 1+ ... + I I. 1Cn_ 1 I:
s
< r < 1,
I
(n-I)Cn=Cln_t+Cln_2C2+···+CllCn-t'
=
onto a starlike domain, then, as z moves around the circle \ z \ ~ r, 0
an
from which we get
I z \ in I z I < 1. Now, let wI denote a member of [(z 1), where z 1 is a point in the disk Iz 1 I < r. Consequently, !tjJ(zl)1 slzll; that is, Ir- 1(twI)!
that disk. Therefore, ItjJ(z)/
1
(23)
l,.
point tw, where 0 < t < 1, also belongs to B. Consequently, the function [-I(t[(z)) = tjJ(z) with tjJ(O) = 0 is defined and regular in the disk I z I < 1 and ItjJ(z)\ S 1 in of B r' Then, wI
+ CltZ +....
band, comparison of tbe coefficients in (23) yields
O. For any point w that belongs to B, the
=
(z)
we have, in accordance with the lemma,
S that map the disk
Let us show that, in this case, the domain B r of the disk
o < r < 1,
f
O. Let [(z) denote such a func
W =
203
§9. SHARP BOUNDS ON THE COEFFICIENTS
IV. EXTREMAL QUESTIONS
202
convex.
sharp inequalities 2) ICnl~l,
n=3, 5, ...
A consequence of this is the fact that, if a function w
(22) disk
I z! < 1
onto a convex domain, then, as
around an arbitrary circle 1) Loewner [1917].
2) Pri:~lov [1924]. The requirement that the function fez) be odd can be replaced
with the condition c2 = 0 (see Goluzin [1938]).
I z I = r,
where 0
z
=
fez) €
S maps the
moves in the positive direction
< r < 1,
the corresponding point w
describes a curve the tangent to wbich alwa ys turns in a counterclockwise direc tion, and, in accordance with
'.j
§5,
this is equivalent to satisfying the condition
IV. EXTREMAL QUESTIONS
204
+ 1 > O.
at ( zj" (z) ) I' (z)
in
I z I < 1.
(24)
By virtue of the maximum principle of harmonic functions, equality
cannot hold here. It is easy to see that the converse is also true, that is, if a function fez)
= z
+
C2 z2
+ .;. is regular in
(24) in that disk, it maps the disk I z Analogously, if a function f (z) = z +
I<1 C
Iz I < 1
CHAPTER V
and satisfies condition
).
univalently onto a convex domain.
2z2
+ • •• is regular in the disk
and satisfies condition (20) in that disk, it maps the disk
Iz I < 1
UNIVALENT CONFORMAL MAPPING
OF MUL TIPLY CONNECTED DOMAINS
Iz I < 1
univalently
onto a domain that is starlike about wo. These conclusions are in accordance with the definitions of convexity and starlikeness that we gave here and in §5.
If tWO functions ,z
I<1
fl (z) = z + •••
and
h (z) = z
§ 1.
onto an annulus
and are connected b}'
In the present chapter, we shall consider the problem of univalent conformal
fj(z) = zf~ (z), then it follows from the equation
at ( z/~ (z») = I. (z)
(25)
Iz I < 1
domain if and only if the function w =
fz (z)
onto a domain that is starlike about w
=
domains. Specifically, it is easy to show that, if a function (= fez) maps a domain B univalently, then at each isolated boundary point of the domain B
univalently onto a convex
maps the disk
Iz I < 1
either f(z) is regular or has a pole and is univalent in the domain obtained from
univalently
O.
We immediately see from this that,if a function w = fez) = z +
C2 z2
+ •••
maps the disk I z \ < 1 univalently onto a convex domain, then, for all n = 2, 3,"', we have ICn I ~ 1; this is a sharp inequality since equality is attained by the function
_z_= l-z
Iz \ < 1
mapping of multiply connected domains onto various domains of special forms known as canonical domains. We make one important general remark regarding such
m(Zl! (Z.») + 1 11 (z)
that the function w = f 1 (z) m~ps the disk
which maps the disk
Univalent conformal mapping of a doubly connected domain
+ •.• are regular in the disk
~ z1& ~'
1&~1.
onto the halfplane ~ (w) > -Y2.
°
valued isolated singularity of the function fez). Let p denote a positive number
(26)
Iz - z 01 < P includes no boundary points of B B· the portion of the domain B contained in the B· is a doubly connected domain that the function (= f (z )
sufficiently small that the disk other than z 0' We denote by disk
CD
°
B by adjunction of all isolated boundary points of B. To see this, let z denote a finite isolated boundary point of the domain B. Then, z can be only a single
Iz -
z 0 I < p.
maps univalently onto a domain in the (-plane containing exterior points. Let a denote one of the exterior poin ts. Then the function F (z) = 1/ (f (z) - a) is
° is a removable Singularity of F (z). If we define the value of the function F (z) at z z ° as the limit of its values as z z we render F (z) regular at z 0 also. If this limit is nonzero, then z ° is a regular bounded in B·; that is, z
=
-->
0'
point of the function f(z) analogously defined at z 0' On the other hand, if the limit is 0, then fez) has a pole at z =
Zo'
In this case, we take f(zo) =
both cases, the function (= fez) maps a neighborhood of the point z
00.
In
° onto a
domain including the point (0 = f(zo)' Now, if fez) is not univalent in the domain obtained from B by adjoining to it the point z 0' there exists a point z 1 € B such that fez 1) = (0' A neighborhood of the point z 1 is mapped by the function
205
§ 1.
V. UNIyALENT MAPPING OF MUL TIPL Y CONNECTED DOMAINS
206
(= f(z) onto a domain including the point (0' Consequently, to every point of a
Let us now show that the domain B" can be mapped univalently onto a circular
< 1(I < h.
"I > 1
two points of the domain B. However, this contradicts the univalence of f(z) in
annulus 1
B. Thus, our remark is proved. It shows that, when we are investigating questions
t = log z " onto the halfplane ~Ht)
of univalent conformal mapping, isolated boundary points can be adjoined to the
corresponding to each point z" in the domain Iz
domain being mapped witllOut causing any complication in the problem.
many points of the form t + 2k TT i, for k = 0,
Let us now begin with the simplest case of the problem posed, namely, the
207
domain B corresponding in the domain B" to sequences that converge to z ~ .
sufficiently small neighborhood of the point (0 there correspond no fewer than
.~
MAPPING A DOUBLY CONNECTED DOMAIN ONTO AN ANNULUS
plane ~(t)
> O.
We first map the domain !Z
> O.
by means of the function
This mapping is not one-to-one because
"I > 1
± 1, ± 2,
there correspond infinitely
The domain B" is thus mapped into a vertical curvilinear strip
case of doubly connected domains. Let us show that every doubly connected
in the t-plane bounded by the imaginary axis and an infinite analytic curve con
domain can be mapped univalently onto some circular annulus, whose boundary
tained in the halfplane ~ (t)
circles may degenerate to points.
the property that, if a point t E B l' then all points t + 2k TT i, for k
Let B denote a doubly connected domain in the z-plane. H it has an isolated boundary point z 0' then, by adjoining it to the domain B, we obtain a simply connected domain that we can then map univalently either onto the disk or onto the (-plane with the point (= that the point z = z
° is mapped into
mapped either onto the annulus 0
00
1(1 < 1
excluded. We can do this in such a way
(= O. The domain B is thus univalently
< Izi < 1
or onto the annulus 0
< Izi
<00.
Suppose now that the boundary of the domain B consists of two continua K 1 and K2 • One of these, let us say K 1 is necessarily bounded. The complement of K 1 in the z-plane is an open set consisting of two disjoint domains. One of these domains, let us say B 1 contains the domain B. The domain B 1 is simply connected. Therefore, it can be mapped conformally onto the disk Iz' I < 1. Under this mapping, the continuum K 2 is mapped into a continuum K~ contained in Iz ' 1 < 1, and the domain B is mapped into a domain B'. Let us now map which ever of the simply connected domain s complementary to K ~ contains B' onto . 1"1 . suc h a way th . z > 1 10 at " z = 00 1S mappe d 1nto z " = 00. Un d er th e domam this mapping, the circle Iz tained in the domain Iz "I
'I
=
>1
1 is mapped into an analytic Jordan curve con and the domain B' is mapped into a doubly con
nected domain B 11 that does not include
00
and that is bounded by this curve"
and by the circle Iz" I = 1. The composite of these two mappings constitutes a univalent mapping of the domain B onto the domain B ". Furthermore, on the bas is of
§3
of Cllapter II, we conclude that this mapping sets up a one-to-one
correspondence between the prime ends of the domain B and the boundary points of the domain B ". Here, just as in
§3
of Chapter II, by the prime end of
the domain B that corresponds to the point z ~ on the boundary of the domain
B", we mean the set of all cluster points of all sequences of points of the
> O.
This strip is a simply connected domain B 1 with =
0,
± 1,
± 2, •.. , also belong to B l' Let us now map the domain B 1 univalently onto a vertical strip B 2 of the form 0 < R (w) < h in such a way that the boundary points t = - ooi, 0, and + 00 i are mapped into the boundary points w = - 00 i, 0, and + 00 i. 1) Under this mapping, the point t = 2TTi is mapped into a point W o on the positive half of the imaginary axis. By means of a similarity transformation in the w-plane if necessary, we can arrange to have Wo = 2TTi. Then, if we denote by w
=
f(t} the mapping function and if we denote by t
=
¢(w) its inverse, we con
clude that these two functions satisfy the relations
!(t+21tl)=!(t)
+ 21tl,
qJ(w+21ti)=qJ(w)
+ 21ti.
(1)
To see that this is the case, note that the functions f(t + 2TTi) and f(t) +
2TTi both map the domain B 1 univalently onto the strip B 2 in such a way that the boundary points - ooi, 0, and + ooi are mapped in both cases into the points - 00 i, 2TTi, and + 00 i respectively. Consequently, by virtue of the uniqueness, these functions coincide, so that we have the first of equations (1). The second
ol equations (1) is obtained in an analogous manner. Finally, let us map the strip B 2 by means of the function (= e W
onto the
h
< 1(1 < e • Again, the mapping is not one-to-one. H we now express ( in terms of z, we obtain the function (= e f (1og Z"), which maps the domain B"
annulus 1
oilto the annulus 1 < 1(1 < e • This mapping is univalent because, by virtue of properties (1), the functions (= ef(log Z") and z"= e¢Oog s) are single-valued h
respectively in the domain B" and in the annulus 1
.,
~.
. " belonging to the half
< I(I < e h •
follows from what we said above that the function (= e f (1og
Z ")
Furthermore, it is analytic in
1) Every strip has rwo infinitely distant boundary points, which we denote by ±roi.
208
§ 1.
V. UNIVALENT MAPPING OF MULTIPLY CONNECTED DOMAINS
the closed region
B"
and that its inverse is analytic in the annulus 1 ~
1'I
~ eh•
MAPPING A DOUBLY CONNECTED DOMAIN ONTO AN ANNULUS
Analogously, we can extend f{z) past the circle Izi = r into the annulus r 2/ R <
Summarizing these results, we have
Iz I < r, and so forth. As a result, we see that the function
Theorem 1. Every doubly connected domain B can be mapped univalently
univalent on the entire plane with the points z = 0 and z =
onto a circular annulus. This mapping defines a one-to-one correspondence between the prime ends of the domain B and the points of the boundary of the annulus. If the domain B is bounded by Jordan curves, the mapping is one-to-one and continuous exclusive of the boundary. On the other hand, if the domain B is bounded by analytic curves, then, in addition, the mapping function is analytic in B and the inverse function is analytic in the closed annulus. As a supplement to this existence theorem, let us prove a uniqueness theorem,
f(z)
Theorem 2. The circular annulus onto which a given doubly connected domain
B is mapped univalentLy is unique up to a linear trans formation: that is, if the domain B is mapped univalently onto two distinct circular annuli, then each of these annuli can be obtained from the other by means of a linear transformation. Consequently, the ratio of the radius of the larger boundary circle to that of the smaller is the same for the two annuli. Furthermore, the function f(z) that maps the domain B onto an annulus in such a way that a given point on the boundary of the annulus corresponds to a given point on the boundary of B is uniquely. determined. Proof. H a domain B is mapped univalendy onto two distinct circular annuli,
-->
0 as z
-->
O. Consequently, it is regular at z
z-plane univalently. Furthermore, since f(z)
f(z)
=
= r, it follows that Izi
Iz I
this last mapping is of the form
a is a real constant. This can be shown as follows: A function
f (z)
can be
extended, by the symmetry principle, beyond the circle Iz 1 = R into the annulus
R < Izi < R2/ r and maps this annulus univalently into the annulus R' < I~ < R '2/ r. Then f(z) can be extended further into the annulus R2/ r < Izi < R4/ r 3, etc. 1) We can always arrange for chis by an inversion che radii of me boundary circles does noC change.
(= 11 ~
under which che racio of
00,
it follows that
=
1, that is, that a
= eia., where a is a real
replace B with a doubly connected domain B' such that B c B' and B ~ B' and B separates the boundary components of B', the modulus increases. Proof. We shall first prove an inequality. Suppose that a doubly connected domain B is contained in the annulus r < Izi < R and that it separates the circles Izi
=
rand Izi = R. Suppose that this domain is mapped univalently onto the
annulus r' < 1(1 < R'. The inverse function z =
cpU) = u +
iv has the expansion
00
Z=I'f(~)= ~
cn~n
n=-oo
in the annulus. H G and g are the areas of the finite domains bounded by the pre images of the circles I~
= r
Q=
~
1(1 =
~ and
g= (where
,=
~
R ~, where r' < r ~ < R ~ < R ' , we have 00
~
mldB=1t
ICI=R~
< Iz 1 < R
,= e a.z , where
--->
Theorem 3. When a doubly connected domain B is extended, that is, when we
< R' can be mapped univalently onto each other in such a way that
=r is mapped into I~ =r,l)
0 and it maps the entire
as z
The ratio mentioned in Theorem 2, of the greater of the radii of the boundary
sufficient for us to show that, if two circular annuli are mapped univalently onto
i
=
f(z) is regular and
excluded and that
circles to the smaller is called the modulus of the domain B.
each other, they can be obtained from each other by a linear transformation. And and r < I~
00
00
constant. This completes the proof of the theorem.
these annuli can be mapped univalently onto each other. Consequently, it will be
to do this, it is obvious that we need only show that, if two annuli r
--->
,=
az, where a is a constant; and since Iz 1 = If(z)1 = r everywhere on the
circle Izi
one which reveals a significant difference between univalent mappings of simply and multiply connected domains.
209
n 1 en III R~2n,
11=-00 00
~ n I en II! r~2nJ
117l dB=1t
IC/ ,~,.~
n=-oo
re i ~ so that J
co
G
R,'2
_ _ ~_ WI~-
g
'2 n:r '2 1 R,1
•
~
n=-oo ~
1tr1
n Ic
Ii (R,'2n-2 -
r'2n-2)
nit
co
~
n I Cn Ii r~2n
~ 0,
11=-00
since n (R ~ 2n- 2
-
r ~ 2n- 2) > 0 for all integers n except 0 and 1. H we now let
, R'1 approach R' ,we 0 b· r 1, approach rand tam
§ 2.
V. UNIVALENT MAPPING OF MULTIPLY CONNECTED DOMAINS
210
-
a
g
with equality holding only when
llHe- 2ie a.)
'ftR.'~
?::--Q'
(2)
'ftr"
¢«() = Co + c.,. But G ~ TTR2 and
g
Z TTr 2.
=
functions
1 F (R~) = R.
sequently, from (2) we have
r
~
r'
J
r
< Izi < R,
the domain
~
+
R. 9C (11
+ ••• ,
1c.1 = 1.
maps the domain Izi
>R
onto the plane with cut at an angle
Lemma 2. If the function, = F (z)
> R, then IF(z) -
aol
on the right is the modulus of the domain B. This completes the proof of the
the image of the domain Iz I > R under the mapping
theorem.
disk
1'- aol ~ 2R. Proof. If
In conclusion, we note that, for the first part of the existence theorem given
Iz 0 I> R,
=
z + R 2e 2i e/ z, which
e to the real axis.
= z + a o + at! z + ... is univalent in > R and the entire boundary of
~ 21z1 in the domain Izi
Izi
,=
F(z) is contained in the
then the fun ction
~=Fl (z) =_I-F{zoz)-~=z+~+ ... Zo Zo ZoZ
above, we can give a proof based (like, the proof of Riemann's theorem) on the solution of an extremal problem: Let B denote a doubly connected domain that is contained in the domain Iz I > 1 and that has as one of its boundary continua the
is univalent in Izi
circle Iz I
entire boundary of the image of the domain Izi
,= f(z)
z + R2e ia/ z. Therefore,
with equality holding, obviously, only tor the function F (z)
B is mapped onto a subdomain of
Out of the family ~ of all functions
=
ffi (e-~i6 ell) ~ R
that annulus, to which (3) is applicable with inequality holding. Then, the ratio
=1.
Chapter II as applied to the
2,
The assertion of the theorem follows because, if we map the extended domain
B' onto the annulus
§ 4 of
~ R2 with equality holding only when F(z)
(3)
¢ «() = c ." where
with equality holding, obviously, only for
la.1
that
211
R2.
Proof. It follows from the area theorem of
Con
-R: > ,R.'
MAPPING ONTO A PLANE
that map the domain
> 1.
Consequently, in accordance with §4 of Chapter iI, the
> 1 under this function is contained
B univalently onto domains of the same type in such a way that the circle Iz I = 1
in the disk
is mapped onto itself, find the one for which the quantity
jz I > R. Then, if we shift from \z I > R to lz I = R, we obtain the second part of
zEB
has the greatest value. The function representing the solution of this extremal
!Iii:
Theorem 1.1) Every domain B in the z-plane can be mapped univalently onto
of given inclination
e to the real axis.
point a of the domain B is mapped into
with parallel rectilinear cuts
Also, this mapping is such that a given
,=
()Q
and the expansion of the mapping
function about z = a is of the form
Let us now investigate univalent conformal mapping of arbitrary multiply
_1_ +Clt(z-a)+ ... or z-a according as a is finite or infinite.
connected domains onto canonical domains of various kinds. The simplest such domain is a plane with parallel rectilinear cuts. In this case, our investigation will be based on the solution of certain extremal problems.
Iz I > R, the quantity llHe -2i e a l ), where
~ 2\zl, where
complement of the domain B / with respect to the plane is a straight line segment
§2. Univalent mapping of a multiply connected domain onto a plane
z + a./z + •••
aol
a domain B' in the extended '-plane such that an arbitrary continuum in the
problem maps the domain B univalently onto a circular annulus.
=
~ 2. In particular, IF .(1)\ ~ 2; that is, If(z) -
the lemma.
M (/)= sup If(z) I
Lemma 1. Out of all functions F(z)
1<:1
z+~-+ ... , z
Proof. It will be sufficient to prove the theorem for the case in which a
()Q,
because, in case of finite a, we can map the domain B by means of the pre
that are univalent in
e is a given real number,
=
liminary transformation z *
is maximized
=
l/(z - a) onto the domain B * containing z *
=
()Q,
by a function mapping the domain Izl>R onto the plane with rectilinear cut that makes an angle
e with the real axis,
1) Beginning with the exuemal property of [he mapping function, which was indicated in the proof given below, this theorem was proved by de Posse! [1931] and Grorzsch [I932].
and only by that Junction. For it,
!
_Ii
_.
212
~;;;;:,~h~_
..-:
.,,'"
.::::".~~,-,
..:",-..:.,._.
,;;
."",,,.
213
§2. MAPPING ONTO A PLANE
V. UNIVALENT MAPPING OF MUL TIPL Y CONNECTED DOMAINS
the existence of a solution of the extremal problem posed is proved.
which reduces the problem to the original case.
Let us now show that the extremal function fo(z) provides the mapping referred
In the case of a simply connected domain B, the theorem is easily proved.
Specifically, if the boundary of the domain B has more than a single boundary
to in the theorem. Let us suppose that the complement of the domain B with
point, then it follows from Riemann's theorem that it can be mapped univalently
respect to its image B' under the function
onto an arbitrary domain constituting the entire (-plane with cut at inclination to the real axis in such a way that z sion of the function
(= "
=
(z) about z
where a-I> O. Then, the function
,=
00
is mapped into z' = 00 and the expan
=
00
e
than a straight line segment of inclination
,=
axis and that it has the expansion Let us show that R(e- 2ie f3l) > o.
Let us turn to the general case. Consider the family ~ of all functions f(z) =
00.
t'filllil'i:
I'*
f
':,
Let us first prove that this problem has a solution. Suppose that the entire the family
3J! are univalent in the domain
Iz I > R.
with Lemma 1, for these func tions we have numbers Bt(e- 2iea l ) are uniformly bounded. of these numbers. H this upper bound is not functions fn(z) € 3J! for which the sequence
Then all functions of
'~
'( ~
1"
SHe- 2i ea l rs R 2; that is, the Let A denote the least upper bound
attained, there exists a sequence of IR(e-2ieal)l converges to A. But,
! I,I if;;'
s 2 in Iz I > R.
'!Ill'
,t,
al
Izi =
10 accordance with Vitali's
f o(z),
which, in accordance with Theorem 2 of
in B o and hence in B. Therefore, fo(z) €
§ 1 of Chapter
'(t) inverse
=
the w-plane with rectilinear cut of inclination tion
,= '(t) - =t + y/t + .•. maps
e with the real axis, and the func
It I > R' onto a different domain.
Yo
< ffi (e-
ii9
(~I
+
11»,
Now, let us look at the function w
=
B univalently and has the expansion z + of z =
w (f o(z )). This function maps the domain (a~O) + (31)/z + ... in a neighborhood
00.
Consequently, w(f o(z)) €
!Jl and we have =
at (e-ii9 II~O») + at (e-ii9 ~I) > A, f o(z)
meets all of
important extremal property of the mapping function, namely, the fact that it maximizes the quantity ~He-2ieal)' 10 the case of finitely connected domains, this theorem can be formulated as
1
follows:
theorem, the sequence of these functions converges inside B 0 to a regular func tion
,=
the requirements of the theorem. At the same time, we have established an
the functions fnk(z) are uniformly bounded in the interior of the domain B o 00.
=
00.
B I can be mapped uni
which, however, is impossible. This shows that the function
R', where
R ' > R. By virtue of the univalence of fnk (z) in B, the same inequality holds also at points of the domain B that belong to the disk Izi < R '.Consequently, obtained from B by removing the point z =
w
,=
t + Yo + y/ t + • . . . The function w('(t)) - Yo = t + (f31 + YI)/t + ••• maps the domain ItI > R' univalently,onto
at (e- ii9 (II~O) + ~1»
that converges uniformly in the interior of the domain Iz I > R to a regular func tion fo{z)/z = 1 + O)/z2 + ... such that !He-2iealO») = A. But in accordance for functions fnk(z) on
in a ne ighborhood of
that is, R(e- 2ie f3l) > O.
fn(z)/z, that is, that the sequence Ifn(z)/zl contains a subsequence Ifnk(z)/z\
s 2R'
+.. ,
valendy onto the domain It I > R' in such a way that the function
at (e-ii9 11)
This shows that the condensation principle can be applied to the functions
with Lemma 2, we have Ifnk(z)\
=
Therefore, on the basis of Lemma 1 we have
Consequently, in accordance
from Lemma 2, for func tions of the family D we have If (z)/ z I
f3 / ,
to the mapping function has the form '(t)
pose the extremal problem: Out of all the functions of the family ~, find the one that maximizes the quantity R(e- 2ie a l ).
Iz I < R.
'+
e
In accordance with Riemann's theorem, the domain
a/ z + .•.
An example of such a function is f(z) == z. Let us
boundary of the domain B is contained in the disk
with the real axis. Let B I denote
the '-plane contains the point 00. Suppose that the function w w(O maps B I univalently onto the w-plane with rectilinear cut of inclination with the real
mapping of the domain B.
in a neighborhood of z
o(z) contains a continuum other
whichever of the simply connected domains complementary to this continuum in
is of the form a-I z + a o + a/ z + ... , «((z) - a o)/ a-I provides the required
that map the domain B univalendy and have the expansion f(z) = z +
,= f e
Theorem 1 '(Hilhert). Every n-connected domain B 'in the z-plane can be
I, is univalent
mapped univalently onto the '-plane with n parallel finite cuts of inclination
D,
which contradicts our assumption that the least upper bound A is not attained by the numbers R(e- 2i eal)' Thus,
with the real axis in such a way that a given point z
I
I
.1:
=
a is then mapped into
e
-----'~.-
214
(=
. ----.--
.._.
.
._.
..
,_...
._~-_"=-==~~~=~-.,_,=.~_.==~_==-,.,==_=======~--
_-.....",.~"'=~_'__..,___..=-~-=,7"_.".,.,.","~_'""
.. .,.. "=.-.-"",,. ,"',=
~--'-_'"".""'--"-,"'.='""
;-" ,",,'C- ._.
§:z.
V. UNIVAL ENT MAPPING OF MUL TIPL Y CONNECTED DOMAINS
and the expansion
00,
I
z_ a
0
(Xl
(z - a)
+...
z+~+ z
or
'-""':';;;';: .~..,. ~""~_"""'''_'~'"''"'-"'"~-'='''''''''''''''""''''~'''''',''''"_'' _. '__ '~"''''''",,_'''",,'''''.'_'","
215
MAPPING ONTO A PLANE
Theor em 2. There exists only one function performing the mapping mentioned
f the mapping function about z = a has the form
+
.==.~~"''"'-'-~-,
m Theorem 1 '.
... ,
Proof. Let us suppose first that B is a bounded domain with boundary con
according as a is finite or not. Some of the cuts referred to may consist of single
sisting. of closed analytic Jordan curves K l '
points.
are two functions (' =
f 1(z)
and ("
=
f 2(z)
.••
,K". Let us suppose that th~re
that map the domain B univalently
onto the plane with parallel rectilinear cuts of inclination () and normalized as
Let us now look at the .question of the correspondence of boundaries under
B
univalent conformal mapping of multiply connected domains. We shall confine our
indicated in Theorem 1 '. Then both these functions are regular in
selves here to finitely connected domains that have only accessible boundary
the point z = a. On K v , v = 1, ... ,n, they assume values that lie respectively . stralg . h tInes I" C\« -i8;-') C\« on certalO -0 e '::. = C v, an d -0 e-i8;-") '::. = c v/I , f or v= 1,"', n;
points. We may assume that the point
00
is in the interior of the domain. Let B
except at
denote such a domain and let K 1, K 2 , ••• , K" denote its boundary continua. Let us map the domain B univalently onto a domain bounded by closed analytic
that is, on K v we have ~(e-i8fl(z)) = c~ and~He-i8f/z)) = c:, where c~ and c ~, v = 1, ••. ,n, are constants. It follows from this that the difference (=
Jordan curves. This can be done as follows: we map the simply connected domain
F(z)
including z =
00
= fl(z) - f 2(z) is regular in B and assumes on K v (for v = 1,"" n) values
lying respectively on certain straight lines d V' Consequently, if we take an
and bounded by the continuum K 1 onto the interior of a circle.
arbitrary point (0 that does not lie on any of the straight lines d v, we conclude
Under such a mapping, the domain B is mapped into a domain B' bounded by a
that arg [F (z) - (0] does not chan~e as we move around any of the curves K v .
circle K{ and the continua K~, ... , K~ . Then, we map the simply connected
Therefore, on the basis of Cauchy's familiar theorem on the zeros of analytic
domain containing the domain B' and bounded by the continuum K~ onto the interior of a circle. Under this mapping, the domain B I is mapped into a domain
functions, it follows that the function F(z)- (0 has no zeros in B, that is, that
B" bounded by a circle K;, an analytic curve K; and the continua K;, ... , K:'
F(z) does not assume the value (0 in B. Thus, all values that F(z) assumes in
Continuing in this way, after n steps we arrive at an n-connected domain
B lie on the straight lines d v' However, this is possible only when F(z) == const
B(")
= 0, it follows that F(z)
whose boundary consists of n closed analytic Jordan curves. The univalent map
in B. Since F(a)
ping of the domain B onto the domain B(") is a composite of successive mappings
theorem is proved in this case.
of univalent domains. Since these mappings define a one-to-one correspondence
assume that it has no isolated boundary points. If there were two distinct func tions performing the mapping described in Theorem 1', we could, by mapping the
aries defined by the mapping of B onto B(n) is also one-to-one. Furthermore,
§ 3 of
°
domain B univalently onto the bounded domain B with boundary consisting of
Chapter II, we_can show that the function inverse to the mapping
function is continuous in B(") except at the point
00.
If the domain B is bounded
only by closed Jordan curves, we can show in addition that the mapping of
closed analytic Jordan curves, find two distinct functions mapping the domain B
B
§ 3 of
°
univalently as indicated in Theorem 1', which, on the basis of what was said
onto BtnI is also bicontinuous. If we now map the domain B(n) onto the plane with rectilinear cuts, then, just as in the proof of Theorem 5 of
= [Jz), so that the
Now, let B denote an arbitrary finitely connected domain. Here, we may
between the boundaries, the correspondence between the points of these bound as in
== 0, that is, f 1(z)
above, is impossible. This completes the proof of the theorem.
Chapter II,
In conclusion, we pre sent an
we can show that the mapping function is regular on the entire boundary of the
~mportant
re lationship for the functions mentioned
in Theorem l' for various values of ().
domain B("). Combining all that we have said, we conclude that, in the pre sen t
Let B denote an n-connected domain bounded by closed analytic Jordan
case, the mapping mentioned in Theorem l' is one-to-one except for boundary
curves K l '
points of the domain B.
...
,K n and let (= j e(z, a) denote a function that maps the domain
B univalently as indicated in Theorem 1 '. Let us show that the following rela
An important supplement to the existence theorem proved above is the follow
tionship holds for arbitrary ():
ing uniqueness theorem:
ii
216
h(Z, a)=eiO(cosf.ijO(Z, a)-lsinBj" (Z, a».
217
§3. MAPPING ONTO A HELICAL DOMAIN
V. UNIVALENT MAPPING OF MULTIPLY CONNECTED DOMAINS
about z
(1)
=
b has the form
2
z 1 b +(7;) +Cll (Z - b)+...
or
Z+ClO+
a; +
... ,
(1)
The difference d(z) between the two sides of this asserted equation is a function that is regular in the domain B and that vanishes at z
=
according as b is finite or infinite. In the case in which the domain B has 'a
a. Furthermore, all the
single boundary point, this is obvious, and then the arc of the logarithmic spiral
values that d(z) assumes on any of the curves K v lie on a straight line C\I (
.,J
e
-i
e':0' J'I = const.
referred to degenerates to a point. 00 the other hand, if the boundary of the domain B is a continuum, let us first map B conformally onto the domain
Reasoning as in the p~oof of Theorem 2, let us show that d(z) '" 0; that is, let us prove (1). This relation, which has been proved for a domain bounded by
Iz 'I > 1
closed analytic Jordan curves is also valid for an arbitrary finitely connected
ping functio n z' = c/J (z) has in a neighborhood of z
in such a way that the point z 1
a+(7;)+Cll(Z-b)+... . zb
domain B. For this, it. is sufficient to map the domain B onto a domain B * bounded by closed analytic Jordan curves, apply (1) to B*, and then return to the
e as
theorem. Suppose that the point z soon as we know io(z, a) and
e and
c, the equation ~ (e - i 110 g
0=
plane with cut along a logarithmic spiral and has an expansion of the form (1). This result for simply connected domains can be generalized to multiply connected domains. For this generalization, we need the analogue of the minimal
= log (, this logarithmic spiral is mapped into the straight line ~(e-iet) = c the real axis, and the ray referred to is mapped into a
property proved in
'{ ,
I
"
0, the logarithmic spiral degenerates
e=
17/2, it degenerates to a circle with center at the origin. H we hold e constant and vary c, we obtain various curves
into a ray issuing from the origin. For
constituting the family of logarithmic spirals of inclination when we speak of logarithmic spirals of inclination
e,
e.
:il·'t )
I.
i:
of Chapter W, which we formulate as a
= z + a o + all z + ... that map the domain univalently in such a way that a given finite point a and the point 00 are mapped into the poi nts 0 and 00 respectively, the quantity SHe -2i e log F '(a» Lemma. Of all functions (= F (z)
Izi > R
Iz I > R
with cut along an arc of a logarithmic spiral of inclination
these are what we mean.
e.
onto the (-plane Here log F '(z)
means the branch that approaches 0 as z - .... 00. We now have
points, it is possible to map B onto the (-plane with cut along an arc of a loga
e in
§3
is minimized by the function F o(z) that maps the disk
In all that follows,
Let us show mat, for any simply connected domain B that has boundary rithmic spiral of inclination
But this possibility
['(00) = 00 with suitably chosen c provides a mapping of the domain B onto the
This last follows, for example, from the fact that, if we shift to the plane
e=
onto the (
As a result, we see that the function (= cF(c/J(z» = fez) with f(o) = 0 and
c defines a logarithmic
property that it is intersected by an arbitrary ray issuing from the origin at an
straight line parallel to the real axis. For
>1
by the condition F (a ') = 0 provides the required mapping.
spiral in the (:-plane with asymptotic point at the origin. This spiral has the
e to
z 'I
this problem has a solution and that the extremal function (= F (z ') normalized
with cuts along circular arcs of concentric circles. For constant
e.
1
'»
logarithmic spirals and, as limiting cases, onto the plane with radial cuts and
t
a is mapped into a point z' =a'. It now
SHe-2i elog F '(a in the class I. This problem was studied in § 3 of Chapter IV (the first application of Theorem 1 with a = 17/2 - e). It was shown there that
mapping of multiply connected domains onto a plane with cuts along arcs of
with inclination
=
plane with cut along a logarithmic spiral of inclination
Univalent mapping of a multiply connecled domain onto a helical domain
e.
Cl_1Z+Clo+~-t... , z
follows from the solution of the problem on the minimum of the quantity
In an analogous manner, we shall find the answer to the question of univalent
angle
or
= b, the expansion
remains to establish the possibility of mapping the domain
iTTI2 (z, a).
§ 3.
is mapped into z' = 00. Then, the map
according as b is finite or infinite. This is possible by virtue of Riemann's
domain B. This yields the relation (1) for B. Equation (1) yields ie(z, a) for arbitrary
=b
Theorem
such a way that given points a and b of the
domain B are mapped into 0 and 00 and the expansion of the mapping function
1.1)
Every domain B in the z-plane can be mapped univalently onto
1) Grotzsch [1931].
II~
218
§ 3.
V. UNIVALENT MAPPING OF MUL TIPLY CONNECTED DOMAINS
a domain B I in the '-plane that includes the points 0 and 00 such that an arbi trary continuum of the complement of the domain B' with respect to the '-plane is an arC of a logarithmic spiral of given inclination e. This mapping maps given points a and b of the domain B into 0 and 00 and the expansion of the mapping function about z = b has the form (z - bfl + a o + a l (z - b) + ... or z + a o + a1z- 1 + ••• according as b is finite or infinite. Proof. We may assume that b = the transformation z·
=
00
MAPPING ONTO A HELICAL DOMAIN
219
> R. If lal > R, this last inequality with z = a implies that la~n)1 ~ 21al. On the other hand, if lal < R, then the point = fn(a) = 0 does not belong to the image of the domain Izi > R under the mapping fn(z); hence, it
domain Izi
'0
belongs to the disk cases,
because, if this is not the case, we can use
la~n)1 ~ 2max(lal, R)
Iz I > R.
in
1'- a~n)1 ~ 2R. =
Therefore, la~n)1 ~ 2R.
M. But then,
,=
Thus, in all
Ifn (x)/zl ~ 2 + M/R
Consequently, the condensation principle can be applied
the sequence lfn(z)/zl, so that it contains a subsequence 1fnk(z)/zl that converges
I/(z - b) to switch from B to a new domain B. and we
uniformly in Iz I > R to a regular function. The corresponding sequence lfnk (Z)l
can then prove the theorem fex this last domain, replacing a and b with I/(a, b)
converges uniformly in an arbitrary closed bounded subset of the domain Iz I > R
and
> R. But since the values of fnk(z) corresponding to points of the domain B that lie in Iz I < 2R belong to the disk
00.
,=
to a function fo(z) that is univalent in Izi
In the case of simply connected domains, the theorem was proved at the
beginning of this section.
I' -
To prove it in the case of a multiply connected domain
a~n)1 ~ 4R and hence to the disk I' I ~ 4R + M, the set of functions fn k(z) is
B, let us consider the family ~ of all functions f(z) that map B univalently in
uniformly bounded inside the domain B with the point z
such a way that a and
and that have an
the uniform convergence, mentioned above, to
An example of
the domain B, and
expansion
f (z) =
00
are mapped respectively into 0 and
z + a o + a/ z + : .• in a neighborhood of z
such a function is f(z)
=
=
00 00.
z - a. We pose the extremal problem: Out of all func
tions of the family ~, find the one that minimizes the quantity z
-->
In this
domains in the t;plane that are complementary to this continuum contains 0 and
domain, all functions f(z) E !Dl are univalent. Consequently, for these functions, the quantity
3'He- 2i B log ['(a))
00
quantity
are mapped into 0 and
3He- 2iB
log
['(a))
00.
¢ (0)
e in such a way that
2iO
logf~ (a)) _
According to Lemma 2 of
§ 2,
=
fn(z)
1'- a~n)1 ~ 2R
=
'+
f30
+
,= ,=
'(t) = t + Yo + y/t + .... Suppose that the point t = a' corresponds to the point f o(a). The function w = ¢ ('(t)) maps the
expansion of the form
A.
domain It I > R univalently onto the w-plane with cut along an arc af a loga
e
rithmic spiral of inclination in such a way that the point t w = O. From the preceding lemma, we have
it follows that the entire boundary of the image of
the domain Iz I > R under the function w
n = 1,2,"') lies in the disk
,=
valently onto a domain It I > R in such a way that the inverse function has an
1, 2"" ,
B lies in the disk I z 1 < R.
Suppose that the entire boundary of the domain
e in such a way that
In accordance with Riemann's theorem, the domain B 1 can be mapped uni
00,
m(e-
¢ (,) maps B 1 univalently onto the w-plane
m(e- 2iO log cp' (0)) < O.
for functions in ~ by A.
=
=
0 and ¢ (00) = 00 and suppose that ¢ (0 has an expansion in a neighborhood of 00. Let us show that
Let us denote the greatest lower bound of the
opposite. Then, there exists a sequence of functions fn(z) € ~, for n -->
=
f3/' + ...
Let us show that this greatest lower bound is attained. We assume the such that, as n
Suppose that the function w
with cut along an arc of a logarithmic spiral of inclination
B' univalently onto the t;plane
with cut along an arc of a logarithmic spiral of inclination
a and
00.
is bounded below by the same number as was
calculated for a function that maps the domain
,=
fo(z) contains a continuum other than an arc of a loga rithmic spiral of inclination e. Let B 1 denote whichever of the simply connected under the mapping
00.
f o(z) holds also in the interior of f o(z) thus proves to be a solution of
Let us show that the extremal function fo(z) provides the mapping indicated
Let us show that this problem has a solution. To do this let us choose a
B' contained in B that includes a and
excluded. Therefore
in the theorem. Let us suppose that the complement of the image of the domain B
00.
simply connected domain
M. The function
00
the extremal problem posed, so that this problem does have a solution.
iHe- 2iB logf'(a)),
where log ['(z) denotes the single-valued branch in the domain B that approaches
o as
f o(z) E
=
z + a~n) + ain )/ z + ... (where
=
a' is mapped into.
m(e- 2iO (log cp' (0) + log~' (a'»)) < m(e- 2iO log~' (a'».
and that Ifn(z) - a~n)1 ~ 21z1 in the
'i.
_~
~_~
_ _.
,__.--.....,.. -----..--
-,.-._.__._-_.. --e.".,
_~
_",,,
."~
_
.~...
_
.,
_
§3. MAPPING ONTO A HELICAL DOMAIN
V. UNIVALENT MAPPING OF MUL TIPL Y CONNECTED DOMAINS
220
Therefore, ~He-2ielog ¢'(O))
< O.
Let us now look at the function w
The correspondence of the boundaries under the mappings referred to in Theorem =
¢ (f 0 (z))
domain B univalently in such a way that a and
= 00
00.
1 ' is studied in the same way as in
are mapped into 0 and
domain B has only accessible boundary points, then the mapping is one-to-one
00
hand, if the domain B is bounded by closed Jordan curves, the mapping function
+ ffi (e-
2i8
Iog/, (a»
is continuous in B except at the point z = b. Finally, if the domain B is
< A,
bounded by analytic Jordan curves, the mapping function is analytic on the bound ary of the domain B.
which, however, contradicts the definition of A. This completes the proof of the
We also have a uniqueness theorem:
theorem. At the same time, we have established the following extremal property of
Theorem 2. There exists only one function carrying out the mapping mentioned
the mapping function, which includes the distortion and rotation theorems for the
in Theorem 1 '.
family of univalent functions that we are considering. 1)
Proof. Let us assume first that the domain B is finite and bounded only by
Of all univalent functions (~F (z) defined on a domain B including have an expansion of the form
F(z)=z+cx o+ CI; in a neighborhood of z =
00,
00
Kn • Let us suppose that there exist two functions (= f l(z) and (= f 2(Z) that carry out the mapping of the domain B mentioned in Theorem 1'. These functions are regular on Kv for v = 1, ... , n
that
closed analytic Jordan curves
+...,
Kl '
.3 (e- iO log/I (z»
function defined in Theorem 1
.3 (e- iO log /2 (z» = c;,
have proved can be formulated in a different way:
where c~ and c~ are constants. Th~refore,itweset (==F(z)==log(f 1(z)/f 2(z)),
Theorem 1 '.2) Every n-connected domain B in the z-plane can be mapped
then the function F (z) is regular in B and the values that it assumes on the
univalently onto the (-plane with n cuts along arcs of logarithmic spirals of mapped into 0 and
00
curves Kvfor v = 1, ... ,n, lie on a straight line segment. We then see, just as
a and b of the domain Bare
in the proof of Theorem 2 of
respectively, and the expansion of the mapping function
const. But f 1(z)/f 2(z)
about z = b has the form z
~
b+CXO+CX1(z-b)+ ... or oz+cxo
may degenerate into points.
e=
that F(z) == const, that is, f 1(z)/f2(z) ==
§ 2.
We mention one relationship regarding the mapping functions which is analogous to (1) of
and
§ 2,
1 as z - .... b. Consequently, f1(z) "" f2(z).
reasoning analogous to that used in the proof of Theorem 2 of
according as b is finite or infinite. Some of these arcs of logarithmic spirals
e= 0
-->
Now, if B is a finitely connected domain, we can prove the theorem by
+ :J + ... ,
From Theorem 1', we obtain as special cases with
=c~,
and
In the case of multiply connected domains, the existence theorem that we
e in such a way that given points
... ,
and, on each of these curves, we have
the quantity ~(e-2ielog F'(z)) with given z E B
e is minimized by the
and given real
inclination
particular, let us show that, if the
in this case also except for points on the boundary of the domain B. On the other
Consequently, it belongs to the family !Jl and
ffi (e- 2iB log~' (a» = ffi (e- 2iB log r.r/ (0»
§ 2. In
if; (z). This function maps the
respectively and it has the expansion z + Co + c / z + ... in a neighborhood of
(=
221
§ 2.
Suppose that B is an n-connected domain bounded by closed analytic Jordan
TT/2
curves
theorems on the existence of a univalent mapping of an n-connected domain onto
Kl '
... ,
Kn and that (== j e(z, a, b) is a function that maps the domain
B univalently as indicated in Theorem I'. Let us show that the following rela
the plane with radial cuts and onto the plane with cuts along arcs of concentric
tionship holds in B for arbitrary
circles. ---1) GcOtzsch [I931a], Goluzin [1937, 1947]. 2) (Koebe [1918].)
e:
log}a(z, a, b)=eIO(cos8Iog!o(z, a, b)-lsinelogj,,(z, a, b» "2
~:
(2)
V. UNIVALENT MAPPING OF MULTIPLY CONNECTED DOMAINS
222
§4. RELATIONSHIPS INVOLVING THE MAPPING FUNCTIONS
where suitable branches of the logarithms are chosen. The difference d (z) between
assume that the domain B is finite and bounded by analytic curves Kl' K 2 , ••• , Kn • Since
the two sides of (2) is a regular function in B and the values that it assumes on the curves
Kv ,
for 1/ = 1, ... ,n, all lie on some straight line ~'He-ie()
Reasoning as in the proof of Th eorem 2 of const
=
§ 2,
223
= const.
:J (10 (z, a»
= canst,
'J (lj fC (z, a» 1"
let us show that d(z):=
= canst,
on K v (1/= 1 , " ' , n) for fixed a € B, that is, since
c, where c is equal to 0 for a suitably chosen branch of d (z). This
+ const j" (z, a)=- j .. (z, a) + const, "2 2"
yields equation (2), which is now proved for a particular form of the domain B.
io (z,
To see this, let us map the domain B onto a domain B. bounded by closed analytic Jordan curves and isolated points. We apply formula (2) to B. and then return to the domain B. As a result, we obtain (2) for the domain B.
a) = jo (z, a)
we also have on the basis of (1)
Some theorems on the existence of univalent mappings of multiply connected
P(z, a)=-Q(z, a)+k.(a),
domains onto various other canonical domains will be given in §6.
(4)
on K v , where kv(a) is independent of z € K v. Analogously, since
§ 4.
Some relationships involving the mapping functions
Equation (2) of
§3
3 (log io (z, a,
enables us to find, for a given n-connected domain B a
e if we know the
function j e (z, a, b) with arbitrary
::J (llog j (z, a, b» 1t
= canst,
"2 on K v for fixed a, b € B, it follows on the basis of (2) that
functions j 0 (z, a, b) and
j TTl 2(z, a, b). But we can also establish relationships between these last func
P (z, a, b) = - Q (z, a, b) +k. (a, b),
tions and the functions j o(z, a) and j TTl /z, a) of §2 v.hich correspond to the same
(5)
on Kv, where kv(a, b) is independent of z E K v .
donain B. To put these relations in the most symmetric form, we set
i U. (z, a) -
b» = canst,
Now, let us look at the integral
P(z,
a)~
Q(z,
a)=+(j~(z, a)+ jo(z, a»
j, (z, a», ) (1)
/J
= 2~i } P (e, z) Pc (e,
a, b) de,
2
taken over the entire boundary
and, analogously,
K=
U~= 1 K v of the domain
B in the positive
direction. 10 accordance with (1) and (2), the functions F (i;, z) and P (i;, a, b) are, for
P(z, a, b)=~ (logj~(z, a, b)-logjo(z, a,
b», )
2
Q(z, a, b)=+(Jogj~(z, a, b)+logjo(Z, a, b».
arbitrary fixed z, a, b € B, regular in B. Therefore, in accordance with Cauchy's (2)
theorem, /1 n
2
h=
d
O. On the other hand, applying (4) and (5), we have
2~{
.=1 n
These relations then take the forms
dz P(z, a, b)=P(b, z)-P(a, z),
I
=
)
:zQ(z, a, b)=Q(b, .z)-Q(a, z).
(3)
(
/
On the basis of the same principles as in the preceding sections, we can
I'~
\ pee, z) dp(e, a, b) ;.
= v~12~i ). (- Q(e,
z)+k.(z»d(- Q(e, a, b)+k.(a, b»
n
='~12~{).
Q(e, z)dQ(e, a,
b)=-2~i)
Q(C, z)QC(e, a, b)dC.
:
O
;
_
~
.
!
l
m
~
=""""' _ _
L
l
l
l
d
~
n,-._
~
__
.
__Jil_C
§4-
V. UNIVALENT MAPPING OF MUL TlPLY CONNECTED DOMAINS
224
(Here, we used the single-valuedness of the function
Q«(,
curves K v') Since
Q'«(,
z) and
Q«(, a,
not know explicitly is called the method of contour integration. I) Relations (4)
a, b) have simple poles in B at the point
and (5) on the boundary of the domain play the significant role in it. We now point out a curious analytic relationship with the preceding values of
last integral we obtain
Q~ (z, a, b) -
/1= =
225
This method of deriving various relationships between functions that we do
b) on each of the
z and the points a and b respectively, by applying the residue theorem to the
Remembering that 11
RELATIONSHIPS INVOLVING THE MAPPING FUNCTIONS
the function mapping the domain B onto a circular annulus with concentric
Q(a, z)+ Q(b, z).
circular cuts. Specifically, let us denote by (= fez) a function that maps a
0, we then arrive at the second of formulas (3).
domain B not containing"" and bounded by analytic Jordan curves K v ' lJ = lJ = 1, ... ,n, onto the annulus q < \(1 < 1 with cuts along arcs of concentric
Let us examine the integral
circles (see Chapter V,
/~=2~1 ~ Q(C,
§6,
the case 8rr/2 of Theorem 7). Suppose that the curves
K 1 and K 2 are mapped onto the circles
z)PC(C, a, b)dC
1(1 =
q and
1(1 =
1. Just as before, we
calculate each of the in tegrals in an analogous fashion. On the one hand, it is obviously equal to _P; (~ a, b).
2~i ~ P (C. z,
On th~ other hand, we can show, just as above, that it is equal to P (b, z)
/3 =
P (a, z).
h = 2~1 ~ P (C, z) (log! (C»c dC,
Thus we obtain the first of formulas (3).
Q; (z,
a, b) is an analytic function in B of the arguments a and b, whereas P: (z, a, b) is an One consequence of formulas (3) is that the derivative
analytic function of the difference function of
Q(z,
a in
a and b. a) -
a), where z and z
B and that P (z, a) - P (z
I,
I
a.
a) and
P; (z,
with (6) and (7), and the fact that
If
we let z' approach z, we see that analogous conclusions hold for the derivatives
Q; (z,
(8)
where z, z 'E B, by two different methods. Keeping in mind this fact, together
belong to B, is an analytic
a) is an analytic function of
(7)
K
Conversely, we conclude from these formulas that
Q(z',
z')(log! (C»c dC,
,
log!(C) = log!(C)
"
+ const on
K.
'~~
and
I
a).
2~i ~
The last result can be derived in a different way, specifically, from the
(Iog!(C»c dC= 1,
KI
relations
2~i
\ (log!(C)cdC=-l,
~
we can prove the follow ing formulas: 2) P~ (z, a) = P~ (a, z), Q~ (z, a) =
'11
Q~ (a, z),
!(z)=!(z') ek2(z,
!
which we can prove by examining the integrals
Is =
2k }p (C,
z) Qc(C. a) dC,
h=
21 ~
P (C, z) PC(C, a) dC
(6)
the, same way as above. That the functions
P (z,
a) and
Q(z,
a) themselves are not necessarily
analytic functions of a is shown by the example when B is the domain \z I > 1.
In this case, we easily see that
P(z, a)
I z-a I-[alil-az'
Q (z, a)
I
I
-Tlil2
I-liz
z- a .
(z)
=
z') -kdz, z'),
ef k l (z) dz - f k a (z) dz
(9) (10)
J.
1) Garabedian and Schiffer [1949 2) In the process, we need to evaluate the integral
2~1 ~ Q (C. z,
Z') (log! (e», dC,
K
and, to do this. we need, by virtue of the multiple-valuedness of the function Q( C z, z '), to make a cut in B connecting the points z and z ' and then replace rhe integral over K with an integral over a closed curve y encircling rhis cut. We then constrict the curve y around the cut. Integration by parts then leads to an integral rhat can be calculated in accordance wi th rhe residue theorem.
226
,
V. UNIVALENT MAPPING OF MUL TlPL Y CONNECTED DOMAINS
Theorem (minimizalion of area). I) Out of all functions f (z) that are regular
We note that z' appears in (9) as an apparent parameter since fez) is, in
§6 of Chapter V, determined up to a constant
accordance with Theorem 7 of
227
§4. RELATIONSHIPS INVOLVING THE MAPPING FUNCTIONS
£n
B and satisfy the conditions f(a) = (), f(b) = 1, where a, bE B, the minimum
value of the integral
factor. In conclusion, we derive an integral formula involving the function P(z, a, b).
B,
For any two functions f(z) and g (z) that are regular in
if,
we define their scalar
product as
B
r
(f,
which is equal to
lI
g) = ~ ~/' (z) g (z) dx dy.
1)= ~ ~ II' (z) I~ do,
77/ P(b,
b, a), is attained only by the function P (2, b, a) 10 (z)= P (b, b, a) .
( ll)
B
Proof. If we represent an arbitrary function f(z) of the type described in the In accordance with Green's formula
~
=
regular in B and that
f (z) g(z) dz = ~ I (z) g (z) dx '- il (z) g' (z) dy
K
(-l iJ/(Z1[W - iJ/(Z1[W) dxdy = -
¢ (a)
=
2i
B
~ ~ f' (z)g (z)dxdy,
(~(z),
B
where z = x + iy. For
(f, g)
we also have the contour representation -
¢ (b)
=
O. For this function, we have in accordance
with formula (13)
K
~~
¢ (z ), we conclude that the function ¢ (z) is
theorem in the form f (z) = f 0 (z) +
1 \
if, g)=-2i "/(z)g(z)dz.
P(z, b, a»=O.
Therefore,
(/, 1> = (/0' 10) + (Cf' 1-;) + (Cf' To) + (~, ~) =
(12)
(/0'
10) + (~,
~).
K
It follows that
If we apply (12) to the case in which g(z) = P(z, a, b) and remember that
(I, 1) ~ (/0' 10),
~ l(z)P~(z, a, b)dz
with equality holding only when (¢'
K
= ~/(z)d ~ (logj:r:..(z, a, b)--logjo(z, a, b»
0, that is, only when f(z) == f o(z). This
completes the proof of the theorem.
K 2
= }/(Z)d~ (-logj-'i(Z, a, b)-logjo(z, a, b»
As our second application of formula (13), consider a sequence of functions
uf
kJ)«() such that kJ)(b) = 0 for v = 1, 2, " ' , that are regular in lJ and satisfy the orthogonality conditions
=-~/(z)dQ(z, a, b)=27ti(f(b)-/(a», K
(k., k.,) =
we obtain the de sired formula
(f(z), P(z, a, b»=1C(f(a)-/(b».
¢) =
I for
'oJ
=
'oJ'
{ o for
'oJ
=1=
'oJ'
'oJ, 'oJ'
and have the property that an arbitrary function
(13)
in
= 1, 2, ...
f (z)
with
f (b) = 0
that is regular
B can be expanded in a convergent Fourier serie s in B:
We point out two applications of this formula. The first of these is the
I) .
"I
------- ------------ --
- . ---
~~--' ~-
-.-
. .---
-~--=-~--=-
- m __ .•_.
--~~-'.,
.• -
"_
" ...
§ 5.
V. UNIVALENT MAPPING OF MULTIPLY CONNECTED DOMAINS
228
.
C._ '
, "_.
'.".
"_n ._._
.,,"_,._"_,.
_._"
CONVERGENCE THEOREMS
229
univalent function f(z), it is necessary and sufficient that the sequence IBn! have a kernel and converge to it, in which case the function <: = f(z) maps A uni
co
I(z) = ~ c.k.(z), C.=(/, k.). .=[
valently onto B.
(It can be shown that such sequences of functions k v (z) exist.)
We note that although the functions are defined at certain points z € A only
Let us apply this to the function fez) = P(z, a, b), where a and b belong to
from some n on, this does not destroy the sense of convergence at such points.
B. Since, in accordance with (13), we have
Proof of the necessity. Suppose that the sequence Ifn(z)! converges uni
c.=(P(z, a, b), k. (z»=1tk. (a),
formly inside the domain A to a univalent function f (z). Then f(z) is regular in
we obtain for P(z, a, b) the constructive formula
A except at the point z = domain B' containing 00.
0:>
P(z, a, b)=7C ~ k.(a)k,(z).
00,
where fez)
=
00,
This maps the domain A onto a
Let us show that the sequence IBnl has a kernel and that B' is contained
,=1
in that kernel. It will be sufficient to show that an arbitrary closed bounded
§ 5.
Convergence theorems for univalent mapping of a sequence of domains
B·
domain
contained in B' belongs to every Bn from some non.
Let 0> 0 denote the distance from
Let us now consider the question of the convergence of a sequence of func
B·
to the boundary of B'. Let us con
tions that are univalent in multiply connected domains in connection with suitable
struct a grid of squares with sides of length 0/2. Consider the domain B. con
convergence of these domains and their images.
stituted by the union of all squares containing points in
B* c B '.
As a preliminary, we introduce the following concepts. Let IBn\, for
00.
including z =
We define the kernel of the sequence 00
IBnl
as the large st domain
every closed subset of which is contained in each
Bn contain a fixed neighborhood of the point z
sequence
I Bn I
=
00.
Bn from some
If(z) -
B·
<:01
> 01
J. c A.
Then
B·
<: =
But, for any point
f(z) in A are domains
<:0 €
B·,
we have
on the boundary of the domain A•. On the other hand, it follows
A.
that there exists an
n > 0 such that Ifn (z) - f(z)j < 01 on the boundary of the domain A. for n > N. Consequently, from the equation
Bn ---t B) if an arbitrary subsequence of these domains has the same kernel B. For example, if B 1 C B 2 C B 3 C ••• , then the sequence of the domains B n has a
In (z) - Co = (/n (z) - I (z»
+(/ (z) -
Co)
• we conclude on the basis of Rouche's theorem that the function fn(z)-
Theorem 1. Let IAn! denote a sequence of domains An' n = 1, 2, '" , in
<:0
has
> N the image of the domain An under the fn(z) contains an arbitrary point <:0 € B·. This in turn shows that
one zero in A., that is, that for n
the z-plane that include the point z = 00. Suppose that this sequence converges to a kernel A. Let Ifn (z)1 denote a sequence of func tions <: = fn(z) such that, for each n = 1, 2, ... , the function fn(z) maps the domain An onto a domain B n including the point <: = 00 in such a way that fn(oo) = 00 and f~ (00) = 1. Then, for the sequence lfn(z)1 to converge uniformly in the interior of the domain 1 ) A to a
mapping
<:=
the sequence
I Bn I
has a kernel, which we denote by B, containing B'
Now, let us consider the functions z
<: -
=
4>n (<:)
inverse to the functions
fn (z). For any closed subset of the domain B, these functions are defined on
it from some non. For sufficiently large positive r, the functions
------
1) The existence of a pole at z = 00 is not significant in the presenr case because uniform convergence of the sequence of the funcrions fn(z) is equivalent here to uniform ~onvergence of rhe sequence of the functions fn(z) --- z.
C B. and
to the boundary of B.. Corre
from the uniform convergence of the functions fn (z) in
We shall say that a
of domains Bn converges to its kernel B (and we shall write
kernel and converges to it.
denote the dis tance from
A· and A •• Here, A· cA. and
B
n on. Obviously, for a kernel to exist, it is necessary and sufficient that all the
domains
>0
sponding to the domains B· and B. under the function
n = 1, 2, , .• , denote a sequence of domains .in the z-plane that include the point z =
Let 01
B*.
defined and univalent in the domain
I ~ :_!-
't._
-
I
expansions
1<:\ > r
9n (<:)
are all
and in that domain they have the
-----_..__._-_..._--------_..._ - - - - - - _ . ""'
"
....
'
'\
::.
"
230
_
(n)
.1:
a\n)
>-
Consequently, from Lemma 2 of §2, we have in 1<:1
I rpn (q -
a\{'l I ~ 2 I~
CONVERGENC E THEOREMS
231
that the function <: = fez) maps A onto the kernel of this subsequence, which again is the domain B.
Thus, Bn
---->
B. This proves the necessity of the con
dition in the theorem.
I,
Proof of the sufficiency. Now suppose that Bn
- ....
B. Let p denote a positive
number such that the domain Iz 1 > P is contained in each of the domains An and
so that
I i') \~ + 2
'l'n
But the convergence of the sequence
Ifn (z)!
\
, "
a~n) 1
I'
r .
to a function
f (z)
to a function
f (z)
implies that the sequence la~n)l converges to a finite limit. Therefore, the func tions ¢n (<:)/ <: are regular and bounded in the domain
1<:1 > r
tion principle can be applied to them, so that the sequence sequence I¢nk
1<:1> r
§ 5.
,'.'/
V. UNIVALENT MAPPING OF MULTIPLY CONNECTED DOMAINS
__._,-----
,
(01 tj
Ifn(z) - a~n)1 ~
has a sub
On the other hand, for all <: E accordance with Lemma 2 of
Bn belonging
to the disk
§ 2, I
1<:\ < 2r,
Ifn(z)1 in the disk
> r.
21z1 + 5p
we have, in
interior of the domain
! ¢n k (<:)1
< 2r.
1
§ 2,
we have
+ 3p. In particular, Ifn(z)1 ~
Iz I < p. Comparing the bounds on Izl2 p, we conclude that Ifn(z)1 ~
Izi ~ p
and in the domain
iii every domain An. This shows that for every closed bounded sub
f n (z)
from some n on are defined and uniformly
Let us show now that the sequence /[n(z)l converges in A. Suppose that there exists a point Zo E A such that !fn(zo)l diverges. Then, there exist two
Therefore, in accordance
00,
subsequences of functions that converge at z
=
z 0 to distinct limits. Further
!fn N(Z)l that converge uniformly in an arbitrary closed subset of A to functions f .(z) and f •• (z), that are regular and univalent in A except at z = 00, and that
¢(<:)
is the inverse of the function <:= fez). It follows that ¢(<:) is independent of the
I ¢n (<:)1
=
more, these two subsequences themselves contain subsequences !fiz)1 and
and ¢ (<:) is univalent in
B. Reasoning word for word as on p. 57, we can show that the function z
!¢n k (<:)1
21 z
arbitrary sequence of these functions.
converges uniformly to ¢ (<:) in the
B exclusive of the point <: =
p, so that Ifn(z)1 ~
bounded on it. Consequently, the condensation principle can be applied to an
Consequently, the functions ¢n (<:) are uniformly hounded in every closed subdomain with Vitali's theorem, the sequence
Izl2
set of A, all the functions
< 4r.
of the domain B that is contained in the disk 1<:1
in
valid at all points of An belonging to the disk
(<:)1
converges uniformly to ¢ (<:) in an arbitrary bounded subset of the domain \ <:1
21z1
5p on lz I = p. By virtue of the univalence of the fn (z), this last inequality remains
that converges uniformly in the interior of the domain
to a limit function ¢(<:)/ <:. On the other hand, the sequence !¢nk
Bn , then Ie - a~n)! ~ 2p by virtue of Lemma 2 of § 2. On the other hand, Ie I ~ p. Consequently, la~n II ~ 3p; that is, the coefficients a~n) are uniformly bounded with respect to n. Again in accordance with Lemma 2 of
and the condensa
!¢n (<:V <:!
1<:1> r
is contained in each Bn. If fn(z) = z + a~n) + at)1 z + ... in a neighborhood of z = 00 and if c does not belong to the domain
suppose that the domain
itse 1£ converges
have different values at z o' It follows from our proof of the nee essity of the con
¢ (<:) maps the domain B onto a domain A I. Reasoning
as at the beginning of the proof but reversing the roles of the domains An and
dition of the theorem as applied to the sequences !B n ,! and !Bn"l that the func
Bn , we can show that A' c A.
that is, onto the same domain B. The function
sequence
chosen, that is, that the sequence
in B. The function z
=
tions
f .(z)
and
f ••(z)
Now, it is easy to show that B' = B. If <:0 E B, then z 0 = ¢(<:0) E A and, that B'
c
z'=/;' (/** (z»=iJ(z)
f(z 0) E B '. This shows that B c B'. But we showed earlier B. Consequently, B' = B. Thus, the function <: = f(z) maps A uni
consequently, <:0
ma p the domain A onto the kernels of these last sequences,
=
maps the domain A into itself and tj; (z) IE z. Suppose that tj; (z) has, in a neighborhood of z =
valentlyonto B.
00,
the expansion
~(z)=z+:~+... ,
If we now take an arbitrary subsequence of the sequence !B n! and the corre sponding subsequence of the sequence Ifn (z)!, which also converges to fez),
where a v , for v 20, is the first nonzero coefficient (which necessarily exists).
then by applying to these subsequences the conclusions derived above, we see
,i;",:, ,Ii,
V. UNIVALENT MAPPING OF MULTIPLY CONNECTED DOMAINS
232
§ 5.
CONVERGENCE. TIIEOREMS
233
Consider the functions l/J2(z), l/J/z), .•. defined in a neighborhood of z = 00 by
domains An> [or n
the expansions
converges, with respect to z 0' to the kernel A. Suppose that the [unctions
,=
a:
W=~2(z)=H~(z»=z+2z +
W = ~a (z) •
•
•
•
= ~ (~2 (z»
•
•
•
•
•
,z •
•
I
+ 3 :~ + •
•
•
= 1,
2, " ' , in the z-plane. Suppose that the sequence IAnl
[n(Z) map the domains An univalently onto domains B n in such a way that
[n(zo)
=
'0
and [;(zo) > 0 [or n
=
1,2, .••. For the sequence l[n(z)1 to converge
uniformly inside the domain A to a univalent [unction, it is necessary and su[
•
[icient that the sequence IBnl have a kernel with respect to
These functions also map the domain A univalently into itself. Since the domain Iz I > P is contained in A, the functions l/Jn (z), for n = 2, 3, .•. , are, in partic > p and the functions l/Jn(Pz)/p= z + naJi'pV+1z V are
We may assume that z 0
ular, univalent in Izi
2,3, " ' , which can be the case only if a v = O. This con tradiction shows that l/J(z) == z, that is, that [.(z) == [ ..(z). This last fact in turn =
shows that the sequence
Un (z)l
and converge to
0 because we can always arrange for this by
= '0 =
making translations of the z- and '-planes. The proof of the first part of the theorem is now based on Theorem I.
valent in Iz I > 1. It then follows as a consequence of the area theorem that Inavlpv+ll ~ 1, n
'0
it. Then the limiting [unction [(z) maps A univalently onto B.
Proof of the necessity. If [n(z)
[(z) is univalent in A, then [; (0) the functions
converges in z o' Thus, the sequence 1[71 (n)\
[(z) uniformly in the interior of A and ['(0) f, O. Consequently, the sequence of
->
-->
converges everywhere in the d,omain A. The uniform convergence of this sequence inside the domain A now follows from Vitali's theorem. This completes the proof
I~ (0)
Fn(Z)=_( I )=Z+ ..
of the sufficiency of the conditions of the theorem and hence the theorem itself.
In Z
One can also prove other theorems on the convergence of sequences of uni which are univalent in the domains A: obtained from An by means of the mapping
valent functions on the basis of the manner in which they ace normalized. We shall
z ' = II z, converges uniformly inside the corresponding kernel A'. Therefore, on
give one of these theorems.
the basis of Theorem 1, the sequence of the images B; of the domains A~ under
Just as above, we shall first introduce ,the following concepts: Let (Bnl,
the mappings
where n = 1, 2, ... , denote a sequence of domains contained in the z-plane; Suppose that a given point z 0 belongs to each of the domains Bn • We define the
ship
kernel of this sequence with respect to z 0 as the largest domain B containing the point z
,=
IB 1 converge s 71
,,=
Fn(z) converges to the kernel B'. Consequently, the sequence
to a nondegenerate kernel B connected with B' by the relation
['(0)/'. The conclusion on the nature of the mapping' = [(z) follows.
Proof of the sufficiency. Suppose that Bn
z 0 and having the property that an arbitrary closed domain contained in it belongs to every Bn from some n on. Convergence to a kernel is defined =
Bn under a fractional-linear mapping of the z-plane is the image of the kernel B (of course, wid~ respect to the image of the point z 0)' This does not violate con
F (z) = n
2, ... and then let Pn approach 1, the
(+)
converge s uniformly in the interior of the domain A' to a univalent function F (z)
kernel of the sequence IBnl remains unchanged and convergence to it is not
such that F (00)
=
00 and F '(00)
=
1. But then the sequence
'/,-,,;
violated.
I
We shall now prove a convergence theorem. Theorem 2. Suppose that a point z
I~ (0) fn
vergence to the kernel. Furthermore, if we perform similarity transformations
= 1,
=
B. Then, in the same notation,
the sequence {B;I has the kernel B' and it converges to that kernel. Consequently the sequence {Fn(z)l defined by
just as before. Obviously, the kernel of the sequence of images of the domains
z ' = p. z on the domains B71 for n 71
- ....
,.I .
z 0 belongs to each o[ a sequence o[
t,t'i,!
f n (z) = I~ (0) F n (;)
Un (z)1
defined by
234
V. UNIVALENT MAPPING OF MULTIPLY CONNECTED DOMAINS
§6. MAPPING ONTO CIRCULAR DOMAINS
converges uniformly in the interior of the domain A to the univalent function
c
f(z)= which maps
Denoting the mapping function by ('" [(z), we then have [(00) '" 00. The func
Kv (v'"
tion [(z) can be extended across the circle
c>o,
F(~)'
235
1, 2, ... , n) into the domain
B v obtained from B by inversion about K v' It maps B 1/ univalently onto the B ~ obtained from B' by inversion about K~. Consequently, the function [(z) maps the circular domain consisting of the domains B, 8 1 , ••• , Bn uni
domain
A onto the kernel B of the sequence of the domains Bn • This com
pletes the proof of the The?tem.
valently onto the circular domain consisting of the domains B', B{, ... , B~. We reason analogously with these exrended domains and continue the process indefi
§6. Univalent mapping of multiply connected domains onto circular domains.
nitely. As a result of such extensions in the z'plane, we obtain a limit domain G,
The continuity method
which the function ('" [(z) maps univalently onto a limit G' in the (-plane. Let
In §§ 2 and 3, when we were solving certain extremal problems, we proved
Z '
theorems on the existence of a univalent mapping of multiply connected domains onto canonical domains. In the case of finitely connected domains, an incompar ably stronger method of proof is the continuity method. The proofs that use this a univalent mapping of finitel y connected domains onto the canonic al domains in question and on a single topological theorem of Brouwer. We shall illustrate this
ii' "
method by proving an important theorem on univalent mapping onto circular
I
of a finite number of inversions about the circles K l' K 2 ,
,Kn- The functions z'", Sk(Z), for k = 1,2,"" exclusive of the function z'", z map the domain B
onto disjoint circular domains B (k), for k = 1, 2, ... , that have no common boundary points and lie each inside one of the circles
points in common (some of the circles may degenerate to points). We first prove
Consider a disk jz -
a uniqueness theorem for circular domains.
aJ < r
covering theorem, the domain
Theorem 1. There exists no more than one [unction that maps a given [initely
connected domain B in the z-plane including
5 (z) denote a fractional linear function composed of an even number of inver
S(z) maps the domain G into itself. Consequently, [(z) is invariant under the group of fractional-linear transformations z'= Sk(Z), for k = 1,2, " ' , composed
method are extremely similar to each other. They are based on the uniqueness of
domains, that is, domains bounded by a finite number of complete circles with no
=
sions about the circles K l' K 2' ..• , K n taken in any order. Then, the function [(S(z)) also maps the domain G onto the domain G' univalently since the function
00
7T
domain in the (.plane, also including 00, and that has the expansion (= z + a 1z- 1 + '" in a neighborhood o[ the point z = 00.
L
K 1, K2 ,
••• ,
Kn •
contained in B. Then, in accordance with Koebe's
B(k)
(for k
=
1, 2, ..• ) contains a disk of radius
(r/ 4) IS'; (a)l. Consequendy, the area of the domain
univalently onto a circular
•••
B(k)
is not less that
(rIS'; (a)I/4)2. Since the sum of the areas of all the domains
B(k
l,
for k
=
1, 2, ... , is finite, we see that the series OJ
~ ISk(a)l~
One can easily see that this theorem is equivalent to
(1 )
k=l
'Theorem 1'. Every [unction that maps a circular domain B ([initely con
nected) univalently onto another circular domain B' is a [ractional-linear [unc
converges. If we now denote by Kk, for k = n + 1, n + 2, ... , all the circles
tion.
that serve as boundaries for the domains Since the two theorems are equivalent, we need only prove Theorem 1 '. Proof. Suppose that the domain B is contained in the z-plane and is
bounded by the circles K 1 , K 2 ,
Kn , that the domain B' is contained in the (·plane and is bounded by circles K {, K ~ , ... , K;, and that the mapping in question maps K v onto K~ (v = 1, 2"", n). Without loss of generality, we may assume that the domains Band B' contain the point 00 and that this point is .,. ,
,J
B(k),
their radii by Rk for k = n + 1, n + 2, '"
for k
=
1, 2, ... , and if we denote
, we can show that the series
OJ
~ Rk
~
i
(2)
k=n+l
converges. To see this, let us delete from the domain G the images of the point
z
=
00
under the mappings z
I",
Sk(Z), k = 1, 2, .... Then all the functions Sk(Z)
are regular and univalent in the remaining portion G * of the domain G. There
mapped into itself.
fore, from the general distortion theorem (see
til !
,1
§ 4 of
Chapter II), we have on
Kv
V. UNIVALENT MAPPING OF MULTIPLY CONNECTED DOMAINS
236 (v
=
1, 2 ... n) for all k = 1, 2,' . . . ,
,
' " •
237
§6. MAPPING ONTO CIRCULAR DOMAINS
' i ; > .
m
0;,.
00
IS;' (z) I ~ M.I Sf. (a) /'
.L; A~(.) < e,
R:(.) ~ e,
.L; .=1
.=1
where Jl v depends on v but not on k or z. Therefore, we get the following expression for the circumference 21T Rk v of the circle Kk v onto which the func tion z' = Sk(Z) maps K v :
21CRk •
•
vergence of the series (2) and (3). Now, for a sufficiently large circle K:
21t1
from which it follows that
J_ \' 1 (z')
21t1
(4)
z
K k(.)
Iz I < R.
But
1 (z') - 1 (z) dz'
\' ~
21ti
~ z' -
z' - z
•
K (·) k
Now let d denote the distance from z to the boundary of the domain G.. It follows from what was just shown that
00
m
k=1
.=1
dz' = _I
z
K k (.)
~ R~~~ M:m·.L; I Si. (a) I 2.
k=n+1
~ z' -
\' I(z') dz'
21t1
• =1
H we square both sides of this inequality and then sum over all k and all v, recalling that each circle K k is the image of a boundary circle of the domain B under exactly one of the functions z' = Sk(Z), we obtain
I
This proves the convergence of the series (2). Analogous conclusions hold for
~ _I \' 1.J 21t1 ~
.=1
m
I(z') dz' ~ ~
z' -
K (.) k
I
Z
---=0
tJ.k(·l
R (.)
d
k
'-
.=1
C'. Consequently, if we denote by Rk the radii of the circles into which the = n
1.J
z' - z
for any point z belonging both to G. and the disk
R" • ~ MR. I Sf. (a) [.
circle s Kk, for k
~ _I
/(z)=_1 } I(z') dz' _
•
n
R,
m
K
co
Izi =
we have, in accordance with Cauchy's formula,
IS;' (z) II dz I ~ M.I Sf. (a) I ~ Idz I ~ M.! Sf. (a) ,. 21CR••
)
where (. is a given positive number. This is always possible by virtue of the con
+ 1, n + 2, ... , are mapped by the function ~ =
~~
r(z) defined
.. !
r !
m
.=1
on G, we conclude that the series
A~(.)
!
m
•
m(·)
<~.
.=1
00
.L;
R~D
But the left-hand member and the first term of the right-hand member of (4) are
k~n+1
independent of the domain G. chosen and hence of (.. Therefore,
converges. It then follows, in particular, that the series
f(z)=~ \' I(z') dz·. 21t1 ~ z'-z
00
~
.1Z
This shows that the function r(z) is regular in the disk
k=n+1
converges, where !'i k is the oscillation of the function
Ak =
max
z', z" E K k
K
(3)
r(z) on Kk,
I/(z') - /(z")
that is,
I.
Let us now denote by G. a circular domain contained in G that contains Band that is bounded only by the circles K k " K k ", •.• , Kk(m)
with indices k', k", ••• , k(m) sufficiently large that
large R, that is, that pole at z
=
00,
Izi < R
for arbitrarily
r(z) is an entire function. Since this function has a simple r(z) is a linear function. This completes the
we conclude that
proof of Theorem 1'. This existence theorem which we now wish to prove consists in the following: ___~eorem 2. 1 ) Every n-connected domain B in the z-plane can be mapped 1) This theorem was proved by Koebe. The continuity method was developed in his article [1918a]. h has been simplified by the present author (see Goluzin [1938 a). Another proof has been given by Keldys [1939).
V. UNIVALENT MAPPING OF MUL TIPL Y CONNECTED DOMAINS
§6. MAPPING ONTO CIRCULAR DOMAINS
univalently onto a circular domain in the (-plane. Among these mappings there is
n-dimensional Euclidean space R~. This means that to every point in the set E
only one normalized mapping that maps a given point z = z E B into (=
is assigned a point in the set E'. The mapping is said to be one-to-one or
238
has an expansion in a neighborhood of the point z
_1_+CI1(Z-a)+ ... z-a according as a is finite
o.r
=
00
and
bijective if to every point of E there corresponds exactly one point of E' and
a of the form
vice versa. The mapping is said to be continuous if, for every sequence Ix(m)\ of
z+~+ ... , z
or
points in E that converges to a point x(O) in E, the sequence of images of these points converges in E' to the image x' (0) of the limit of the original sequence.
infinite.
Before proceeding with the proof of this theorem, let us recall a few concepts
Consider n-dimensional Euclidean space, in which a point is an n-tuple of finite real numbers (xl' X2' ••• ,X n ). We denote this space by Rn and we denote a point in it by a single letter, for example, X or by an expression of the form indicating in parentheses the specific real numbers defiriing it,
these numbers beirig called the coordinates of the point x. Two points are thought of as coinciding if and only if their corresponding coordinate are equal. The distance between two points x(1)(x(1) ... x(1») and x(2)(x(2) ... x(2») in the space
Rn is defined as the number
l' 'n p(x(1), x(2») .
1"
= ,"",," l"'" k= I
(x k(1 ) - x(2»)2]lh k • An
E Rn whose distance from a is less than p. A neighborhood of a point a E R n is defined as any open ball with center at a. A neighborhood is suffic iently small if the radius of the ball is sufficiently small. A set E
c R"
X
is said to be bounded if
it is contained in some ball. A point a E R n is said to be a cluster point of a
E of Rn if every neighborhood of the point
a includes at least one point
of E other than a. A sequence of points x(m) ERn' for m = 1, 2, ... , is said to converge to a point x(O) C R n if p(x(m), x(O») ---> 0 as m --> 00. A point a belonging to a set E
c Rn
E eRn onto a set E' c
R~ is also continuous. A one-to-one continuous map
ping is also called a topological mapping. Obviously, the image of a closed set under a topological mapping is a closed set. The following theorem of Brouwer on topological mappings of arbitrary sets is known as the theorem on preservation of a domain. Theorem 3 (Brouwer). The image of an interior point of a set E eRn under a
topological mapping is always an interior point of the image set E'; equivalently, the image E' of an open set E eRn under a topological mapping is an open set.
7).
open ball of radius p with center at a is defined as the set of all points
subset
It is easy to show that the inverse of a bijective continuous mapping of a closed set
from n-dimensional geometry.
x(x I' " • , x n )
is called an interior point of E if some sufficiently
small neighborhood of a is contained in E. A set E eRn is said to be closed if
We postpone the proof of this important theorem to the following section. The proof of Theorem 2 by the continuity method is based on it. Proof of Theorem 2. By virtue of the results of
§ 2,
any finitely connected
domain can be mapped univalently onto the plane with parallel straight-line cuts at an arbitrary given inclination () and with suitable normalization. Therefore, it will be sufficient to prove Theorem 2 for domains constituted by the z-plane with rectilinear cuts parallel to the real axis and for a '"
00.
Let us denote by ~ the family of all n-connected domains B consisting of the z-plane with n finite rectilinear cuts parallel to the real axis (numbered from 1 to n) and let us denote by !IJI' the set of all n-connected circular domains B' in the (-plane with boundary circles numbered from 1 to n. Two identical domains
it contains all its cluster points. A set E eRn is said to be open if it consists
Band B' wi th boundaries numbered in a different manner are considered as
only of interior points. It is said to be connected if any two points in it can be
distinct.
connected by a continuous curve cont ained in E. By a continuous curve in Rn , we mean a set of points in the space sented in the form Xk = ¢k (t) for
239
Rn the coordinates of which can be repre
a:: t:: band
is a continuous function of t in the interval
a::
k", 1, ... ,n, where each ¢k (t) t :: b. All the properties analo
Let us consider normalized (as indicated in the theorem) univalent mappings of domains B E ~ onto domains B' E
ii' such that identically indexed boundary
curves of the domains Band B I are mapped onto each other. We consider domains in the families ~ and ~' as corresponding to each other if they can be
gous to the properties of point sets in a plane that were expounded in the geo
mapped one onto the other as indicated. Then, from Theorem 1 of the present
metrical introduction can be established for point sets in the space Rn •
section and Theorem 2 of
Now, consider a mapping of a subset
E of Rn onto a subset E' of another
§ 2,
we conclude that
...........
,"~"'-"'t""_~
.._.
--=-_~==~:,::,-.~_=~::-_~;::._
.,.-"
~;«'..
-
_~~
.;.~ ..
=._,
~-'
. '----
'~€::";';'~';;;;;;;:-O;;;;:· _ i_T":'·'''''~'''.§i.","·Vif''''''''";c~~~
(a) to every domain
B€
where B;
!fI there corresponds no more than one domain
,I',
B' € ! fI',
(f3) to every domain B' € !fI' there corresponds a unique domain B € !fl. Our problem is to show that to every domain B € ~ there corresponds exactly one domain B' €
."....~~*'~
;;;;:;"-''''-'..__.'
- .",..
~"'_·:;.A;;¥.;;r~",:;)-::
§6. MAPPING ONTO CIRCULAR DOMAINS
V. UNIVALENT MAPPING OF MULTIPLY CONNECTED DOMAINS
240
...;l''.:;\.5;.>:;;o.~"·,,,~· ~~
B;
=
f.
241
B~. h then follow s on the basis of the same convergence theorem that
B~. But if
Bn - .... B. and B~ - .... B;, then we conclude, again on the basis
of the convergence theorem, that the domain B. can be mapped normwise onto the domain
B;,
which proves the assertion made.
From this point, the proof of the existence theorem is fairly simple. We are
~' .
The families !D1 and ~1 consist of domains defined by 3n parameters. For
now dealing with a one-to-one continuous mapping of the open set
n'
of points in
the parameters of the domain B we may take, for example, the coordinates of the
3n-dimensional space R~" onto a subset (proper or otherwise) of the set !Il in
origins of the cuts and their lengths. For the parameters of B' we may take the
another 3n-dimensional space R 3". Consequently, Brouwer's theorem can be
coordinates of the centers and the radii of the boundary circles. If we represent
applied. According to Ithis theorem, the image !Dl o of the set !Dl' is an open set in 3n-dimensional space. Let us show that !llo coinc ides with !Il.
the domains Band B' by points in 3n-dimensional Euclidean spaces R 3n and
R;n and denote these points also by Band B', we conclude immediately from geometrical considerations that the point sets B and B' corresponding both to the domains B € ~ and t() the domains B' € !ll' (and we shall denote these also by !fI and ~') are connected open sets in R 3n and R ~n respectively. If we set up between the points of the sets !ll and !Dl' the same correspondence as exists between the domains of the families !Dl and !Ill', we conclude on the basis of (a) and (f3) that to every point B' € !fI' there corresponds exactly one point B € !JJl and that to every point B € ~ there corresponds no more than a single point B'€!IlI'.
Let B 1 denote a domain in the family 1lI that can be mapped onto the ,domain
R' in the family ~'. Such a domain can be obtained by a mapping of an arbitrary domain in ~' onto the domain in !Dl corresponding to it. On the other hand, let
B 2 denote an arbitrary domain in !JI for which we have not yet determined whether it can be mapped. Let us connect the point in the space R 3" representing the domain B 2 with the point representing the domain B 1 by a continuous curve L contained in
n.
Let us proceed along this curve from the point B 1 to the point
B 2. If the domain B 2 could not be mapped onto a domain in !Il', then, as we proceed along L from B 1 to B 2' we would encounter a point B. such that all the domains in 1lI corresponding to points of the curve L between Bland B. could
Let us show that this correspondence is continuous, specifically that
B 1 - .... B as B{ - .... B' (that is, as the boundary circles of the domain B{ ap
'proach coincidence with the boundary circles of the domain B'). Here, B 1 corre sponds to B; and B corresponds to B'. If a sequence of the domains B' did
be mapped onto domains in !JI' and the domain B. either could not be mapped onto domains in ~ I or would be a cluster point of the set of point s of L corre spondirig to domains that cannot be mapped onto domains in !Dl'. But this is
not converge to a limit domain, we could choose from it two subsequences IBnll
impossible because, by virtue of the property of the limit domains mentioned
and lB n2 1 that converge to the domains Band B o' where B f. B o• On the basis of the convergence theorem, we conclude that the domain B' is mapped normwis(
above, the domain B. can be mapped onto a domain
belongs to 1lI 0 ' but then it cannot be a cluster point of domains that cannot be
onto B and also onto B 0' which, in accordance with (fJ) implies that B = B o·
mapped onto domains in !JI' because 1lI 0 is an open set. Thus, all points of the
Here, we assumed that the boundaries of all the domains Blare bounded, but
curve L from B 1 to B 2 belong to the set !Dl o. But then, again on the basis of the property of limit domains, B 2 € !Dl o.
this follows from Lemma 2 of
§ 2.
We mention also the following property of the limit domains: if B. €
1lI and
if there exists a sequence of domains Bn € !Ill such that Bn - .... B. and the Bn
can be mapped normwise onto B~ € ~', then the domain B. can be mapped
normwis e onto B; € !Ill'. This is true bec ause the sequence of the domains B ~
converges to a domain B; € llI' since otherwise would there would be two sub
sequences IB~ll and IB~2l that converge respectively to domaUls B'. and B~,
n'
and hence the point B.
Thus, we have shown that the set 1lI 0 coincides with !Dl and, consequently, for every domain in ~ there exists a domain in
n'
onto which the first can be
mapped univalently and normwise. This completes the proof of Theorem 2. Since the proof that we have given used only the uniqueness theorem and a few trivial properties of circular domains, the same procedure can be used with
__. _... _ """" ..•_,_
".~
,'_'"
~
__._,__ ._
._::::~_"
__:::._'::'_'_'_'=-'
'_
__"::'...
""::_
__::"'":.':--:,:-"_-:'_.~
r~
•• ~':'''~:'_ _
'''''='''''"""_«.,",,,_.
§6. MAPPING ONTO CIRCULAR DOMAINS
V. UNIVALENT MAPPING OF MULTIPLY CONNECTED DOMAINS
242
IzI = 1 as
243
equal success in proving theorems on the existence of a univalent mapping of
includes the point z = 0 and has the circle
finitely connected domains onto various other canonical domains as soon as we
can be mapped univalently onto the disk 1<:1 < 1 wi th cuts along arcs of loga
have proved the corresponding uniqueness theorem for them.
rithmic spirals of given inclination () in such a way that jzl = 1 is mapped into
For example, we can prove in this way the following theorems of Koebe, I) which generalize Theorems 1 and l' of
§ 2 and
Theorems 1 and l' of
§ 3:
Theorem 4. For a giVM point a E B, there exists a unique function mapping an arbitrary given n-connected domain B in the z-plane with boundary continua K l'
... ,
K n univale ntly onto the (-plane cut by n rectilinear cuts with given
inclinations J.I =
On to the real axis in such a way that the continua K v , for 1, ... ,n, are mapped respec tively into the cuts with inclinations ()v, the ()1"'"
point is mapped into (= "", and the expansion of the mapping function about z
=
1(1 = 1
one of its boundary con tinua
and the mapping function (= fez) is normalized by the conditions f(o) =0
and ['(0)
> O. This mapping is unique.
Theorem 7. Every n-connected domain B contained in the annulus q < that has as two of its boundary continua the circles mapped univalently onto an annulus q' < 1(I
Izi =
1(1 =
1 and
1(I =
Izi
=
Iz 1< 1
q can be
< 1 with cuts along arcs of loga
rithmic spirals of given inclination () in such a way that mapped into
1 and
Iz I =
1 and
Iz I =
q are
q'. This mapping function is unique up to a con
stant factor. We note that Grotzsch succeeded in proving analogous uniqueness theorems
a has the form 1
z-a
+
lX l(Z
-a)+ ...
or
Z+~+ z
for extremely broad classes of canonical domains and at the same time established
... ,
by the continuity method theorems on the existence of a univalent mapping of arbitrary finitely connected domains onto such domains (d. Grotzsch [1935]). We conclude our investigation of questions concerning univalent mappings of
according as a is fin ite or in finite.
Theorem 5. Every n-connected domain B in the z-plane with boundary con tinua Kl' ..• , K n can be mapped univalently onto the (-plane with n cuts along arcs of logarithmic spirals of inclinations ()1' ••• ,
respectively to the radial
()n
1, ... ,n, are mapped respectively into arcs of incl ination OJ), given points a and b in B are mapped into 0 and"" respectively, and the expansion of the mapping function about z = b has the form directions in such a way that the continua K v , for
z 1 b+lXo+1X1(Z-b)+ ...
or
J.I =
z+cx.o+ :' +
multiply connected domains with a consideration of the possibility of univalent mapping of any two given n-connected domains onto each other. In the case of simply connected domains, Riemann's theorem tells us that, with very few excep tions, this is always possible. The situation is different in the case of n-con nected domains, where n
> 1.
For it to be possible to map two such domains uni
vaLently onto each other, it is obviously necessary and sufficient that there exist two circular domains onto which they can each be mapped univalently and which can be univalently mapped one onto the other. We can take these circular domains
... ,
of a "special" form, specifically, annuli of the form q <
Izi < Q with
n - 2
circular cuts in the in terior since an arbitrary circular n-connected domain can
according as b is finite or infinite.
always be mapped onto a domain of such a form by means of a suitably chosen
The uniqueness of the mapping functions referred to in these theorems is proved in just the same way as Theorem 2 of
§ 2 and
Theorem 2 of
§ 3 were
mapped univalently onto each other, then by Theorem 1 " the mapping function is fractional·linear and, according to the symmetry principle, it maps the points 0
proved. By the continuity method, Koebe [191sa] proved, in particular, the following
and "" into each other. Consequently, this function is of the form z '= az or
z ' = a/ z. In the first case, the following quantities must coincide for the two
theorems:
Theorem 6. Every n-connected domain B contained in the disk
fraction al-linear function. But if two circular domains of the spec ial form can be
Iz I < 1
that
1) See Koebe [I9IBa] and Goluzin [I938a~ Krylov [1938] has given a different proof
of these theorems.
circular domains that we are considering: the ratios of the radii of all correspond ing boundary circles, the angles between the rays extending from the origin to the centers of the circular cuts, and the ratios of the distances of these centers
,«
-~-~-~~ENTMAP~mG OF :=~y CON~~~~~~S~~- -~---'~-~O_~~~~~:-PR=~:F~O=~S~~R=~--~~~-- ···,~s
from the origin; that is, the values of one quantity in the case n
'iii!. 2 and of a l l l g ' c .
=
do not lie in a single plane. The set of points x(x1"'"
'~~ ,
3n - 6 independent quantities determining these domains in the case n > 2 m u s t " $ :
n
IS
be the same for the two domains. In the second case, an analogous situation holds
x i =
for one of the circular domains in question and for the domain obtained from the other by the transformation z I =
II z.
Conversely, if these requirements are satis
1JAJXY\
i = I, 2, .•• , n, Aj
j=O
is called an n-dimensional simplex and the points x(k), k
I.
for it to be possible to map two n-connected circular domains of a special form
are single-valued continuous functions of xI' ••. , x n •
2, one, and
0, 1, ...
=
,n,
are
The numbers Ai in (1)
To see this, remember that, since the points x(k), k = 0, 1, ... ,n, do not
> 2, 3n - 6 suitable real quantities determining one of these domains
in the case n
(1)
j=O
called its vertic es. 1) Let us denote this simplex by
=
1JA j = I,
?:' 0,
fied, the two domains can obviously be mapped univalently onto each other. Thus, onto each other, it is necessary and sufficient that, in the case n
x n ) whose coordinates
are determined from the formulas
lie in a single plane, the determinant
coincide with real quantities determining the other. These quantities are called the moduli of the circular domains in question and of all the domains that are
(0)
mapped univalently onto them.
\0)
x l ' ... , X n '
" '/-
(l) (1) Xl , ... , X n '
~
i-'·~
H we partition all n-connected domains into classes, putting into a single (n)
XI ' ....
class all domains that can be mapped univalendy onto each other, then each such class will contain circular domains of a special form and, ip accordance with what
=
2 on one and in the case n
system of n + 1 linear equations in the Ai that has a unique solution. Con
> 2 on 3n - 6 real parameters.
sequently, the Ai are obtained as linear functions of the coordinates i
§7. Proof of Brouwer's theorem 1) To prove Brouwer's theorem as stated in the preceding section,
= 1, ...
,n; hence, they are continuous functions of the
Xi'
Xi,
for
The set of points
of the simplex I that are obtained from (1) when one of the Ai is equal to 0 is w~
need some
called an (n - I)-dimensional face or simply a face of the simplex
additional information regarding the space Rn • A set of points x E Rn with coordinates x
I
cannot be equal to O. This means that the system of n + 1 equations in (1) is a
was said above, all classes of n-connected domains constitute a set depending in the case n
(n) Xn ,
I.
In general,
the set of points of the simplex I that are obtained from (1) if k of the numbers l ' . . . ,x n
satisfying a given linear
Ai are equal to 0 is called an (n - k )-dimensional face.
equation
An (n - k )-dimensional face of a simplex I is itself a simplex in (n - k)
+ asxs +...+ anxn = 0,
a1x1
dimensional Euclidean space, specifically the (n - k)-dimensional plane passing through those vertices of the simplex I that lie on that face. The point x E I
where
Ql'
a 2,
•• ,
,an are not all zero, is called an (n - I)-dimensional plane or
simply the plane Rn -
1•
obtained from (1) for AD = Al = ... = An = I/(n + 1) is called the center of the
The set of all points common to two (n - I)-dimensional
simplex
planes the left-hand members of the equations of which are linearly independent constitutes an (n - 2)-dimensional plane in R n • In general, the set of all points common to k planes under the same condition constitutes an (n - k)-dimensional
,.
""i"
¥
"~"'~~;r
l
I.
By virtue of the continuity of the Ai as functions of xl' " ' , x n
(which we showed above), it follows that an arbitrary point (1) for all nonzero Ai is an interior point of the simplex
X
E
I. obtained
from
I.
Let us show that a simplex I can be partitioned into a finite number of . ',"i"'I;J" .,'~l; I:· plane in Rn • An (n - k )-dimensional plane in Rn can be regarded as an (n - k) simplexes the diameters of which (the diameter of a figure being the greatest :'~~' dimensional Euclidean space. distance between any two points of it) is less than an arbitrary number f >0 in '~: ~ ;',',7); Let x(k)(x(k), ••• ,x(k)), k = 0, 1,"" n, denote n + 1 points in Rn that 1
1) (Sperner [I928].)
n
I~.
, 7/0
I
1) In the case n = 1, the simplex is a line segment; for n = 2, it is a triangle; for n = 3, it is a tetrahedron.
246
§7. PROOF OF BROUWER'S THEOREM
V. UNIVALENT MAPPING OF MUL TIPLY CONNECTED DOMAINS
eO).
such a way that, if two subsimplexes have points in common, then the set of all
point
common points is a common face of some dimension or else a vertex of these
~J'
simplexes. Such a partition can be made as follows, for example: let us look at
of the numbers
all possible permutations
io' il"'" in
... J'
'n
Let us now look at the question of the dimensions of the simplex ,
keeping the notation given above for it. Let d denote the greatest
!xV) - xV ')1
of the numbers 0, 1"", n. Let us
denote by ~iO,il'''',in the set of points x € with Aio SAil:S ...
J'
0' l'
I '"I
.(0)
~ that have the representation (1)
:s \n' .Let us show that
(j'l
-Xl
for i
I--n+l"':' I "\'1 «}) (j'l XI -Xl )
~iO,il'''',in is a simplex. Let
(since one of the numbers xli) for , - x~i'1, , Therefore, if
X
(xl' " ' , x n } €
n
xi·) n -k + l'
elk) .=k I =-~--
i-, l " ' , n.
le}Ol-xII=
(2)
x/v xi
~ and
I! n
Xi
=(n--~+l)el')-(n-'1)el'+l), -"I
nd " ~n+l' J =0, .. " n,
i=
0,1, " ' , n, must be equal to 0).
= ~7=oAiX~i),
i
= 1",', n,
then
n
I d Ai(e~O)-xY'l) ~n+l ~
nd
A;=n+l'
/=0
In particular, for points ~ (k 1, k = 1, 2, ...
(jnl_dn)
0, " ' , n.We then have
I
i=O
From (2), we have (. )
n
I
i, i'=
1,"', nand
=
1=0
~ ( k) (~ik), .•. , ~ ~k)} (k = 0, 1,'" , n) d~note the center of a!1 (n - k}-dimensional ~ . h' (J k ) . I ,x (Ik + I) , ... , x(J n ). Th en, f ace 0 f t h e simp ex ..:. wIt vertIces at x
~
247
,n,
we have
I e!O)-e(k)I~~ , '~n+l' '1=0, I, ... , n-l, Reasoning analogously with the (n - I}-dimensional face of the simplex
•
~i o,i 1"" ,i n with vertices at ~ (l),
Consequently,
••• ,
~ (n), we can show that
_n_d<~ l e\l)_e(k)l
X·= I
(. )
n
~ Ai"t1 x.J· = ~ ~ A't!')
l. 'J~t' .=0 '=0
for the points ~(k 1, k = 2, 3, ... , n.
~
We proceed in this way with faces of fewer dimensions. As a result, we see that all the numbers I~~i) t:~i , - S,
where A~= (n+ I) Al o'
A~=(n-'1+l)(Al.-Ai._l)' n
n
d for the simplexes ~J'
'1=1, ... ,11,
.=0
for i = 1, ... , nand
i, i' = 0,
1, .•.
,n,
do
J'
0' I'
••• J'
'n
do not exceed nd/(n + I}. H we now repeat
this partitioning process for the subsimplexes with each of the simplexes
n
~ A~= ~ Ai. = ~ A.= I.
.=0
')1,
not exceed nd/(n + I}. Consequently, the quantities corresponding to the quantity
~iO,il"",in' then with the simplexes obtained from these, etc., the simplex ~
.=0
is partitioned
afte~
p stages into subsimplexes having in common only faces of
It follows that a point x belongs to ~i o,i 1"" ,in if and only if it belongs to the ••• , ~(n). Therefore, the set ~io.il"",in
various dimensions and having the property that the difference between corre
simplex with vertices ~(0l, ~(l),
sponding coordinates of the vertices of each of them does not exceed
coincides with this simplex. The simplex ~ is therefore partitioned into (n + r)l
(n/(n + I}}Pd. in absolute value. Let ~' denote any of these subsimplexes with
simplexes ~iO,il'... ,jn' Two simplexes ~iO,il"",in and ~io,il '''',i~ have in common only those points x for which there are equal numbers among the numbers
vertices
y(k)(y(k), .,. , I
Y (k)} for k = 0, 1, •.. n
n
y, y'
Ao' A!, ... ,An' that is, for which certain of the· A~ are equal to 0. These points
EE', Yi =
~io,il"",in and ~jo,i{'''·,j~. n particular, all these simplexes contain the 1
.:1
.
.'.'
~:'
Then, if n
~ AjYV), Y; = ~ A,Sill,
1=0
constitute the common faces of different dimensions of the simplexes
,n.
;=0
248
§7. PROOF OF BROUWER'S THEOREM
V. UNIVALENT MAPPING OF MULTIPLY CONNECTED DOMAINS
249
The coordinates of the vertices of these subsegments are the numbers
we have n
IYi -
Y;
I= I~
(), j=O
n
j -
A;) y}Jl
I= Ij=O ~ (~j 9 (yyl -
y}Ol
-
we proceed from the vertex of
I
1
°
and 1. H
with coordinate 0, the first of the ak that we
encounter with different coordinates has coordinates (0, 1), the next has coordi nates (1,0), then (0, 1), etc. (Here, we are indicating the coordinates of points
n
~ (n-t- ltd! (Aj+Aj) =2 (n-t- ltd
met successively as we move along 1,) The last of these ak must necessarily
1=0
bave coordinates (0, 1) because the second end of
1
must have coordinate 1. This
proves that the number of segments ak with distinct coordinates is odd. Let us
and
p (Y, y') =
{.! 1=1
(Yi - y;)2
now suppose that the assertion has been proved for an (n - 1) -dimensional
~ Vn2t~ +- I) 2P d 2
simplex and let us prove it for an n-dimensional simplex. Let tk denote the
~2vn (n-t-
It
Dumber of faces of the simplex ak with coordinates (0, 1, ... , n - 1). H tk > 0,
d.
then the simplex ak has coordinates (0, 1, .•. , n - 1, a), where 0::; a ::; n. Here,
(3)
.;;' .if a = n, we obviously have tk = 1. On the other hand, if a
< n,
then tk = 2
It follows that the diameters of all the subsimplexes after a suffic ient number p
~Cibecause the coordinates of all n + 1 faces of the simplex ak are obtained from
of stages in this partitioning process will not exceed an arbitrary given number
,
f>
0. This is the desired partition of the simplex
1.
(0, 1, .•. ; n - 1) only by deleting the number a when encountered twice. Let e denote the number of all simplexe s
To prove Brouwer's theorem, we need first to prove two lemmas. Lemma L
Let
1
(0, 1, .•. , n - 1, a) by discarding a single number and we obtain the numbers
denote the number of all ak for which tk
denote an n-dimensional simplex with vertices xU) [or
Ej , [or.i
(3)
2. Then, from what was said above,
~ tk=e+2j.
(4)
k.=1
0, 1, "', n, in such a way that the [ollowing three conditions are satis[ied: (1) each point x € 1 belongs to at least one o[ the Ej ; (2) x(j) €
Ej ;
=
p
j = 0, 1, ... ,n. Suppose that the points o[ 1 are distributed among n + 1 closed sets
ak with coordinate s (0, 1, .•. , n) and let [
=
On the other hand, if a face of a subsimplex ak lying inside
belongs to two of the simplexes ak. Let g denote the number of such faces
[or k Ie- j. Then, the sets Ej' [or j = 0, 1, ... ,n, have at Ie ast one common point.
1
belonging to distinct ak' A face of a simplex ak lying on one of the faces of the
into a finite number of subsimplexes
simplex
aI' a2' ... , ap in such a way that any two of these simplexes have in common
< v2 < ... < v q ,
intersections
simplexes. Consequently,
sn
nE
j,
j
=
0, 1, " ' ,
n -
1, that satisfy
Sn
will have an odd number h of faces
Sk
with
coordinates (0, 1, ... , n - 1). Thus, we have
p
assign the n numbers that are the coordinates of the vertices of ak in it. Let us
~ tk=2g+h.
k=1
show that the number of simplexes ak of which all the coordinates are distinct 1 because then
is an In - I)-dimensional simplex the points of
Our assertion is considered proved for the case of (n - I)-dimensional
let us assign to it the
n + 1 numbers which are the coordinates of its vertices and to each face let us
=
n
sn
condi tions (1), (2), and (3) of the Ie mma.
number VI as coordinate. To the simplex ak let us assign as coordinates the
is an odd number. This assertion is true in the case n
But
Sn'
which belong to the
Ej for j = 0, 1, ... ,n. Each
vertex of the simplex ak belongs to one or more of the sets Ej • H it belongs to the sets E VI' E v2' ••• , E vq ' where VI
1, on the other hand, will have coordinates (0, 1, ••. , n - 1) only if
this face lies on
only faces of some dimension or other. Let us show that among the ak there exists a simplex containing points of each set
has coordinates
(0, 1, ... , n - 1), it appears twice in the sum l~=Itk because each such face
Ej does not contain a single point on the [ace s j with vertices in x
Proof. Let us partition the simplex
1
1 is a
Comparing (4) and (5), we obtain e
line segment which is divided into subsegments ak by a finite number of points.
=
(5)
2 (g - [) + h; that is, e, the number of
simplexes ak with coordinates (0, 1, ... , n), is an odd number. Consequently,
.;.~
II:.
§7. PROOF OF BROUWER'S THEOREM
V. UNIVALENT MAPPING OF MUL TIPLY CONNECTED DOMAINS
250
fewer than n + 1 sets E k and this point is a boundary point of E (that is, not an
there exist simplexes with coordinates (0, 1, ... , n). Each such simplex contains
Ej, for j = 0, I, ...
points be longing to all the sets
,n,
interior point of E), then, by changing the sets E k in an arbitrarily small
because its vertices
neighborhood of the point a, we can arrange to have every point of the set E
Ej •
belong to distinct
Now, suppose that we are given a sequence of positive numbers
1, 2, •.. , that approaches
°
as k
-->
00.
(k,
for k
belong to no more than n of the sets E k •
=
Proof. Suppose that I is a simplex of arbitrarily small diameter and that a
I into a finite number
Let us partition
(1'
Suppose that among the subsimplexes a~ is a simplex containing
points of all the sets
Ej ,
for j
=
,n.
0, 1, " .
simplex exists. Let us now partition
Ej •
of the sets
I and
are closed, the point a belongs to all the sets
Ej , for j
ak. =
Since the sets
f>
Ej
assigning to
Suppose that
0, 1, " ' , n. Let us form closed sets
with center at an arbilrary point x E: D con
Ej ,
for j
=
AJ
O
Ek • Let us now cover S 1, 2, ... , with diameter less than
),
simplex as the union of n closed sets with a unique point in common, each of
where
which contains one of the vertices of the simplex. Let us take one of the vertices
0, 1"", n, by
of the subsimplexes, for example b. Suppose that it belongs to the simplexes I for j
for all j = 0, 1, ...
,n,
=
which would contradict the
j,
1, 2, ... ,h. The point b belongs to only one of the closed sets into which
each of the simplexes sets, let us say,
F/,
I
j
is divided. Consequently, b is contained in h such
for j
=
1, 2, ...
,h,
equations
that lie respectively in
I j•
Let us set
h
n
Fk
n
;=0
Ej,
for j
U
F~.
We reason in the same way for all the vertices of the subsimplexes. Then, the set
j=O =
=
')=1
~ Aj = ~AjO)= I. For the same reason, the sets
=
show that such a covering exists. Let us partition S into (n - I)-dimensional
Ej all points of the set I for which Aj ~ AiO) in the representation I lies in at least one of the sets Ej because, other
< AiO)
f
simplexes with diameters less than d2n and then let us represent each such
(1). Each point of the set
wise, we would have Aj
such that every ball of radius
sists of points belonging to no more than n of the sets
I into sets Ej as indi Ej have a unique common point a, which is I has the representation (l) and that corre
sponding to the point a are the values of the parameters Aj equal to =
°
with a finite number of closed sets F j, for j
0, 1, ... ,n. This com
Let us point out one way of partitioning the simplex
Ai O)';' 0, for j
n E. The set D does not include any points common to more than n Ek. Consequently, every point xED has a neighborhood whose inter
( such that each point of S will belong tp no more than n of the sets F j • Let us
cated in Lemma 1 such that the sets
I.
I).
the Heine-Borel theorem, generalized to Rn , it then follows that there exists an
pletes the proof of the lemma.
an interior point of
(that is, the set of points on all of the faces of
section with E can be covered by no more than n of the Ek.ln accordance with
a an arbitrary neighbor
hood of which contains infinitely many of the simplexes
by taking an arbitrary simplex,
Define D = S
We continue the process. As a result, we
Rn , there exists in I a point
I
denote the boundary of
the sequence of diameters approaches 0. In accordance with Weierstrass's theoren generalized to the space
I
tion of a, and then performing a similarity transformation on it about a. Let S
From what we have shown, such a
I into a finite number of simplexes of the
obtain a sequence of simplexes a~, a'z, ... each of which is contained in
We can get the simplex
displacing it in Rn in such a way that some interior point of it occupies the posi
same kind but with diameters less than (z. And suppose that a~ is a simplex containing points of all the sets
I.
is an interior point of
of simplexes having in ~ommon only faces of different dimensions with diameters less than
251
S is completely covered by the sets Fk, and each point xES belongs to no more than n sets F k because an interior point of a subsimplex belongs to no more than n of the sets F k into which it is partitioned and, consequently, to no more than n of the sets F k. On the other hand, a point on a face of a subsimplex can belong to only those F I. that contain the vertices of that face and hence to no more than n - 1 of the sets F k. Furthermore, one can easily show that the diameters of all the sets F k is less than (. Thus, the sets F k provide the
0, 1, ... ,n have the point a as their
unique point in common. Furthermore Ej obviously contains the point xCi) and does not contain points of the face s j' We shall use this fact in what follows. Lemma 2. Let E denote a closed bounded subset of R n. Suppose that E is covered by a finite number of closed sets E k such that each point of the set E
belongs to at least one of them. If E includes a unique point a belonging to no
1
252
V. UNIVALENT MAPPING OF MULTIPLY CONNECTED DOMAINS
~7. PROOF OF BROUWER'S THEOREM
required covering for S. Let us now combine the sets Fk with the sets E k as follows:
1) If
253
E' assigns to the sets Ej closed set·s E/ in l ' that have the unique com common point a'. If a' were a boundary point of E' and hence of 1', then, in accordance with Lemma 2, by modifying the sets E/ in a neighborhood of a', we onto
F k does not contain points of D, we combine it with anyone of
Ek. 2) On the ·other hand, if Fk does contain points of D, we combine Fk with one of the sets Ek to which such points belong. Let us show that each
the sets
could arrange for the sets
E/, j = 0,
1, ...
,n,
to have no point in common. With
point xES still belongs to no more than n of the sets E k. This is true because
such a covering, the corresponding modified closed sets Ej have no common
a point xES that does '!ot belong to D belongs to no more than n of the sets F k
points and at the same time satisfy the conditions of the lemma, which, however,
and hence now belongs to no more than n of the sets Ek' On the other hand, if
is impossible. The contradiction proves Brouwer's theorem.
xED, it belongs to no more than n of the original sets, that is, to E l' E 2' ••• , E f> where f.:s: n. On the other hand, x belongs to certain of the sets Fk, let us say to F l' F 2 " ' " Fg , where g ~ h. The sets FI"'" Fg are contained in the ball of radius (with center at x. But, from our construction, this ball con :1,
tains points belonging to no more than n of the original sets E k, let us say to the sets E l' E 2' ••. , Eh , where f.:s: h ..:s: n. From 2), the sets F l' .•• , Fg are combined with certain of the sets E h
••• ,
Eh' Consequently, a point xES can
belong to no more than n of the modified sets Ek , namely to certain of these sets that contain the original sets E I' ... , Eh' Thus, after we adjoin the sets Fk to the sets E k , every point xES belongs to no more than n of the sets E k • Since a is a boundary point of the set E, the simplex 1 contains an interior
\"'
point c that does not belong to E. Let us distribute the points of the sitnplex 1 and heJlce let us redistr ibute the points of the set K
=
E U1 among the sets Ek E v2' .•• , E v p ' where
as follows: If a point xES belongs to the sets E vI' p .:s: n, then the entire segment cx (one-dimensional simplex) belongs to all these sets and only to them. With this new covering of the set E U 1, the point c is the only point that can belong to more than n of the sets Ek • But c does not belong to E. Consequently, E does not contain points belonging to more than n of the sets E k • This completes the proof of the lemma. Brouwer's theorem. A topological mapping of a set E C R ll onto a set E'
c
R~ maps interior points of E into interior points of E •
Proof. Let a denote an interior point of the set E and let 1 denote ail arbi trarily small simplex contained in E and including a as an interior point. Corre
E', and corresponding to the simplex I 1'. Let us subdivide 1 into n + 1 closed subsets Ej • j = 0, 1, ... ,n, in such a way that
sponding to the point a is a point a' in is a set
each Ej
l' c E'.
Let us show that a' is an interior point of the set
contains only one vertex and no point of the opposite face and in such
a way that a is the only point belonging to all the sets Ej • The mapping of E
,.,
I
···"---r .-
--------.-T-.i'q.. -'·-~~ - T---
~~~#iI~.--.--_._--
f-:-'-~C
. -Ii .-
~
.... "".
--~.;:;;:;L:;iii;
255
§1. MAPPING OF A MULTlH..YCONNECTED DOMAIN
without loss of generality that case, the function t
=
c:
and b are finite and let us observe that, in this
log «z - a)/(z - b)) can be extended into B along an arbi
trary path and it maps the domain B onto the t-plane with the point The inverse function z
CHAPTER VI
=
excluded.
00
¢(t) is regular in that plane. Therefore, if there existed
a function (= f(z) that maps the domain B conformally onto the disk function (= f(¢ (t)) would be regular in
1t I <
00
1(1 < I,
the
and it would not exceed unity in
absolute value, and this cannot be the case.
MAPPING OF MULTIPLY CONNECTED DOMAINS ONTO A DISK
Let us turn now to the proof of the basic theorem.
Theorem. 1 ) Every domain B in the z-plane with mor.e than two boundary points can be mapped con formally onto the disk \ (I
§ 1.
Confonnal mapping of a multiply connected dom,ain onto
([or a basic element of the mapping function) a point (= 0 and the positive di
a disk In
§ 1 of Chapter
rection of the real axis. This mapping is unique.
II, we showed that no regular function can map a multiply
connected domain bijectively onto the disk
< 1 in such a way that corre
sponding to a given point Zo € B and a given direction at that point there are
I(I < 1.
Proof. We note first of all that the domain B can be mapped onto a domain
Let us now consider, in
mean an initial element of a function and all possible analytic continuations of
B* contained in I(I < 1 in such a way that to each point of B* corresponds only one point of B. To see this, let a, b, and c denote three boundary points of the domain B. By means of a fractional-linear mapping, let us map the z-plane onto
that element into B. As we shall now show, among these functions there is one
the t-plane in such a way that these three points are mapped respectively into 0,
addition to regular functions defined on a multiply connected domain B, functions that can be extended into B along an arbitrary path. Here, by such a function we
with the property that all values that it assumes in B lie in the disk that each value in the disk
1(I < 1
I(I < 1
1, and
and
00.
The domain B is then mapped into a domain B* that does not include
the three points 0, 1,
is assumed at exactly one point of the domain
00.
Therefore the function (= '\(t), inverse to the modular
B. We shall call the correspondence that this function sets up between points of
function t = fl(z) (see §6 of Chapter II) can be extended into B* along an arbi
the domain B and points of the disk I(I < I a conformal mapping of the domain B onto the disk 1(I < 1. The conformality follows from the fact that all elements
a domain B*. Under this mapping, to every point of the domain B* there corre
trary path and the values that it assumes all lie in the disk
I(I < 1
and occupy
quently, their first derivatives do not vanish anywhere (that is, at any finite
sponds as preimage only one point of the domain B* and hence only one point of B. By means of a fractional-linear transformation of the disk I (I < 1 into itself,
point). The mapping function, although it is regular at every point of the domain
we can always arrange for the point (= 0 and the positive direction of the real
B, is not univalent in B because, if it were, it would set up a one-to-one corre
axis to correspond to the point z 0 and a given direction at z o. (Other points in
of the mapping function are univalent in the corresponding domains and, conse
spondence between the domain B and the disk
1(I < 1,
the disk
and we have seen that
this cannot be. In the case of a finite domain B, the inverse function can be ex tended into the disk tion in tlIat disk.
1(I < 1
\(1
< 1 may also correspond to
zo.)
Proof of the theorem is now reduced to proving it for the case of a domain
along an arbitrary path and hence is a regular func
contained in the disk
I z I < 1.
The given point z 0 € B is the point z
=
B
0 and the
given direction at that point is the direction of the positive half of the real axis.
Before proceeding to a proof of the possibility of such a mapping of a multiply connected domain onto a disk, let us first mention the exceptional case when such a mapping does not exist. This is the case when the domain B, constitutes the entire z-plane with two exceptional points a and b. To see this, let us assume
254
,
.j, I,
1) This theorem was discovered by Poincare. Rada noted [1922-1923a] that the idea of the proof of Riemann's theorem that was given in §2 of Chapter II can be used to prove this theorem also.
VI. MAPPING ONTO A DISK
256
In this case, let us consider the set ~
=
§L
If(z)l of all functions fez) with the fol
lowing properties: 1) they are defined by basic elements about z
f(O)
=
0 and
f' (0) > 0;
=
it follows, on the basis of Rouche 's theorem, that the function fnk (z) assumes
0 for which
the value c at two points of the domain B, which is impossible. This proves our
2) they can be extended into B along an arbitrary path;
assertion.
3) the values that these functions assume in B as a result of analytic continuation all belong to the disk
1(I
< 1; 4) every value of ( in the disk
1'I
Let us show now that the function fo (z) provides the mapping mentioned in
< 1 is assumed
the theorem; specifically, that it assumes in B every value ( in
by every function at' no more than one point of the domain B. An example of such-a function is the function fez)
=
to the contrary, that there exists a number c such that
az, where 0 < a ~ 1.
everywhere in B. Let us denote by t
We pose the following extremal problem: Out of all functions fez) € ~, find the one maximizing
f' (0),
there exists a disk
I
function F (z)
ment of each of the functions fez) E: ~ is regular and less than unity in absolute shows that the set of numbers
f' (0)
n = 1, 2,"', such that lf~
Suppose that (=
is bounded above. We denote the least upper
In (z) €
(O)l converges to
1 z I < P to a regular function fo (z), which is not a constant, since f ~ (0) = a> O. If we now extend these el~ ments into B along different paths, we see step by step in accordance with
W·'
=
zll <E
and
Iz - z21 <E
~ 1
> 0 on the circles
r.•
that Ifnk (z) - fo (z)!
<m
and
Iz - z21 = E.
1
c
= (Ink (Z) -
10 (Z)
1'1
<1
bijectively Ollfo itself. Conse
don of fez). In particular, if we take two distinct elements of the same multiple
both of which map the domain B conformally onto the disk
I(I < 1.
It follows
from what has been said that these branches can be expressed each as a fractional linear function of the other. Thus, all branches of a multiple-valued mapping func Oidon can be represented as a fractional-linear function of just one of them a,nd the
>
r
Let kdenote an integer such
Y+ (/0 (Z) -
Now let (= fez) denote the function providing the mapping indicated in the
...alued mapping function fez) at the point Zo and extend them along all possible
corresponding fractional-linear functions map the disk
I(I < 1
into itself.
This conclusion enables us to study more deeply the nature of the multiple
on these two circles. Then from
f nk (z) -
in the disks
paths contained in B, we obtain two functions, that is, two branches of fez),
have no common points, that they are contained in B,
Iz - zll = E
are two functions providing this mapping.
quently, it must be a fractional-linear function. Thus, F(z) = (af(z) + (3)j(y fez) + 0);
and that these elements assume the value c neither in the interiors of these m
h (z)
that is, any other mapping function can be expressed as a fractional-linear func
c. Here,
disks, except at the center, nor on their boundaries. Suppose that Ifo(z) - c
and (=
the function (' = F(f- 1 «(n maps the disk
fO(ZI) and fO(Z2) are the values at the points zl and Z2 of certain elements of the function fo (z) that are obtained from the basic element by analytic continua tion along certain paths 11 and l2 contained in B. Suppose that the disks Iz -
II (z)
Onto a disk but not satisfying the same normalization conditions as (= fez). Then,
~ 1 at no more than one point. Le t us suppose, to the contrary, that there
fo (z2)
> 1. We see that the > a, which is impossible.
0 and ¢' (0)
IJJl and F' (0) = f~ (0) ¢ '(0)
l>receding theorem. Let (= F (z) denote any other function mapping the domain B
everywhere. Let us show that (= fo (z) assumes in B every value in the disk =
along an arbitrary path
It.,~,,
the corresponding analytic continuations of the function fo (z). Consequently, the
'I
=
I(I < 1
~: the theorem.
Vitali's theorem that the sequences of analytic continuations obtained converge to
1
I (I < 1. Suppose,
and fez) .;,. c
h ([11 «(' n are 1(I < 1 and ,(' 1< 1, respectively. Therefore, the function ( ' = ¢«() maps the disk 1(1 < 1 bijectively onto itself in such a way that ¢(O) = 0 ri~iand ¢' (0) > O. But then, ¢«() == (; f 1 (z) = h (z). This completes the proof of r regular
that converges uniformly in the interior of the disk
exist in B two distinct points z 1 and z2 such that fo (z 1)
<1
Then the function (' = II ([i «(n = ¢ «() and its inverse (=
~,
a. According to the condensation
Ifo(z)j
1
1
principle, the sequence of basic elements of the functions fn (z) has a subsequence
function fo(z) can be extended into B along an arbitrary path and
¢ ([0 (z» E:
c
It remains to prove the uniqueness of the mapping mentioned in the theorem.
(0) ~ 1/ p. This
bound by a. Obviously, a? 1. There exist s a sequence of functions
=
I
¢ «() the function used in § 2 of Chapter
that does not pass through c and that ¢ (0)
z 1 < p, where p > 0, that lies entirely in B, the basic ele
f'
=
TI. We note that this function can be extended into
calculated for the basic element of the function. Since
value in this disk. Therefore, by the Schwarz lemma, we have
257
MAPPING OF A MULTIPLY CONNECTED DOMAIN
valuedness of the mapping function fez). Let fo(z) denote one of the branches of the function fez). We denote by ('
C)
=
S«() = (a( + (3)/(y( + 0) or more briefly,
by S the fractional-linear function mapping the branch fo (z) into the branch
,4
i~
~'M
_Ii
_::liiiiiUiiIlWlIliI&:&\W4\liiiiJi£WU
RW*1i!i&iMt'IIIIlW
~J2kiiiiiiii
)Q~~1'"
JliMF
§l.
VI. MAPPING ONTO A DISK
258
f* (z); that is, f* (z)
MAPPING OF A MUL TIPL Y CONNECTED DOMAIN
259
If a transformation 5 € G other than the identity transformation has two fixed
5fo(z). Let us show that the set G of all "transformations" 5 corresponding to all possible branches of the function fez) constitutes a group. =
points on the circle
1(1
=
1, it is called a hyperbolic transformation. If it has
Let us extend the branch 5 do (z), 51 € G, along a closed path such that
only one, it is called a parabolic transformation. Let us show that, if 5 is either
extending the branch fo(z) along this path yields the branch 5z!0(z), 52 € G. We
a parabolic or a hyperbolic transformation, the sequence of transformations 5, 52,
thus obtain a new branch 5 I 5z!0(z) of the function fez). Therefore 5152 € G.
53, ... does not include the identity transformation.
Analogously, if we extend the branch fo (z) along a closed path
th~ extension
of
Let us suppose first that 5 is a hyperbolic transformation with fixed points
fo (z) along which yields a branch 5fo, 5 e G, though in the opposite direction, we obtain a branch f*(z) = 5*fo (z), where 5* € G and we again obtain from 5fo(z) the branch fo (z) and at the same time the branch 55*fo (z); that is, we have fo (z) = 55*fo (z), so that 5* = 5 -1. Consequently,S -1 € G. Thus, G is a group.
(I and (2. If we define the transformation t = T(() = (( - (1)/(( - (2), we see
T5T- I (() is a fractional-linear transformation with fixed points a and Therefore, (' = a( and, consequently, T5(() = aT((). This last shows that
that (' 00.
=
the transformation ('
=
5(() is obtained from the equation
We note that the neutral element of this group is the identity transformation.
t' - t l . ~-t t'-t =a~, a=const.
With regard to the function z = ¢ ((,) inverse to (= f(z), when B does not contain "", it can be extended into
1(I
2
< 1 along an arbitrary path and hence is
1(I
Now, if the point ( in (1) lies on the circle
< 1. From this we conclude that, in the general case, its only singu larities in the disk 1(I < 1 are poles; that is, that it is a meromorphic function in 1'1 < 1. It follows from the relation z = ¢(f(z)) that ¢(5fo(z)) = ¢(fo(z» or ¢(5() = ¢((), where 5 € G. This shows that the function ¢(() remains invariant under transformations belonging to the group G. Therefore, it is an automorphic function
regular in
also lie on the circle
=
1, then the point (' must
1. Therefore, the arguments arg((('-(I)/(('- (2))
=,
at the points (' and ( are equal. From this we conclude that a
a ~ 1 because we would then have ('
> O. Furthermore,
which cannot be. Thus, the transforma
5 can be obtained from equation (1) by solving that equation for ('. Here, a n is positive and different from unity. The transformation (' = 5 ( (where n = ± 1, tion
We noted above that all the transformations belonging to the group Gleave
I(I
1(' 1=
1(1
and arg ((( - (I) (( - (2)) indicating the angles subtended by the segment (1(2
with group G. the circle
(1)
"-"2
=
± 2, ... ) can obviously be obtained from the equation
1 unchanged. Let us show now that the fixed points of all these
transformations lie on the circle
'(I
=
1. By fixed points of the transformation 5,
we mean points ( satisfying the equation 5( =
(.
t'-C1=an • ~-tl.
A fractional-linear trans forma·
~'-'2
(2)
t - t2
tion other than the identity transformation has no more than two fixed points. Let Therefore, it is always different from the identity transformation. Furthermore, it
us suppose that a given transformation 5 € G has a fixed point a such that
Ia I < 1.
follows from (2) that the sequence of the functions 5 n ( and 5-
Then, 1/ a will also be a fixed point of 5. But the equation 5a= a
implies that 5( ~
a in
a sufficiently small domain
a < I(I - al < 8.
n
Suppose that
I5( - al > m > a on the circle 1(- al = O. Let (3 denote an arbitrary number in the domain 0< I( - aj < m. Since 5(() - (3 = (5(() - a) + (a- (3), it follows from Rouche's theorem that the function 5 (() - {3 has a zero y in the disk I' - al < O. Here, y is distinct from a and (3. It follows from 5 (y)
=
=
can be nODe in the domain
1'1 > 1.
We proceed analogously in the case in which 5 is a parabolic transformation
(' =
=
that the transformation ("
DO
1(\ =
T5T- I (() is a transformation with unique fixed point 00. Therefore, (+ a and, consequently, T5(() = T(() + a, where a = const. This shows
that ('
1(\ < 1 and hence that there
Therefore, all the fixed points lie on the circle
§2.
with fixed point (I' By making the transformation t = T (() = 1/(( - (1) we see
¢(() maps a sufficiently small neighborhood of the
fixed points of the transfonnations 5 € G in the open disk
converge as
to the constants (I and (2' This convergence is uniform on an arbi
00
be necessary for
{3 that ¢ (y ) = ¢ ((3).
point (= a univalently. This contradiction leads to the conclusion that there can be
+
(
trary closed set not including the points z 1 and z 2. This last conclusion will
But, for sufficiently small 0 and m, this last cannot be since ¢' (a) ~ O. Conse quently, the function z
-->
n
=
5( is obtained from the equation 1 1 ~'_~I = t-'I
+a,
1.
.~
i'
(3)
i!{ajW;:;;1ffi)j¢j"'MM:t"~Wi\t":~'4it;*'~t-,j;¥"\l'~'t,;;;,,-\t,,,.,~,,-\:;,,,\,:",, "i:Vi~,¥-'U!i:;i'i.-;i~~;I,£{~':;Eiltj;'~"'\#"':"~.J;}.l:jfu~~m::c'* ":rlt:';:2f'if-'y,r~¥Jrti1i~'gt"(~~~"'"i.l)V',"1z'ji""J'l.J.:!j!@i"
260
VI. MAPPING ONTO A DISK
§I.
where a 1= 0 because, in the opposite case, we would have (' impossible. On the other hand, the transformation (/ is defined by the equation
1
C' - C1
=
1
C- C1
=
which is
= (,
sn( (for n
=
261
point equivalent to it and cannot be reduced to a point. Thus, for us to proceed
±l, ±2,"')
in B along a closed path with some element of the function fez) and return to the original point with the same element of fez), it is necessary and sufficient that it be possible to reduce this path by a continuous deformation to a point without
+ na.
(4)
getting outside B. I) In conclusion, let us pause to construct a Riemann surface connected with
Therefore, once again, it eannot be the identity transformation for any n. We note
the mapping that we have been considering. We confine ourselves to the case of
that the sequences of the functions sn( and s-n( converge uniformly to (Ion an arbitrary closed set including the point (I as n
-->
a finitely connected domain B.
+00.
If the domain B is n-connected, we can render it simply connected by means
This property shows, in particular, that G is an infinite group, that is, that
of n - 1 suitable transverse cuts (broken lines). We then take an infinite set of
it contains an infinite number of distinct transformations and that fez) is an infi
copies of the domain B so cut and consider them as lying one above another. We
nitely-many-valued function in B. On the other hand, the function ¢«() assumes
choose anyone of them as the basic domain and denote it by B o. We adjoin to
1(' < 1 every value in B at an infinite number of points. All points in the disk 1'1 < 1 at which the function ¢ «() assumes the same value are called equivalent
in
Bo along each edge of each cut one of the remaining copies. We make this con nection-this gluing-along opposite edges of the cuts. As a result, we obtain
points. For every point a in the disk 1 a 1 < 1 there is an infinite set of points equivalent to a. These are all obtained from each other by transformations of the group G. Since these points are zeros of the function ¢ «() - ¢ (a), they do not have points of accumulation in 1(I < 1. It follows, in particular, that the group G
a (2n - I)-sheeted surface B 1 lying over the domain' B and having (2n - 3) (m - 2) free edges of cuts, that is, edges along which no gluing has been made. Then, we again adjoin along each of these edges one of the as yet unused copies of the cut domain B in such a way that we again perform the gluing along opposite edges of
is countable.
the cuts. We then obtain a new surface B 2 with a finite Dumber of sheets and
Since the function fez) is not single-valued in B, the question arises as to
some number of free edges of cuts. If we continue this way indefinitely, we finally
what conditions must be satisfied by the path l contained in B and issuing from
obtain an infinitely-sheeted Riemann surface Boo.
and returning to a point z 0 in such a way that as we proceed along it with some element of the function
MAPPING OF A MULTIPLY CONNECTED DOMAIN
f (z)
If we consider the initial element fo (z) of the mapping function fez) defined
we return to the point z 0 with the same element.
Consider the case in which B does not include
00.
in the domain B 0, the analytic continuation of this element into B is identical
If, as we proceed along l from
to its analytic continuation onto the surface Boo. As the result of all possible
z 0 with some element of fez), we return to z 0 with the same element, the corre sponding point (= fez) issuing from some point (0 returns to (0 after describing a closed path l' in the disk 1(I < 1. The path l' can, by a continuous (for example. similarity) deformation be reduced to a· point that does not leave the disk 1 (I < 1.
analytic continuations of fez) onto the surface B oe , we obviously obtain a single valued function on that surface. Consequently, the function (= fez) that we studied above also provides a one-to-one mapping of the surface Boe onto the disk
Obviously, under this deformation, the preimage l as it is continuously deformed
1(I < 1.
The surface Boe is called the universal covering surface of the
domain B. The modular surface constructed earlier (see §6 of Chapter II) is, in
also reduces to a point without getting outside B. Let us show that, conversely,
this terminology, the universal covering surface of the domain consisting of the
if the path l can, by a continuous deformation, be reduced to a point without ever
plane with the points 0, 1 and "" excluded.
getting outside B, then, by proceeding along it with an arbitrary element of fez), we always return to the original point with the same element of fez). This is true because, otherwise, under such a deformation, the corresponding path l', While being continuously deformed in ,(,
< 1,
will go from the point (0 to some other I) If B includes
j
00,
an analogous condition must be satisfied on the Riemann sphere.
262
~.
-
MAPpmG =TO
=: . --",'----::=::0: OOONDAro~ OF
§2. Correspondence of boundaries under a mapping of a multiplyO~
connected domain onto a disk~,;
In
§ 1,
,,~~
we examined a function fez) that maps a given multiply connected
domain B conformally onto the disk
I"
< 1 only inside B. If we now let the
to
point z approach the boundary of the domain B, we can easily show that the sequence of the correspon~ling points' approaches the circle In other words, for arbitrary from the point z
c
I:
1'1
";'1 uniformly.
.,
~
> 0, there exists a 0> 0 such that, if the distance
corresponding point 'from the circle
cB
I"
=
1 is less than
f.
'2,
cover the domain B l ' This shows that the inverse function
'1 which
is regular and bounded in B b approaches a constant uniformly, on a sequence of arcs connecting the points
It I < 1
'1 and
'2'
If we map the domain B 1 onto the disk
and use Lemma 1 of §3 of Chapter II, we see that the
tion. If
'0
='
const,
1'1
I" < 1 _ f.
=
1 is
1'I = 1, the curve A that we considered above 1'I < 1 whose only point in common with the circle
is its fixed point on
must be a closed curve in
To see this, let E
corresponding to points of the disk
approach 0 continuously, the corresponding curves A, all passing through and
which, however, is impossible. This shows that 5 must be a parabolic permuta
B to the boundary of B is less than 0, the distance of the
denote the set of all points z
263
~'
'0'
For values of z inside I, the corresponding points , lie inside
the curve A. But as r
->
'0,
0, the curve A is constricted to the point
'0
Conse
Then we need merely take for 0 the distance from the set E to the boundary of
quently, as we let z approach a, the branch fo (z) approaches
the domain B. When we make a more thorough study of the behavior of the function
we approach a with another branch 5fo (z) of the function fez), the corresponding
fez) as z approaches the boundary of the domain B, we shall confine ourselves
point' = 5fo (z) approaches the point 5'0' Thus, as we let z approach a, the
to the case of a finitely connected domain B. Such a domain can easily
be mapped
uniformly. 1£
only limiting values of the function fez) can be the points 5,0' where 1'0
1=
1
and 5 c G. When the entire boundary of the domain B consists of isolated points, it fol
univalently onto a domain bounded by closed analytic Jordan curves and isolated points. The correspondence of the boundaries under such a mapping has already been studied. Therefore, for our purposes, it will be sufficient in the future to
lows from what was said above that, as we let z approach the boundary of the
consider a domain B with a boundary consisting of closed analytic Jordan curves and isolated points.
domain B, the function fez) can have as its limiting values only values belong ing to a countable set of numbers, which are points on = 1. It follows that, as
Let a denote an isolated boundary point of the domain B, which we may assume to be finite. Suppose that the closed disk
Iz _ a I ~ r
, describes a path in the disk
Iz _ a 1= r
one time in the positive direction, the branch fo(z) of the function fez) becomes
that terminates on the circle
some point different from these points, the corresponding point z
contains no boundary
points of the domain B other than a. As we go around the circle I:
"I < 1
I"
=
1'1 =
1 at
•
a path that cannot terminate at a definite point. Thus, in the present case, the
.
function
the branch 5 0 f o Cz), where 50 ~ 1 is one of the transformations belonging to the group G considered in §1. Furthermore, if we go n times around I not neces
1'1 = 1 1'1 < 1.
does not have definite limits on
a countable set of points on that circle from
as , approaches any but
Now, let K 1 denote a closed curve constituting part of the boundary of B 1 •
sarily in t~e same direction each time), we end up at the original point with the
Let uS encircle K 1 with a Jordan curve I contained in B and defining, together
element 50 n fo(z). Let us show that the transformation 50 is a parabolic trans
with K b a doubly connected domain B' consisting only of points belonging to
formation. Let us suppose, to the contrary, that 50 is a hyperbolic transforma
B. If we go around the curve I with branch fo (z) once in the positive direction,
tion with two distinct fixed points
'1 '2 and
on
1'1
=
I"
we arrive at the original point with a branch 5do(z). On the other hand, if we go
1. If we choose a point
on I and move around I an infinite number of times with branch fo (z), the corre sponding point , describes a curve A in the disk circle
1'1
=
<1
< I z - al < r
the other), we arrive at the otlglnal point with branch 5
that approaches the
1 and, more specifically, approache,s the points
continue fo (z) in the domain 0
around I several times (possibly some times in one direction and some times in
'1 '2' and
If we
1'1
< 1. If we
Here, 51 is a
around the curve infinitely many times with the branch fo (z), the corresponding
along an arbitrary path, we see that
all values that the function assumes in the process lie in one of the domains, let us say B 1, into which the curve A partitions the disk
t fo (z).
transformation of the group G other than the identity transformation. If we proceed point , describes in the disk
'2
1'1 :s 1
a Jordan curve A that terminates at the
fixed points '1 and of the transformation 51 (which, for all we know as yet, may coincide). If we extend f (z) inside the domain B' , we obtain a multiple-
let r
i
°
264
§2. CORRESPONDENCE OF BOUNDARIES
VI. MAPPING ONTO A DISK
valued function in that dOlllain. If, by means of a cut, we map the domain B' into
265
of the group G. There are also countably lIlany of these.
I(I
a simply connected domain, all the branches of the multiple-valued function will
4) all remaining points on the circle
be single-valued in that domain and will lIlap it bijectively onto domains adjacent
This classification was made from the point of view of the behavior of the
to each other and filling up completely one of the domains, let us say B 1, into which the curve A panitions the disk
I(I < 1.
There is a continuous one-to-one
correspondence of the boundaries under these mappings. Therefore, the arcs corre sponding to the curve K 1 iie on the circle
I(I
=
1 and together they cover the arc
(I (2. In addition to the arc (I (2 there are other arcs on
1('
= 1 and these arise
in the same way from other branches of the function f(z) than Si'nfo(z) and from
=
function ¢«() in a neighborhood of the circle
1.
1('
=
1. If the function ¢«()
approaches a definite lilllit as the point (on the circle
I(I
=
1 approaches a
point a on that circle, the point a belongs to class 1) or 3) because then the point z = ¢«() approaches some point on the boundary of the domain B as (--+ a and, consequently, the point ( corresponding to it approaches either an interior • point of one of the arcs of the type (1(2 or one of the points of the type (0' Let
other curves constituting parts of the boundary of the domain B (with one excep
us show that almost all points on the circle
tion to be mentioned below). These arcs cannot overlap because, if they did,
is, that the sum of the lengths of all arcs of the type (I (2 is equal to 2". Let
1('
1 are points of class 1), that
=
domains analogous to the domain B 1 would have common points whereas either
us arrange these arcs in a sequence {~lk) ~k)l, for k = 1, 2, .. '. Let Uk «(), for
domains analogous to the domain B' have no common points or the branches of
k
=
1, 2, ... , denote a harmonic function that is bounded in
1(\ < 1,
that is equal
I(I
the function f(z) considered in them do not coincide at a single point in them. It
to 1 on the arc (~k) (~k\ and that is equal to 0 at all other points on
follows that the arc (1(2, which does not reduce to a point, cannot cover com
This function can be constructed in accordance with Poisson's formula 1)
I(I = 1; that is, (I l (2' The only exception is the case when B is a doubly connected domain one boundary of which is an isolated point. In this case, the group G consists of the powers of one of its transformations which pletely the circle
Uk
1('
<1
2
a curve that
+r
2
de, C= re 'rp.
(1)
00
~ Uk (C).
(2)
k=l
For (= 0, this series obviously converges. Funhermore, if we denote by sn«() the sum of its first n terms, we see that
Sn
«() S S n +1 «() in I(I < 1. It follows,
terminates at an interior point of one of the arcs obtained according to the type
in accordance with Harnack's theorem, that the series (2) converges uniformly
of the arc (I (2. On the other hand, as the point z approaches K along a curve
inside the disk
that encircles K infinitely many times, the corresponding point (describes in
1(1 < 1
I(I < 1
and that its sum
2"
1 \
s (C) ~ 21t ~.
It follows from what has been said that, if the boundary of a multiply con
nected domain B does not consist entirely of isolated boundary points, then
I(I
=
1 fall into four classes:
1) interior points of all arcs defined according to the type of the arcs (I (2' Corresponding to each such point is a unique nonisolated boundary point of the domain B, and at each of these ¢«() is regular. 2) points of the type (I and (2 are fixed points of hyperbolic transformations
S
«() is a harmonic function in I(I < 1.
Funhermore, it follows from (1) and (2) that
a curve that terlllinates at one of the endpoints of these arcs.
points on the circle
in
\(1 < 1.
sequence
S
~
l - rl ..
. , . de = 1
If we now let ( approach an interior point of the arc (lk)(~k), the
k«() --+ 1 and, since
S
k«() S S «() S 1, it follows that
S
«() --+ 1 as
( approaches an interior point of that arc. Furthermore, if we map the disk \ (I
<1
into itself by means of transformations of the group G, the arcs (lk) (~k) are then mapped into each other (because
¢ «() has definite limiting values only on
of the group G. There are countably many of these points. 3) points of the type of (0' that is, fixed points of parabolic transformations
1.
t~klc(:)
B
approaches a point on K 1 while describing a continuous curve, the corresponding point ( defined by an arbitrary branch of f(z) describes in
~ 1 2 l -(6r ) - rcos -rp
Consider the series
is a parabolic transformation. Thus, when we exclude this last one, we may assert that the transformation S 1 considered above is hyperbolic. If the point z €
1 (C) = 21t
=
1) This formula will also be proved in Chapter IX.
o=;=o="'~"'--
---_.. _---_._-
;;'·'Z';"''';''C;'.:r,''~'''~:.'''-'>;•.-i;,,;,;;,;,:;;:,;,.:·;,,;;o;;'''''"~~·'''I'·.~.
266
VI. MAPPING ONTO A DISK
"""'-:;;;.....;="-,,,...
=_-""'-,""'"
267
with either sign chosen is then regular and bounded in the neighborhood in ques
these arcs). Consequently, the function s «() is invariant under transformations
I"
..· ---":;;";,,,;.,,;,;;'';'~':;.,;; --
§3. DIRICHLET'S PROBLEM AND GREEN'S FUNCTION
of the group G. This last shows that, if we use the function z disk
"="""".~c,-"""""",,,",,,,,,,,,,,,",,;:';''';;,~;:;:',;;,-.;:::,.~
=
tion of the point z = a, and, with suitable choice of the value for F (a), regular
¢ «() to map the
at the point a itself.
< 1 onto the domain B, we obtain in B a single-valued bounded harmonic
Obviously, F(a) is nonzero. Therefore, the function u + iv
function is identically equal to unity in B because, otherwise, it either would
that a function u(z) that is harmonic and bounded in a domain B is harmonic at
min~mum inside B or would approach an extreme value as z approaches isolated boundary points, which, in accordance with the maximum
points.
principle and the beginning of the present section, is impossible.
I(I < 1.
00
S
~
«(t) (~k»)
de =
\~
21t
k=1
so that Lk=llength
The maximum principle of harmonic functions has an important generalization: tIf a function u(z) is harmonic and bounded above in a domain B (that is, its
But
values in B do not exceed some finite constant) and if all its limiting values, as
OJ
(0) = '\'1 ~
~ '\'1 length 21t ~
1
its argument approaches any except finitely many boundary points of B through
fr(k)C{k) \'0 1 2 ,
values in B, are no greater than M, u (z) ~ M everywhere in B. An analogous
"=1
C(k)c(k) 2
=
all isolated boundary points when it is extended in a suitable manner to these
attain a maximum or a
Consequently, s «() '" 1 in the disk
=
a and the function u(z) is harmonic at z
a. It follows
function s (f(z)) that is equal to unity at all nonisolated boundary points. Such a
is also regular at z
generalization holds for functions bounded below.
277. This completes the proof.
=
Proof. We may assume that the domain B includes the point
00.
Let al" ..
.. , an denote the exceptional points referred to and let H denote the diameter
§3. Dirichlet's problem and Green's function
of the boundary of the domain B. For arbitrary
A finite real function that, together with its first two derivatives, I) is single
f> 0, consider the function
n
valued and continuous in a domain B of the z-plane and satisfies Laplace's
v(z)=M - u (z)
equation in B is said to be harmonic in B. Such a function can be regarded as
+e !tog
I _
H ~,.Il
k=l
the real part of Some analytic f unction that can be extended in B along an arbi
which is harmonic in B. As z approaches boundary points other than al"", an,
trary path. A function is said to be harmonic at a point if it is harmonic in some
this function approaches nonnegative limits since each term in the sum above has
domain including that point. We note that, if u (z) is a harmonic function in a suf
a nonnegative value. As z approaches one of the points ak, this sum approaches
ficiently small neighborhood of a point z
+ 00, but u(z) is bounded above. Consequently, the limit of v(z) as z approaches
=
a with the point a itself deleted and
if it is bounded below in that neighborhood, that is, if u(z) that is, if u (z)
S. M),
it will also be harmonic at z
is extended to include the point z
=
= a
~ m
(or bounded above,
. aoy of these points is + 00. Since v (z) is obviously not a constant, it cannot
if its domain of definition
lJave minima inside the domain B. Therefore, v(z) n
a in a suitable maoner. To see this, let v(z)
ll(z)~M +e !tog~
denote the harmonic conjugate of U(z) and let w denote the increment in the func tion v (z)
as z moves once around a sufficiently small circle
sufficiently large circle
Iz I =
R in case a .
=
Iz - al
= f
~
0 in B; that is,
,.
k=1
(or a
This holds for arbitrary
(0). The function
f
> O. By letting
f
approach 0, we obtain u(z):S M,
which completes the proof.
+ 2" (u+lvj
Let us new pass on to Dirichlet's problem. Suppose that the domain B has a
F(z)=e- '"
boundary
K consisting of a finite number of closed Jordan curves without com
mon points. The Dirichlet problem for such a domain consists in 1) The derivatives here refer only ro derivatives at finite points. In what follows, the harmonic functions mentioned may at rimes be multiple-valued, bur in such cases an arbi trary branch of such a multiple-valued function will be single-valued in an arbitrary suf ficiently small neighborhood of each point in the domain B.
fin~ing
a func
tion u(z) that is harmonic and bounded in B, that is continuous in B except possibly at countably many points on the boundary K; and that assumes prestated
~:j.
·
1 ~
.! .
i
values at points of continuity on the boundary K. The boundary values are subject
268
269
§3. DIRICHLET'S PROBLEM AND GREEN'S FUNCTION
VI. MAPPING ONTO A DISK
I(I < 1.
If we now return to the z-plane, the function V
«;) becomes
only to the condition that they constitute a function on K that has a finite or
identical in
countable set of points of discontinuity. Let us show that this problem always
the function u(z), which is single-valued and bounded in B and which satisfies
has a unique solution.
the Dirichlet problem posed above. It also follows from what was said above that
I(I < 1,
Let us map the domain B onto the disk
just as in
§ 1.
this problem has a unique solution. We note that if certain of the curves constitut
Consider the
ing the boundary K degenerated into points, the Dirichlet problem for the domain
1(' 1
1 belonging to the class (1) of § 2. Let V «(') denote a function whose values coincide with the values u tz ') at points
set of points (' on the circle
'=
z' of the boundary K that correspond under this mapping to the points ( on I(' I '= 1. This real function V «( ') is defined, bounded, and continuous almost everywhere 1) on the circle
I(
circle the condition V (S «('»)
1 '= '=
1 and satisfies almost everywhere on that
V «(), where S is an arbitrary transformation be
longing to the group G of §2. Thus, we arrive at the problem of finding a function V «() that is harmonic and bounded in the disk \ 'I
< 1 and that coincides with V «() at all points of
continuity of the function V«() on
1'1
'=
1.
obtained would not always have a solution. To see this, note that the function u(z) which provides a solution to the Dirichlet problem for such a domain is, by
I'.. .:. virtue of what was said above, a harmonic function at all isolated boundary points. ."
'~·.Therefore, it is uniquely determined by its values on the remaining boundary ,; curves. This shows that the values of the function u (z) at isolated boundary
'iCf points cannot be chosen arbitrarily. :J:'
Thus, we have proved the existence and uniqueness of the solution to the
i) Dirichlet problem for an arbitrary finitely connected domain B bounded by Jordan
~;curves and for arbitrary given values on the boundary K that constitute either a I.
{(continuous function or a function with a finite or countable set of points of disl;,
Consider the function
!'continuity. To find the solution of an arbitrary Dirichlet problem for a given
2",
1 \
19
U(C)=2~ ~ U(e ),
"
l - rs
.f: domain B, it is sufficient to know the solution of anyone particular Dirichlet
.
fl'.\
I
_.dB, C=re''l',
{problem. Here, it is a question of finding the Green's function. The Green's func where the integral is in the sense of Riemann. By virtue of the kn5>wn properties of a Poisson integral, this function is harmonic and bounded in the shift to
1 (' I
'=
I(I < land,
'if tion for a ~,
under
1, it coincides with V «() at all points of continuity of the
domain B is defined as a real function g (z, () satisfying the following
(three conditions: 1) for every (€
B, it is a harmonic function of z in the domain
. B except at the point z'= (; 2) g(z,() approaches +00 as z --- ( in such a way
1/1 z -
~ remains bounded if ( is finite and in
function V «(i). On the other hand, any function V «() possessing these proper
that the difference g (z, () - log
ties is unique. To see this, suppose that there exists another such function V 1«()'
such a way that the difference g (z, (0) - log I zl is bounded if .' '= 08; 3) as () approaches the boundary K, the function g(z, () approaches O. In accord
Then, for arbitrary ( in
1'1
< 1, we have, in accordance with Poisson's formula
210
1 \
UI (C) = 2~ ~ UI
('9)
Re
RS - r
B
RB-2Rrcos(6 _ 'f')
+ rB
dB C_ ,
-
ilnce with the maximum principle, g(z, () must be positive everywhere in B. If
l
re ,
where r < R < 1. If we now take the limit as R --- 1, remembering that we can take the limit under the integral sign (Lebesgue's theorem), we see that VI «() ==
V«() in
\(1 < 1.
It follows from the uniqueness properry that V «() remains invariant under
a Green's function g(z, () exists, then, for finite ( the difference g(z, () log (1/\ z -(I) when assigned the app~priate value at z'= (, at the point z '=
values -log(1/\z - (I>; that is, this difference is a solution of a particular Dirich let problem for the domain B. Conversely, if u(z) is a solution of this Dirichlet problem, then the function log
(1/1 z - 'I), + u (z)
is obviously a Green's function
transformations of the group G. In fact, the functions V(S«()) and V«() are
for the domain B. An analogous situation holds for
both solutions of the problem that we have just examined and hence they are
sheeted mapping of the domain B by means of a function z*
1) In the present case, with the possible exception of a countable set of points on
I ~I = 1.
(
is harmonic in B and continuous in B and assumes on the boundary K the
('=
00. Under a single
[(z) onto another domain B* bounded.by Jordan curves, a Green's function g(z, () is transformed into the function g([-l(z*), [-1«(*»), which, obviously, is a Green's function for the domain B*. This shows that, if we know a Green's function for one domain, '=
270
VI. MAPPING ONTO A DISK
§3. DIRICHLET'S PROBLEM AND GREEN'S FUNCTION
we can, by means of univalent mappings, obtain Green's functions for various
It is obviously equal to 0 at points on the real axis and it coincides with u (t) on
other domains of the same order of connectedness. In the case of a simply con nected domain B, there is no Green's function, as the function cates, where t
It 1<
the disk
-log
I f(z)j
271
the upper boundary semicircle. Consequently, U (t) coincides with u (t) on the
indi
'entire boundary of the upper half-disk and therefore throughout this entire half
f(z) is a function that maps a domain B univalently onto 1 in such a way that the point z = , is mapped into t = O. For =
disk. If c:I>(t) is an analytic function in
It I (
p and its real part is the function
U(t), then the difference c:I>(t) - F(¢(t)) has real part equal to zero in the upper
multiply connected domains, there is no such simple relationship with conformal
'~half-disk. Therefore, this difference must be a constant in 1 t I < p. This last fact
mapping.
.
shows that the function F (¢(t)) can be extended in a neighborhood of t .
In the case of a simply connected domain B bounded by a closed Jordan
=
0
1beyond the real axis and, in particular, it is a continuous function at the point
curve, an analytic function P(z, ,) the real part of ~hich is a Green's function
!k = O.
Returning to the z-plane, we conclude that the function F(z), considered
g (z, ,) for a domain B is obviously continuous on B except at the point z
rin
B,
is continuous at the point zoo Since Zo is an arbitrary point on the bound
= "
If, in addition, the boundary of the domain B is an analytic curve, then p (z, ,) is regular on that curve. It turns out that these propenies also hold in the case of finitely connected domains. Let us consider first a finitely connected domain B bounded by arbitrary closed Jordan curves. Let g(z, ,) denote a Greenfg function g - ih. Let us
=
1;( (z) l~,.:
=
c:I>' (t)/ ¢ I (t)
take an arbitrary points Zo on one of the boundary curves of the domain B and let us describe a circle y about it sufficiently small that it intersects'the curve and there are no other boundary curves either inside the circle y or on it. The part of
B except, of course,
~
0 in a part of a sufficiently small neighborhood of the point
~,;zo contained in B. This means that the zeros of the function F I (z) cannot
~! duster on the
of B and let h (z, ,) denote its harmonic conj ugate. As we shall see, the last function is non-single-valued in B. Consider the function F (z)
f'ary of the domain B, this means that F(z) is continuous on
li"at the point ,. Furthermore, since c:I>' (0) is obviously nonzero, it follows that
f,:'
boundary of the domain B. Consequently, F I (z) and hence
~,bave only finitely many zeros. Finally, since the function w
~ of the
ne ighborhood of the point t
=
=
p~ (z, ,)
c:I>{t) maps the part
0 lying above the real axis into the part of
I,: the neighborhood of the point w = c:I> (0) that lies to the right of a vertical line
passing through the point w
"1,
=
c:I>(0), it follows that, as we move along the bound
the domain B lying inside y consists of simply connected domains. We denote
"!aryof B in a neighborhood of zoo the function h(z, ,) always changes in the
by B 0 whichever one of these domains has the point Zo on its boundary. Let us
,rsame direction. More precisely, as z describes the boundary of the domain B in
map B 0 onto the halfplane t
=
0 (t) > 0
in such a way that z
O. If we denote the mapping function by z
regular in the domain
o(t)
~
=
on
t
1
=
tthe positive direction, h(z, ,) increases. Now, if Zo is an interior point of an analytic arc on the boundary of B, then the function ¢ (t) is regular at the point
and its real part u (t) is also continuous in
0 and is nonnegative. Noting that u (t)
ficiently close to t u (t)
0 (t) > 0
=
z 0 is mapped into ¢ (t), the function F (¢ (t)) is =
=
= O. But then, the function
0 at points on the real axis suf
0, we describe a circle It \
p sufficiently small that
=
0 on the diameter lying on the real axis. Let us now define a function U(t) I
=
p as follows: U (t) coincides with the function u (t) on the upper semi
circle and it assumes at points in the lower semicircle the same values as the function u (t) assumes at the conjugate points in the upper halfplane with with oppos ite sign. The function U (t) is obviously continuous on the circle
I" =
p•
Consider the integral
,)
is regular at
the point z o'
'g,
By analogous reasoning, we can obtain a ~.,' functions:
g~neralized
theorem on harmonic
Let u(z) denote a function that is harmonic in a domain B bounded by closed ~aTtalytic Jordan curves. Suppose that u (z) is continuous in B and that its values On
the boundary of B constitute an analytic function of a parameter determining
points on the boundary (that is, a function that can be represented in a neighbor hood of each boundary point in the form of a series of powers of the parameter in
2"
1 \ '0 U(t)=2'IC ~ U(pe l ) .
F (z) and hence the function p (z,
pi ,,
rl It,
_.\
I
_.de,
,
t=re'l'.
o It defines a function U (t) that is harmonic in
question). Then an analytic function F(z) the real part of which is u(z) is regular in B.
I t I < P and continuous onl t I :$ p.
Proof. Just as above, we shift to the t-plane. Then, the values of u (z) on
the boundary of the domain B in a neighborhood of a point Zo can be represented
conclude that the last integral approaches 0 as
by a Maclaurin series of powers of the real parameter t. In the complex t-plane,
integral, we have
this series defines a function that is reguiar at t
=
273
§3. DIRICHLET'S PROBLEM AND GREEN'S FUNCTION
VI. MAPPING ONTO A DISK
272
0 . With regard to the middle
f -->
0 and real on the'real axis
I
(in a neighborhood of t = 0). When we return to the z-plane, this fast function be
._0
lim
comes a function 'I'(z) that is regular at the point Zo and equal to u(z) on the boundary of the domain B. in a neighborhood of the point z o' The difference
\ ~
\z-CI=I
d
dg
U •
ds~lim HO,,_t'~ _.
d log
U
I z - CI ds d.
-lim
(
d.
U1,_CI
~-2
._olz_~I=1
F(z) _ 'I'(z) is regular in the portion of the neighborhood of the point Zo lying in B, and its real part is continuous, including boundary points of the domain B, at .hus, if we take the limit in ( 1) as
which points its value is O. Resting on this last remark, we can again show that
•
this difference is regular at the point zoo But then, the function F(z) itself is
f -->
O. we obtain
1 \
u(q=2~
dg k u(z) dn ds.
(2)
regular at the point z o· Let us now show how, once we know a Green's function for the domain B, we
This formula is known as "Green's formula". It expresses the values of the func
can find the solution of an arbitrary Dirichlet problem for that domain.
tion u(z) inside the domain B in terms of its values on the boundary of B. Since
Let us first consider the case of a domain
closed analytic curves. Let u(z) denote a solution of a Dirichlet problem for the domain
the function p(z, ()
B with boundary K consisting of
B with given boundary values which define on K an analytic function of
=
g - ih is regular on K and hence dg/dn
=
dh/ds (one of
the Cauchy-Riemann conditions), we can put the preceding formula in a different , form:
u (q =
a real parameter determining points on K. Green's formula I)
2~ ~
rt (z)
dh (z,
q.
(3)
K
\ \ ,
\ ( dU l
~ ~ (Uarzl - Ulau) dx dy = -
~ u dn
B.
Let us show that formula (3) holds for a finitely connected domain B bounded
1(.
is applicable to the functions u(z) and Ul(z)
o~tained
dU)
Ul dn ds
-
by arbitrary closed Jordan curves and for an arbitrary given function u (z) on the
g(z, () and to the domain BE from B by subtracting the disk \ z - (\ < (.2) Here, the integral on the
left is over the domain
4i
boundary K, where the integral on the right should be understood as a Stieltjes
=
integral. Let g(z,. () denote a Green's function for the domain B and let S> 0
that on the right is over the boundary K E of the domain
denote a number less than the distance from any of the zeros of the function
BE in the positive direction; ds denotes an element of arc on KE; and d/dn de
p;(z, () (where p
notes the derivative with respect to the inner normal. Since Au
>'balf the distance between different boundary curves constituting K. On the set
and since u 1
=0
=
Au 1 = 0 in BE
on K, the preceding formula reduces to ' } ,
dg U dn ds
+
\
~ Iz-CI=I
dg
U dn
ds -
~
du g dn ds
=
g - ih) or the point (to the boundary K and also less than
consisting of all points of the domain B whose distance from the boundary K is
= O.
/) or greater, the function g(z, () has a positive minimum
(1)
K A of points of the domain B defined by the equation g(z, ()
Iz- 1=1
= (,
Consider the set =
>.., where 0
<
A < Ao. This set consists of analytic curves. To see this, let Zo denote a member
Here, the first integral is independent of (. Remembering that the function
g (z, () is of the order of magnitude of log ( on the circle \ z - (\
.\0.
of
we easily !
KA. The points of KA in a neighborhood of Zo have a parametric representa
z (t) that can be obtained by solving the equation p (z, () = A + it. Since (z, () ~ 0 for z = z 0, this parametric representation defines an analytic arc in a neighborhood of zo0 Furthermore, the subset K>.. of the set K A consisting of tion z
=
p; I) This formula is also valid for a domain B including z = De, as we can see if we apply it first to the domain BE obtained from B by subtracting the domain \ > II E, let E approach 0, and then proceed as described in the text. 2) For finite ~. In the case ~ =00, we take the domain \ > II E.
zl
zl
points at a distance no greater than [) from a given boundary curve L of the do main B consists of a single closed analytic Jordan curve surrounding L because,
",,_."_::- _"_"
._~".
274
"'..__.
::,..~
..__
. -,<.,. ,:.--
.-
"""""._ -~~
_._:.:;~~~':~?"'::-":'":'~~~~:'~"-"':':'::"~~~"-§T:4':~~~":::'-:':::'?=~~':~'#
We note that univalent mappings of the domain B transform formulas (3) and (5) into the corresponding formulas for the image.
tion and on the boundary of which g (z, () would be equal to '\, whic~ is impos
In conclusion, let us prove the symmetry of a Green's function; that is, let
sible since we would then have g(z, () == ,\ in B. It follows that the inequality
g(z, () > ,\ defines in B a domain BA including the point z = ( with boundary K,\ consisting of closed analytic Jordan curves approximating the boundary of the domain B. ~ow, suppos~ that u (z) is a function that is harmonic in B and con
liS prove that g (z, () = g
obviously a Green"s function for the domain B,\ and since the function
0'=
2~
\
II
(z) dh (z, C)
z). Let us apply formula (5) to the function u (z)
which is harmonic in B and continuous on
Ii.
=
Since
0 on K, we obtain
+ log Iz -
~ log Iz' - CIdh (z', z). K It 'ollows from this formula that g (z, () is a harmonic function of ( in B except CI= 2'11
at the point (= z, in a neighborhood of which it also has a singularity of the
= 2'1t ~ II (z). (0)) do.
;).
«,
~
as parameter determining points on K,\, it follows that
(C) =
=
g (z, C)
h(z, ()
increases as z moves around K,\ in the positive direction, so that we may take
(I,
g(z, () + log Iz -
i,g(z, ()
tinuous on B. Let us apply formula (3) to the domain B,\. Since g (z, () -,\ is
II
275
§4. APPLICATION TO A UNIVALENT MAPPING
VI. MAPPING ONTO A DISK
otherwise, B would contain a domain in which g(z, () would be a harmonic func
0'
~1'i:::'-::';''.~~'2'::;;;-;''
(4)
K).
same type as the function g«, z). Therefore, the difference d(z, ()
",g«,
taut, as we let z approach K, we have lim d (z, ()
tion u(z) at points on K and K,\ corresponding to the same value of
I h,ave
will, for
sufficiently small positive '\, differ from each other by an amount less than
g(z, ()
z) is a harmonic function of each argument throughout the entire domain B.
But u (z) is continuous on B. Therefore, if we also express the points on K in terms of a = h (z, (), we conclude that, for every f > 0, the values of the func 0'
=
= -
lim g «, z)
s 0;
that is,
d(z, () SO for every ( in B. On the other hand, as we let (approach K, we limd (z, ()
=
lim-g (z, ()
?: 0; that is d (z, () ?: 0 in B. Consequently,
; d(z, () = 0, that is g(z, () = g(' z) for every z and ( in B.
f.
Consequently, the integral in the right-hand member of (4) approaches the integral
§4. Application to a univalent mapping of multiply connected domains
over K as ,\ --. O. By taking the limit, we obtain
We can use the solution of the Dirichlet problem to give new proofs of theo
1l(C)=2~) U (zo (0)) do,
, rems on the existence of a univalent mapping of finitely connected domains onto
i; certain
that is,
canonical domains, for example, onto the plane with rectilinear parallel
!
U
(C) = 2'1t ~
i,
U
(5)
(z) dh (z, C).
Finally, formula (5) remains valid when the function u(z) has a finite number of points of discontinuity on the boundary K but remains bounded
(I u (z)1
< M) in
B. To see this, let us include the points of discontinuity on K in the interiors of small arcs (z(O'~), z(O'k)), suchthatIIO'~ -O'k\
I~ (U(ZdO))-Il(Zo(o)))dal~; +i.\ -->
cuts or onto the plane with cuts along arcs of logarithmic spirals of equal incli nation. Let us stop here to prove Theorem l' of §2 of Chapter V on the mapping of an n-connected domain B in the z-plane onto the (-plane with finite parallel straight-line cuts of inclination () to the real axis. As we know, it will be suf ficient to consider the case when the domain B does not include z
=
00
and is
bounded by closed analytic Jordan curves K 1> K 2 , •• " Kn , where K n is assumed to be the external boundary.
lll(z).(o))-u(zo(o))lda.
K
But off these arcs, u(z,\(O'))
~ll
u(zo(O')) uniformly. Consequently, for sufficiently
Consider n - I functions uk (z), for k
=
1, ... , n - 1, that are harmonic in B.
Uk (z) is equal to I on K k and equal to zero on all K k , for k' ~ k. For each k, let vdz) denote the harmonic conjugate of uk(Z). The func
Suppose that each
small '\, the last integral is also less than d2. This proves formula (5) in the
tions Vk(z) are in Eeneral multiple-valued in B but, in accordance with §3, they
present case.
are continuous on B. Let us denote by wk,l the increment in the function Vk(Z)
276
277
§5. MAPPING OF AN n-CONNECTED DOMAIN
VI. MAPPING ONTO A DISK
as z moves around K I in the positive direction (with respect to the domain B).
cannot be the case. Thus, the univalence of fez) in B is proved and so is the
One can easily show that the determinant IWk,l1 of the numbers wk,l' is nonzero.
fact that the image of the domain
B under the function (= fez) is a plane with
e with the
real axis. The function f(z) + c
Let us suppose, to the contrary, that this determinant is zero. Then there exist
parallel rectilinear cuts of inclination
real constants ~k , for k = 1, •. " n - 1, not all equal to zero, such that
yields the same mapping, and for some c we have, in a neighborhood of z
I~~lAkwk.1 = 0, l = 1"" , n - 1. The function
the expansion I
n-l
z_ a
/(z) = ~Ak(Uk+I'Vk)' which can be extended into
B along an arbitrary path, returnS to its original
f'~
value as z moves around each of the curves K I encircling K n • But then, this function is single-valued in B and hence is regular in B. The real part of the
_~(e-iBI(z - a» on the boundary K
=
(z - a)
+..., 08.
By analogous reasoning, we can prove Theorem I' of §3 of Chapter V re along arcs of logarithmic spirals of equal inclination. §5. Mapping of an n-connected domain onto an n-sheeted disk
fez) == const in B. But this is impossible since the real part of the function fez) vanishes on the curve Kn and is nonzero on at least one of the curves Kk for
Iw k,ll ;, o.
Now, suppose that uo is a harmonic function in
(XI
a,
8'Uding the mapping of a finitely connected domain B onto a plane with cuts
function fez) is constant on each of the curves K k , for k = 1, ... , n. Then, reasoning as in the proof of Theorem 2 of §2 of Chapter V, we can show that
k = 1, ... , n - 1. This contradiction proves that
+
is finite and the expansion z + a1z- 1 + •• : 1£ a =
k=1
=
In conclusion, let us look at the possibility of mapping an n-connected onesheeted domain
B onto an n-sheeted disk
I(I
< 1. By an n-sheeted disk, we
. ,mean here a Riemann surface lying entirely over the disk
B that is equal to
I(I < 1
and having over
each point of that disk exactly n points (a branch point of order p is considered
U~=IKk and that Vo is its harmonic con
as p points). This mapping is assumed to be effected by a bijective regular func
jugate. Let us choose real constants Ak' k = 1', •.• , n - 1, such that the function
tion. If we do not wish to resort to the concept of a Riemann surface, the mapping
n-l
should be understood in the sense that, if (= f(z) is the function performing the
'Vo+ Ii,",,} ~ Ak'Uk
mapping, then the function fez) - (0 has exactly n zeros in B for arbitrary (0
will be single-valu~d in B. This is possible since IWk,ll;' O. Let us show that
such that 1(0 I < 1 . (with a zero of order m counted m times) and has no zero at
the function
all in n-I
f(z)=z
I
a+ei61(uo+I'Uo+
~Ak(UIi+I'Vk»)
Theorem. Every n-connected domain B in the z-planethat has no isolated
boundary points can be put in one-to-one correspondence with the n-sheeted disk
1i=1
. ,(I < 1 by means of a regular function. If B is bounded by closed Jordan curves
is univalent in B. This function is regular in B except at the point z = a and assumes values on K I (for l = 1," " n) that lie on some straight line d l at an inclination with the real axis. This shows that, if (0 does not lie on the
e
straight lines d l , l = 1, ••. , n, the increase in the function arg (f(z) - (0) as z moves around the boundary of the domain B in the positive direction is equal to Zero. On the other hand, this increase mUSt be equal to 2" times the difference between the number of zeros and the number of poles of the function fez) - (0 in the domain B. Since f(z) - (0 has a single simple pole in B, namely, at z
B for any (0 such that 1(0 \ > 1.
=
a,
~.
tK1'" "
K n, then the function defining this correspondence is continuous in B
t;.and it can be normalized in such a way that, at n given points al"", an, where {, ak € K k , for k = 1, ... ,n, it assumes a single given value a such that \ a\ = 1 ill
'and, at two given points band c of the curve K n that are distinct from the an
and that are distributed along with the an in a definite order on K n , assumes
. given values f3 and y distinc t from a and distributed, along with a, in the same qrder on the circle
1('
= 1. Under this normalization the mapping is unique. (Here,
it follows that fez) assumes the value ~ at exactly one point in B. If the func
"order" refers to the direction of motion around the boundary of the domain
tion fez) assumed any value (1 at more than one point in B, this would also be
the boundary of the n-sheeted disk.)
true of points sufficiently close to (I' which, by virtue of what was shown above,
B and
i;II~I~'~'"'~tl~~Jll!lll1llll'_>!JIl/;Illi·.t~~~~~'';;>~W~~'!ftj!lm l/lN;it1.~..,.
§s.
VI. MAPPING ONTO A DISK
278
Proof. As always, it will be sufficient to consider a domain B that does not include
00
and that is bounded by closed analytic Jordan curves Kk,
k'=
1, ... , n,
where K n is the outer boundary. Suppose that the curve K k (where k,= 1, "', n) is defined by the equation z and that the point
Zk
Instead of the disk
'=
zk(t),
for 0
:s 1.
Suppose also that zk(O)
Kk in the positive direction as
(t) moves along
1'1 < 1_ and the
~ t
= ak
t increases.
n-sheeted disk mentioned in the theorem, we
q
r,_ =7_'
;
IlftiIDWIIll ~~:lI;~
MAPPING OF· AN n-CONNECTED DOMAIN
279
past the real axis and it is a function tbat is regular on that axis except at the point (= O. It follows tbat tbe function f(z) neighborhood of the point z
'=
(l/l7i) log (z - ak) is regular in a
In tbe same way we can prove that the function
= ak.
f(z) + (l/l7i) log (z - b k ) is regular in a neighborbood of tbe point z
=
bk • It fol
lows from what was said above that the function v(z) approacbes +00 uniformly, at the same speed as the function
_77-
1
log
Iz -
as
akl,
z approaches
through
ak
can obviously choose an arbitrary simply connected domain G that can be mapped
values in B, and tbat v(z) approaches
onto tbe disk and the n-sheeted surface over it respectively. For such a domain
approaches bk through values in B. This
floperty enables us to sbow that the function w = f{z) maps the domain B onto
lying over the (w
'=
f{z)
=
function w
'=
u + iv)-plane, we take the strip G: 0 < !lHw) < 1 and seek a u + iv that is regular in the domain B and that maps B onto
the n-sbeeted strip lying over G and having the property that w -~ +
00
i ~s
Z
approaches points ak through points in B. To construct the function w
=
f{z), let us choose on each of the curves K k ,
for k = 1,' . " n, a point b k = Zk (tk), where 0 < tk < 1. These points, together with the points ak partition the curves K k into arcs K~: z = z k (t) for 0 < t < t k F
and
K;::
z = z k (t) for tk
< t < 1.
Suppose that u (z) is a bounded harmonic func
tion in B that is equal to zero on the arcs
K~
and equal to 1 on the
a~cs K;:.
function
-1T-
I
log
(l/Iz - bkl),
tb~ n-sheeted strip 0
< lR (w) < 1.
-00
uniformly at tbe same speed as the
z
as
Let Wo denote any point in tbe strip 0
<
. lit (w) < 1. Since the function f(z) does not assume the vaiue Wo in sufficiently small neighborhoods of tbe points ak and b k for k,= 1,"', n, tbe number of zeros of tbe function f(z) _. Wo in B is equal to (217)-1 times the increment in arg [f{z) - wo] as z moves around tbe boundary K '= U~=IKk of the domain B in the positive direction, where, in ueighborboods of the points ak and bk, tbe patb is along small arcs contained in B. But as z moves around the curve K lc (for k = 1,"', n) in the positive direction, beginning at some point Zo € K k , tbe corre
Let v (z) denote the harmonic conjugate of u (z) (defined up to a constant term).
sponding point w = f{z) firSt moves downward along the imaginary axis and then
In general, the function v(z) is not single-valued in B. Let
moves along a sufficiently distant curve to the line
denote the increase in v(z) as z moves around K k
for k
1, "', n, in tbe positive direction. Wk>
=
lR (w)
'=
1 and then moves up
ward along tbat line. It then moves along a sufficiently distant curve back to tbe imaginary axis and returns along it to the initial point zoo Tbis shows that the
Here, in a neighborhood of the points ak and bk , the motion is along small arcs contained in B, The numbers W k depend on the choice of the points b k' If we
increment in arg [f{z) - wo] as z moves around the curve K lc in the positive
succeed in choosing the points b k in such a way that all the numbers
direction is equal to 21T. Then, the increment in arg [({z) - wo] as z moves
equal to zero, then the function f(z)
=
W
k
are
u + iv will be single-valued and hence
around the entire boundary of K intbe positive direction is equal to 277n. Conse
o in B is equal to n no mat
regular in B. Let us show that, in such a case, the function f(z) serves as the
quently, the number of zeros of tbe function f(z) -
required mapping. To do this, let us investigate first the behavior of f(z) in a
ter wbat the value of Wo; that is, the function w
=
neighborhood of the points ak and b k • Let us map the simply connected domain bounded only by the curve K k and containing the domain B onto the halfplane
domain B bijectively onto the n-sheeted strip 0
< lR (w) < 1.
'2'(() > 0 in such a way that the points ak and bk are mapped respectively into o and 08. The mapping function z = ef>{() is regular at -;; = 0 and has an expan sion z = ef> {()
=
ak
+
C 1 (+
.. " where
CI
-f.
0, in a neighborhood of that point.
values 0 and 1 on tbe positive and negative halves of the real axis respectively. But the function (1/1T) arg ( has this same property. Therefore, the difference
f{ef> {()) - (1/ rri) log ( can, in accordance with the symmetry principle, be continued
f{z) does indeed map the We assume that the
function f(z) is single-valued in B. Let us show tbat values t b .. " t n for wbich all the numbers WI.···. are equal to zero do exist. Choosing t n arbitrarily in the interval 0 us consider the different tk such that 0
This transformation maps u (z) into a function tbat is harmonic and bounded in a domain of tbe (-plane tbat contains the real axis on its boundary and tbat has tbe
W
Then u(z), v(z), and
Wk>
k
=
:s tk :s 1 for
k
< t n < 1,
t l " " , tn_I'
set
V(Z)=V(Z, tt> ... , tn-I),
Wk=Wk(tt> .. " tn-I),
let
= 1,2,' ", n - 1.
1,"', n, are functions of
U(Z)=ll(Z, tt> ... , tn-I),
W n
k= 1, ...• n.
We
280
§s.
VI. MAPPING ONTO A DISK
Let us show that there exist numbers t I>
•• "
Wk(tl> ... , tn-I)=O.
t,,_1
such that
that t!. v. k
I) The functions (U k (t I>
S I.
• ",
k= 1, 2, ... , n.
(1)
=
dh(!.,
Z)=2~.! (h(bk,
iLtty of the domain is equal to O. On the other hand, this increment is equal
z)-h(ak' z»,
S tv S I
4) It follows from properties 2) and 3) that (Uk(tI>"', t"_I) (for k= I,'"
••• , n - I) is a decreasing function of the parameter tk'
for
5) The function (Uk(tI>"', tn-I) is positive for tk = 0 and negative for I (under the assumption that all the other numbers t l' •• " t n _ 1 are different [from both 0 and I). To see this, note that for tk = 0, the function u is equal to (unity on K k and less than unity in B. Consequently, au/an < 0 on K k • Also, we iLtk
au = d.\ dv= d\ - au oydx+ ax dy k
It.
where the integral is over an arbitrary closed analytic curve
S tv S I,
on the basis of the formula
for v = I, ••. , n - 1.
t,,_I) is an increasing function of the parameter tv for v ~ k. To see this, note that, for t~ > tv' the difference 2) Each (U k (t I>
= jdV=- } ~~dS k
•• "
11•• k=(J)k(t~, ... , t., ... , tn_I)-wk(t h
t~, ... , tn_I) is the increment in the function vv{z) as z moves around the contour :K k • This function is the conjugate of the function U v (z), which is harmonic in B and van ... ,
ishes on K except on the arc of the curve K v between the points z v (t) and
Now, by using properties 1)-5), let us show that the system (1) has exactly (one solution. It follows from properties I) and 5) that, for every system of values
t2, ... , t,,_1> there exists a unique value t 1
Uv + iv v vanishes at some point on K k and hence the image of the domain B under the function w = f)z) would get OUtside the strip 0< ~(w)
11•• k = ), dv. k
= } 0%8' ds = k
~:; ds
- ) k
= '1 {t2,
... , t n -1), 0
< t 1 < I,
such
that (U 1('1' t2" . " t,,_I) = 0 with 'I (t2," " tn_I) is continuous with respect to
t2, .• " t,,_I' Since, in accordance with propenr 2), (U 1 VI' t2' ... , t,,_I)increases with increase in t2'" " t,,_1 and. in accordance with property 4), decreases
with vanishing of the derivative au/as, would mean that the derivative of the =
(3)
k
that (Uk> O. Analogously, for tk = I, the function Uk is equal to 0 on K k and ipositive in B. Consequently, au/an> 0 on K k and, on the basis of equations (3), 1,: !we have (Uk < O.
zv{t;), on which it is equal to 1. Since u v > 0 in B, we have du/an> 0 on Kk • This is a strict inequality because vanishing of the above derivative, together function fv{z)
=
can show just as in 2) that this has to be a strict inequality. Therefore it follows
Ck contained in B
and encircling only the curve K k • Consequently. each (Uk (tl" .. , t"_I) is a con tinuous function of t v in, the interval 0
to
; th~ left-hand member of equation (2).
k=1
t"_I) is determined from the formula
.
(2)
To see this, note that if the domain B can be converted into a simply
t,,_I). au/ax, and au/ay are continuous functions of z,
= O.
k=1
!valued in it and consequently its increment as z moves around the entire bound
n
tI>···. t,,_1 as z varies in B and each tv varies in the interval 0 v = I •.. '. n - 1. But (Uk (tI>
n
!connected domain by means of suitable cuts, the function v{z) will be single
tn-I)=2~.! ~ ••••
t"_I) increases with increasing
.. "
~ ..., tn-I)
t,,_I):
I, .. '. n - 1. This is true because, in
k=IKk
and hence u{z, t I>
"
3) We have
t,,_I) are continuous functions of each of the
where v n
u(z. tl> ....
and this means that (U k (t l'
tv for v ~ k.
To do this. we mention first a few properties of the functions (U k (t 1> •• variables tv for 0 S t}J accordance with §3.
< 0,
281
MAPPING OF AN n-CONNECTED OOMAIN
~ with increasing :i. "
,that
'1' it follows from the identity w! ['tl
'1
(tlb
••• ,
tn_V,
.ti ,
... ,
tn_I]
== 0
(t2," " tn_I) increases with increasing tv for v ~ 1.
Let us define the functions
(J)k (t~,
... , tn_I) = Wk ('tl (tlb
••• ,
tn_I), t 2,
... ,
tn_I), k
=
2, ... , n.
r ..~-_-- -_·_·-·_"~ __"""-"-"'--,""~-~----""__ ~---viiM'iR"iiiO"''''''''""''''''''-----'----·_-'''''''''----';;;-"-~,,,,,,~=«=>~,-,,, ""'''''T'''''''~''-''-'''''''''---~''''''"'''''"',,"'''''''''''';''ii'''''"'''''''''''''''''-'''''_~'''''''''''''''''''''''--;;'''''''''''''''''''~~''"''''''''' ......
282
VI. MAPPING ONTO A DISK
..-"...-..;;-._,,....,...-"""""..........,~""' ....................."""""
§6. SOME IDENTITIES
283
These functions are continuous with respect to t2' " ' , tn_I' For each k, the
and c on
function w~ (t2,"', tn_I) obviously increases with increasing til for
w = f(z) is uniquely determined. This completes the proof of the theorem.
II
~ k.
Furthermore, it follows from (2) that, for all t2' " ' , tn_I'
Kn are mapped respectively into
+ ioo, 8, and - ioo and the function
The proof that we have given is due to Grunsky. 1) The question arises whether the idea of the proofs of numerous existence theorems can be used in the
n
~ Wk(t2,
present case to establish the existence of functions mapping an n-connected
tn_I)=O,
"0'
k=2
domain B onto an n-sheeted disk by using some extremal property of these func
It then follows that W~ (t2,' ", tn-I)
(lor k
=
2,"', n - 1) increases with
tions. It turns out that this is possible and that in this connection the problem
If'
(a)!, where a is a finite member of B,
increasing tk' Finally, it follows from property 5) that w~ (t2, .• " tn_I) is posi
centers around the maximum value of
1. Consequendy, the functions W~(t2>'''' tn_I) 1)-5), with the number of functions and the number of variables decreased by one. We reason in the salIle ",ay with
out of all the functions f(z) that satisfy the condition f(a) = 0, that are regular
tive for tk
=
0 and negative for tk
=
satisfy conditions analogous to conditions
these functions: there exists a unique value t2
=
T2 (t3"", tn_I)fin the interval
0< t2 < 1 that satisfies the equation W~(t2,"" tn_I) = O. Here, T2 (t3"", tn_I) is continuous and increases with increasing til for II > 2. Now let us consider the functions
Wk (ta, ... , tn_I)
= wi. (t2 (ta, .0.,
tn_I), t a, .•• , tn_I), k
= 3,
0", n,
and continue in the same way. After n - 1 steps, we arrive at the uniquely de
such that Wok
tn_I
(Tb ••• ,
=
canst, t n_9= t n _2 (tn_I), ... , t l = "I (t 2 , • tn_I), Tn_I) = 0 for k = 1, .. " n - 1. But then, in accordance with 0
a function w = f(z) mapping the domain B bijectively onto the n-sheeted strip in such a way that the given points ak €
mapped into w
=
will be explained in Chapters IX and X.
§6. Some identities connecting a univalent conformal mapping and the Dirichlet problem In §4 of Chapter V, by using the method of contour integration, we estab lished certain relationships connecting various functions that map a given finitely we shall continue the application of this method 2 ) by introducing into our study
.,
equation (2), we also have w n (Tl> •• " Tn_I) = O. This proves that there exists a system of values t 1> •• " tn_1 satisfying equations (1) and hence that there exists
o < ~w < 1
Chapter XI. It uses a number of boundary properties of analytic functions which
connected domain univalently onto canonical domains. In the present section,
fined system of values
tn_I =
'in B, and that do not exceed unity in absolute value. The proof is given in §3 of
K k , k = 1, 2, •. "
n, are
+ ioe.
appropriate harmonic functions defined as the solutions of certain Dirichlet pro blems. Specifically, suppose that B is an n-connected domain that does not in clude "" and that is bounded by n closed analytic Jordan curves KII for II = 1,,·· ... , n. Suppose that w)z) (for II = 1, ... , n) is a function that is harmonic in
B, equal to 1 on K II , and equal to zero on all KIJ.' for 11 ~ functions wlI(z), for
and the constant term up to which the single-valued function f(z) was defined. Suppose now that, in addition to the point an, two arbitrary points b and care defined on the contour K n and suppose, for definiteness, that the points an , b, and c occur in that order as one moves in the positive direction around K n. If
z n (en), for 0 < t n < 1, we fix that value t
=
t n as indicated above and from it
00.
(z)= 1,
term the single-valued function v (z). We choose this term in such a way that the
K n is mapped into a given point D on ~w
=
O. Then, the points an, b,
(1)
.=1
because the sum on the left is a harmonic function in B and it is equal to 1 everywhere on the boundary K = U~=IKII' The harmonic function to w)z)
(II =
;:;;)z)
conjugate
1, ... , n) is not in general single-valued in B. Let (;:;;II(z))KIJ. as
usual denote the increment in the function ~II(Z) as z moves around the curve KIJ. in the positive direction (that is, positive with respect to the domain). Sup
we can find the system of values t I, .• " tn_1 which determines up to a constant point b €
We note that the
n
only arbitrariness was in the choice of the value of t n in the interval 0
=
II.
1,"', n, satisfy the relation
~
Let us now prove the uniqueness of the mapping. In the treatment above, the
c
11=
1) Gnmsky [19371.
2) See Garabedian and Schiffer [19491.
---
==~.~=~--:-~.-~
-T~;;;';'-~~~_-===-~=-~'~
§6. SOME IDENTITIES
VI. MAPPING ONTO A DISK
284
285
Transforming the second of tbe integrals (6) in the same way, we obtain
pose that
=
(iii. (Z»K,.
2tcP,..•, '"",
If we now set wv(z) = w)z) + i;';;)z) (for tion w v (z) satisfies the conditions
1) w. (z)
=-
w. (z)
II
+
= 1,
'I
clL., on K,., c,..•
... , n. 12 =
= 1"", n), then the analytic func
- 2~1 ~
Q (I:, u, v) W~ (C) dC
n
+ "'~
1 k,.
(u, 'V) P,.. •'
To evaluate the remaining integral, we make a cut y in B from u to v.
= const,
(2)
Since the integrand in the domain so obtained is single-valued, the integration
1.1.= 1, ... , n,
over K can be replaced with integration over an arbitrary closed analytic Jordan
2) (w. (Z»K,. = 2tcIP,..•,
(3)
"I,
curve C contained in B and containing the cut y but no point of the boundary of
{'
3) w~ (z) is regular in
B.
B. Integration by parts then leads to the following:
Turning to the contour-integration method, we shall use the functions P(z, u),
Q(z,
Q(z,
u), P (z, u, v), and
u, v) introduced in §4 of Chapter V. With respect
to tbese functions, we recall that tbey satisfy on Kv
P (z, u) = P(z,
-
Q (z, It)
(II =
+k;(ii),
1, " ' , n) the c~nditions
j
j
P (C, u) w~ (C) dC,
n
P (C, u) dw. (C) = n ,.
=-
2~1 \ ;
- ,. ~I 2~i
__
~I 2~i }
Q(I:,
+!
k", (u, v) P,..•
",=1 n
=~M-~M+!~~0~r ,.=1
n
w.(v)-w.(u)= ~ k,.(u,v)PIL ••, '1=1, ... , n.
(8)
,.=1
12 =
2~1
j
P (I:, u, v) w~ (C) dC.
functions w;(u) and wv(v) - wv(u) (defined as solutions of Dirichlet problems)
(6)
,.
j,. (- Q
( - Q (I:, u)
ll)W~(C)
(C, u)
pings). With an eye to finding inverses of these formulas, let us show tbat the
+ k,. (u»
n
~ P,.. • A,.=O, '1=1, ... ,
+ k,. (u» dw. (C)
-W~ (u)
+!
If this system has nonzero solutions A for /l W
(9)
=
1,"', n, then the function
n
~AILW",(Z) ,. = 1
is single-valued and consequently regular in B. Also, it assumes on each curve
K v (for
k,. (u)P,.. ""
II
= 1,"',
n) values lying on some rectilinear segment. By the same
reasoning as was used in the proof of Theorem 2 of §2 of Chapter V, we can
,.=1
As a result, we arrive at the formulas (for u € B)
show that tbis function must be constant in B. But then the function "i.~'=IAj.Lwjz)
n
w~(u)= ~ k",(u)P,..•, '1=1, ... , n.
n.
IL=I
+ ,.=1 ! k,.(u)P,..• =
is equal to n - I and let us at the same time clarify
consider the system of linear equations
dw. (C)
n
dC
II P j.L, vii
. the nature of the indeterminateness of the inverse problem. To do this, let us
---n
,.~I
in terms of the functions k j.L (u) and k j.L (u, v) (defined in terms of univalent map rank of the matrix
formulas (2) and (4) to transform the first of these integrals, we obtain
= 2~i
'l(1:, u, v) w. (C) dC
As a result, we obtain the formulas
10 accordance with Cauchy's theorem, each of these integrals is equal to O. If we use
II
j
Formulas (7) and (8) are interesting in that they provide expressions for the
Now consider the integrals
2~i
2~i
(5)
wbere kv(u) and kv(u, v) are independent of z in K v ' We note tbat tbe functions kv(u) and kv(u, v) will appear explicitly in the relationsbips that we derive below.
11 =
12 =
n
(4)
'V)=- Q(z, u, 'V) +k.(u, 'V),
ll,
,
(7)
will also be a constant. Since this constant assumes the value Avon Kv (for
286
VI. MAPPING ONTO A DISK
§6. SOME IDENTITIES
v = I,···, n), it follows that A} = A 2 = ••• = An. This shows that the only pos
sible system of solutions of equations (9) is the system A}
= A2 = ••• =
with u, v €
An.
B. If we evaluate these integrals first by using the residue theorem
directly and then by using (4), (5), and (10 ") as above, we arrive at the following
Now, on the basis of (1), we have
formulas:
n
~ ow... (z)= const in B,
(10)
... =1
n
p;(u, zo)=Q(zo, u)+P(zo, u)- ~ kp.(U)ill... (ZO)' ... =1
so that the increment in the sum on the left as z moves around any of the curves
P ('V, zo) - p (u, zo) = Q (zo,
lI,
'V)
in question is equal to 0; that is
+ P (zo,
n
~ kp. (u, 'V) ill... (zo).
'V) -
",=1
d'we separate the real parts in (11), we obtain the curious formula
From this we conclude that the numbers Al
=
A2 =
••• =
system (9). This shows that the rank of the matrix
An do indeed satis,fy the
IIP,u., vii
n·
g('V, zo) - g(u, zo) = log If~ (zo, u, 'V) 1- ~ log p... (u, 'V)' ill... (zo).
is equal torn - 1.
2
Furthermore, it follows from what we have shown that the functions k,u. (u), for 11 = I," . , n, can be determined from formulas (7) and that the functions kJu, v), for 11 = I, ... , n can be determined from formulas (8), in both cases up to a com mon term. On the ot~er hand, the differences k,u.(u) - k,u.'(u) and k,u.(u, v)
Here P,u. (u, v), for 11
Let us now turn to the Green's function g(z, ,) for the domain B. We de note the corresponding analytic function of z by P (z, '), so that g (z, ,)
lR(p (z,
,n. Obviously, the function p (z,
point z
= ,
(12)
",=1
I, ... , n, are the radii of circles on which lie respectively the images of the curves K,u.' for 11 = I, ... , n, under the mapping w = irr /2(z, u, v). =
By using the symmetry of the Green's function and replacing Zo with z €
B,
we can also rewrite (12) in the form
I, ... , n, are uniquely determined by these formulas. This
last fact is significant, for example, for formulas (9) and (10) of §4 of Chapter V.
g(z, 'V) - g(z, u) =
m(logj ,,(z, ll, 'V»
n
~ log p... (u, 'V)' ill... (z)}.
-
"2
1'0=1
Then, together with (11) we have the formula n
=
= log j" (z, ll, 'V) - ~ log p... (u, 'V) . W ... (z). 2" 1'0=1 The purely imaginary constant term on the right is not written because we have
p (z, 'V) - P (z,
,) has a logarithmic singularity at the
and is not single-valued in B. The multiple-valuedness is easily
shown:
ll)
{Dot normalized the imaginary parts of the functions p (z, ,) and W,u. (,) in any way.
(p (z,
~»K. = ) •
dp (z, Q=
.
~
/JI (p 5z, e» ds
.
K
The actual result is obtained as a relationship of the analytic representation of functions that are meromorphic in the domain B and real on the boundary K.
= - I ) Og~: C) ds= - 21tlill. (C).
These functions are known as Schottky functions. Let fez) denote such a func tion that has in B only simple poles ak, for k = I," " N, with residues rk'
•
Furthermore, suppose that fez) -is regular on K. Consider the integral
Thus, the function p (z, ,) possesses the properties
1) p(z, C)=- p(z, C)+const on 2) (p (z, Q)K = - 21tlill. (Q; 3)
lI,
(10' )
n.
... =1
=
(11) n
~ p .... • =0, p.= 1, ... ,
k,u.' (u, v), for 11, 11'
287
p; (z, Q
K.;
1I0") (10"
•
is regular in B except at the simple pole
Z = ,
1
with residue -1.
/6 =
2~i
From the residue theorem we have N
Consider the integrals
Is = ~nl )
P (e, u) p; (C, zo) d~ and
} P (C, u, 'V)f' (q dC.
)
I" = 2~1 )
P (c,
lI,
'V) p; (c, zo) d~
/6 =
~ rk P ; (ak' u, 'V). k=1
On the other hand, by using (5) and the relationship
288
VI. MAPPING ONTO A DISK
§6. SOME IDENTITIES
If we multiply (16) by PI)' Sum with respect to /l iJ-,
fez) =f(z) on K. ('J = I, ... , n),
=
1, .. " n, and use relation
ships (7) and (10), we obtain the conditions N
we obtain
1& =
289
~
~(~rkW~(ak»)=O, 'J=I, ... , n.
+
1 \ . - '~I 2ni ; (- Q (~, ll, v) k. (u, v» df (C)
•
=2~i
}
Q(~, u, v)f'(C)d"
It is a remarkable fact that conditions (17) on the numbers ak and 'k are not only necessary but also sufficient for the function (14) constructed from them to
N
=
(17)
k=1
k~lrkQ;(ak' tt,
be, for suitable A, a Schottky function. To prove that conditions (17) are suf
v)+f(u)-f(v).
ficient, suppose that they are satisfied. Then, by using (7), we can transform (The last integral is calculated by use of the cut y connecting u and v, just as before.) As a result, we obtain the relationship
..
them to the form
N
fell) - f(v.) = ~ (rkQ; (a k, tt, v) - rkP; (ak' u, v».
",=1 (13)
k=1
By using formulas (3) of §4 of Chapter V, we transform (13) to the form
k=1
0, v=l, ... , n.
(18)
If we consider (18) as a system of equations of the type (9), we conclude on the basis of what was proved above that this system has only equal soluti!!ns, and
N
this leads to n relations (16) with suitable A. But then, it follows from the iden
f(u)- h(rkQ(u, ak)-rkP(Il, ak) k=1
N
n
~ Pp..• ~ (~rkklJ. (ak») =
tity (15)as applied to z € KI) that the function feu) defined in accordance with N
formula (14) is real on KI)' for all v
=f(v)- ~(rkQ(v, ak)-rkP(v, ak»'
=
1, .. " n, so that fez) is a Schottky function.
Let us stop for an application of this result. Let ak, for k
k=1
=
1, .• " n,
denote arbitrary points in a domain B. From the identiry Since the left-hand member depends only on u and the right-hand member only on
n
~w~(z)-O, z
v, we conclude that the function feu) has a representation in the domain B of
EB"
• = 1
the form
we conclude that the system of n linear homogeneous equations
n
f(u)= ~(rkQ(u, ak)-rkP(u, ak»+A,
n
h A.~ (eiOw~ (ak»
(14)
k=l
= 0, k
= I, ... ,
n,
.=1
where A is a constant.
has, for arbitrary (), the nonzero system of solutions AI
It turns out that the numbers ak and 'k, for k
=
1, ... , N, cannot be arbi
trary. It is easy to establish certain relationships between them. For z € KI)' we obtain from (14) on the basis of (4) N
f(u)= ~(rkQ(u, ak) k=1 N
=
L
w;(ak))11 is less than n. But
then the system of equations
+ rkQ(u, ak)-rkk.{ak»+A
k=.{
also has nonzero real solutions in the 'k> for k
N (15)
k=1
~ (~rkk. (ak) - A:)=o, 'J= 1, ... , n. k=l
(e
An
~ "" i9w.(ak»=O, ' v=l, ... ,·n, L..Jrk..J(e =
1, " ' , n. Consequently, condi
tions (17) are satisfied for the numbers 'kei6 and ak, for k = 1,,", n. There fore, there exists a constant A such that the corresponding function feu) defined
in accordance with formula (14), which, in the present case, is the function
Since fez) is real on KI)' we have N
11o
= ••• =
n
=2 hffi(rkQ(Il, ak»- hrkk.Jak)+A. k=1
Consequently, the rank of the matrix
i6
(16)
n
n
k=1
k=1
f(z)= ~rk(eiOQ(z, ak)-e- i9 p(z, ak»+A= ~rkeIOj_O
. c-·c..,.--,..'··..,._·,'-"._··,·"'".,.,.."...,.,·""·"",..
290
,.""'...,.,,,·,,",.,._",".,_",-",,.,,,_,..-,~-~,-"_
...,.,-.. . . __
,,,-_-,-,,,,-~,,",.~",""""',""""'
The only poles that this function can have in B are included among the (arg [fez) -
en K'
where
m::; n,
_ ,,"-
":"~
291
and are equivalent to the complex conditions
[this last on the basis of formula (1) of §2 of Chapter V], is a Schottky function.
m,
,.c""_~",_~,,,,,,-,,,,,,,,,,-.-_-
§6. SOME IDENTITIES
VI. MAPPING ONTO A DISK
If the number of these poles is
__
N
ak'
~rkw~(ak)=O,
then, by evaluating the increment
"=1, ... , n.
(22)
"=1 The question regarding the sufficiency of conditions (22) for the possibility
we easily conclude that fez) assumes every value not lying on
suitable intervals of the real axis at exactly m points of the domain B. Conse
of construction of the corresponding function F (z) by means of formulas (21) is
quently, the Riemann surface consisting of m sheets of the w-plane with suitable
not answered here.
cuts lying over the rearaxis is the single-valued image of the domain B under the bijective function w = fez).
Remark. The functions that we nave been considering include, for example, the functions of §S that map the domain B onto an n-sheeted disk. However it
In conclusion, let us look at functions that are regular in the region B with modulus equal to unity everywhere on the boundary K. It follows easily
is possible to exhibit examples of simpler functions of this kind. Specifically, ellch of the two functions
from Cauchy's theorem on the number of zeros that if such a function F(z) has
I(I < 1.
exactly N zeros in B, it maps the domain B onto the N-sheeted disk Suppose now that the function F(z) has in B only simple zeros 1, ... , N, and that
F(z) _ 1 k z-+ak z-ak-r-' k= 1, lim
ak,
/1 tz) =
for k
2n-sheeted disk
... ,N .
(9)
I(F(Z)-F~Z»)'
-,
k=1
F(Z)~~.~.r,.P(z. aJ+C, ak)+D.
I
(20) for all real
-I-
I
',0
(24)
e.
This means that, for all z, u E: B, the ratio on the left side of
(24) cannot assume purely imaginary values. And since this ratio approaches 1 as z
---+ U,
we conclude that it assumes only values in the right halfplane. But
then, we have in B
Q; (z,
(21)
u) =
~ (j~ (z, u)
+j'~ (z, u» -=F O. 2
J
:.'f (~r"w~ (ak») = 0, ffi (~rkw; (ak») = 0, "=1, ... , n,
j~ (z, u)
i " 11 (Z, u) --r- tan \1, "2
Thus, the function
Q~
(z, u) has in
B a unique double pole
z = u and has no
zeros at all.
In the present case, conditions (17) take the forms
k-I
1, " ' , n.
2"
k=1
k=1
=
Therefore,
where A and B are constants. Furthermore, from (20) we obtain
N
(23)
")'
j~ (z'. ll) = eiO (cos aj~ (z, ll) - i sin aj' " (z, u» -=F O.
F(z)- F(z)=- 2(rkQ(z,ak)+rkP(z,ak»+B,
N
u, v)
(~_ ..
u € B.
)
N
F~Z)= l>"Q(z,
n'
In the first place, it follows from formula (1) of §2 of Chapter V that, for z,
F~Z)= !(rkQ(Z, ak)-rkP(z, ak»+A, 1 "=1
p; (z,
We confine ourselves to proving this assertion only for the first of the func
them, keeping (19) in mind, we obtain the following formulas:
I
/9 (z) =
tions (23), assuming, as above, that the domain B is bounded by closed analytic
are obviously Schottky functions in the domain B. If we apply formula (14) to
F(z)+
..\'
(~
I(I < 1.
Jordan curves K.",. v and
u)
n'
where u, v E: B,- is such a function and each of them maps the domain B onto the
=
Then, the functions
F(z)+F~Z)
P; (z,
·- ;·:·:~-:· :~-~-
l
"'.0.
,,;,
~
t-
In the second place, if z = zv(s) is the equation of the curve K v and s is the length of arc on K v measured in the positive direction with respect to the domain B, t hen, when we differentiate (4) with respect to s, we obtain
r
292
\
VI. MAPPING ONTO A DISK
P'z (z. (S), u) z~ (z) = It follows that on KJ)'
(z, 1ft (z) 1= p; Q; (z,
I
Q; (z. (S),
ll) Z~ (S).
(25)
I-I 1=1.
(8) Z' (8)
u) u) -
Z'
(26)
On this basis, we conclude that the functions P; (Z, u) and no zeros on K. To see this, suppose, for ex.ample, that a €
Q; (a,
K. Then, in accordance with (26), we have P; (a, u)
j; 12 (a, u) = O.
have j ~ (a, u)
u)
Q; (z, =
U) have
THE MEASURE PROPERTIES OF CLOSED SETS IN THE PLANE
0, where
0 .and hence j~ (a, u)
=
CHAPTER VII
=
Therefore, in accordance with formula (1) of §2 of Chapter V, we
=0 for all e. This means that, for all e. the mapping ,= j iz, u)
maps the point a into the endpoint of the boundary cut of the image of the domain
B. On the other hand, if in a neighborhood of z j~ (z, u)
= Cl (z -
a)
=
a we set
+ ..., I "2" (z, u) =
d l (z - a)
The transfinite diameter and ~eby~ev's constant
In contemporary questions in the theory of functions of a complex variable, a
significant role is played by certain specific ways of measuring closed sets in
+ ...,
the complex plane. We begin with one of these ways, which was proposed by
we conclude from the geometrical interpretation of the argument of the derivative of an analytic function that the ratio c 1/d 1 is purely imaginary. Therefore, since
je (z, u) = eiO (Ct cos 9 -ld t sin 9) (z - a)
•
§I.
+ ...,
Fekete. 1) Let E denote a closed bounded infinite set of points in the z-plane. For n points z l' Z 2' .• "
e,
(arg
Q; (z,
P; (z, u»
K =
u»
K = -
B,
V(z l'
417 (n - 1).
z) is equal to the Vandermonde determinant
Z l' .. , , Z •
"
"
z) is a continuous function of its arguments and the fact that the set
E is closed. Finally, let us define which is regular in the
=
exactly 2n zeros in B. This information, Cauchy's theorem on the number of zeros, and property (25) enable us to conclude that the function ~ =
ft (z)
assumes
< 1 at exactly 2n points and it does
(2)
does not increase with increasing n. Let ';1' ';2' ... " ' , ';n+l denote a system of points belonging to the set E such that
Let us show that d
IV(';I' ';2' "', ';n+l)!
not assume any other values at all; that is, it maps the domain B onto a 2n sheeted disk. This completes the proof of the assertion. 1 )
2
d V'! (n-l) , nil'
has exactly 2n - 2 zeros in B. But then the function fl (z) has
I"
(1)
. longing to the set E. Such a maximum exists by virtue of the fact that
477, it follows from what we have proved thac
in the domain B every value in the disk
n ~ 2.
n Let Vn = Vn (E) denote the maximum value of IV( z l' " ' , z)1 as z l' " ' , Z n range over all systems of n distinct points be
(arg Z'Z' (8») (8) K= 2 (arg z' (S»K= 41t (n - 2).
P; (z, u),
Zl)'
k,l=l k
of the numbers
This last equation shows that the function region
Zn)=
As we know, the product V (z l' .• "
Thirdly, from (25) we have
Since (arg
Zg, ... ,
and this contradicts the univalence of the
function je(z, u) in the domain B. This proves the assertion.
P' (z, U») (, arg Q' (z, u) K=
n (Zk It
V(Z!>
it follows that the derivative j ~ (z, u) has at the point z = a a Zero of no less than second order for some real
E, we define
zn €
V(e , ., ., t
n
= Vn+t' Then, since
eMt ) = (E t -
Eg) (e t - Ea) •• , (E'l
we obtain 1) In a similar way. Garabedian and Schiffer [1950] have constructed functions rhat map rhe domain B onto an (I-sheeted disk. 1) Fekete [1923].
293
"T'"
EIi+1) V (eg,
••• ,
En+1)'
(3)
294
§1.
VII. MEASURE PROPERTIES OF CLOSED SETS
Vn+1 ~ I ~1 -
~ 1·1 ~l
-
~s I '"
I ~l -
~n+l l' Vn'
subset of the z-plane. Since the sequences of their coefficients converge when
(4)
the sequence of these polynomials themselves converges, it follows that the limit
function is a polynomial t n (z)
Analogously,
V n+l E; Iej -
ed ·1 e'l - eal ... I~'l -
1
~n+tI· Vn'
~n:l ~ l'en~l ~ ell··j ~n:l ~ e~ I'...'1 ~n+~ ~ e~ I: ~n.
Together with t (z), let us also consider the polynomial
.
n
VII
n
, . smallest maximum modulus by •
':;
!;,
Vn+1 ~
2 2 v
Tn'
This limit is
'=
d(E), d
'=
There is a connection between the transfinite diameter d of a set E and degreepolynomials pn (z) '= zn + c lzn -1 + ... + C n and consider the maximum of the modulus of each of them on E. Let us show that there exists a polynomial m (E) denote the greatest lower
where D is the diameter, in the usual sense, of the set E. Define
lim
'=
converges
n.
k
to mn'
on E, we have \in(z)\
«a
Let us represent these polynomials in accordance with Lagrange's
n+1
_ '\'1 -
(z - Zl) (z - Z2) "" (z. - zd (z. - zs) .=1
=
(z - ZV_I) (z (z. - Z._I) (z.
ZV+!) ••• (z - ZII+') (z ) z.+I)'" (z. zn+t> PII. k .,
where z I' " ' , zn +1 are points in E (the same points for all the polynomials). It follows from this representation that the polynomials p
n,
k (z) are uniformly
bounded in absolute value on ail arbitrary bounded subset of the z·plane. Conse quently, in accordance with the condensation principle, the sequence of them contains a subsequence of polynomials that converges uniformly on every bounded
?
Tn
< a + (. Then,
+- ()n. If k and l are positive integers and Zo is a
fixed point belonging to E,then, for all z in E
\ (z - zo)l [tn (Z)]k \ < D1(0: It follows that m n k +1 ~
D1 (a + ()nk 'tnk+l
(z)
'tn
n-+oo
We have a ~ f3.We shaH show that the opposite inequality, namely {3 ~ a, also
formula: PII, k
lim
(1.,
holds. Let ( denote any positive numbu and choose n such that
(z),
of the type described such that the sequence of their moduli on E
1, 2, .• "
'tn =
n
bound of these maxima. Then, there exists a sequence of polynomials p
k
zEE
n ...... oo
another constant, known as l:eby'rev's constant. For fixed n, consider all nth
t n (z)
n/maxl (z - zot I < max lz-zo \ < D,
number of points, it is natural to set d(E) == O.
'=
. We shall also call the polynomial n
zEE
lim n-+OO d n . In the case of a set E consisting of a finite
whose maximum modulus is minimum. Let m n
~
E,
'tnE;
called, in accordance with Fekete, the transfinite diameter of the set E and is denoted by d
+ .•.
for n == 1, 2, .•. converges. We note that this sequence is bounded since, for
Furthermore, since dn obviously does not exceed the diameter of the set E 00.
zn
n
n
(6)
Zo €
--+
'=
l:eby!tev's polynomial for E.
Let us define T == V~- and let us show that the sequence of the numbers
which was what we wished to show. for any n, it follows that dn approaches a finite limit as n
t n (z)
••. + C , which has the smallest maximum modulus on E out of all the polynomials II (z) == zn + ••• + c , which have zeros only on E. In this case, we denote the
pn
for
+ .. , + c n , whose maximum modulus on E is
n
+ 1 V2n+1 \1,n+1 Vn+lE; n , n+1
zn
. called (;eby'Sev's polynomial for E.
(5)
n
"n-I
'=
obviously m • This is the polynomial with least maximum modulus on E. It is
Multiplying (4) and (5), we obtain
so that
295
TRANSFINITE DIAMETER AND CEBYSEV'S CONST ANT
Now, if
T
t:I. as n nv --+ t-' v
--+
00,
<
+ st k.
and
I Dnk+l ((1,
nk
+ s)"k+l.
(7)
by setting n v == nk V + l l! , where 0
< l l !< n,
we
. conclude from (7) that {3 ~ a + {. Since { is an arbitrary positive number it fol lows that f3 ~ a. Thus, f3 == a; that is, the sequence ber
T to
IT v I
converges. The num
which it converges is called the l:eby'Sev constant of the set E.
Let us show that
T
==
d. Here, we may assume that the set E is infinite. As
a preliminary, let us prove the inequalities
296
mn~ VV: ~(n+ l)m n. 1
-
Therefore, if the points X,
IV(e I , " ' , en)1
=
logarithms converges
~ )(
~1
.~) ~~
X -
e
(
•••
~ )
x-
~II
=
V (x, V (~
e
ell)
lt ••• ,
n
en belong to E and if e
, "', 1
, "', 1
to log
en are such that
< Vn+ IIV. n
1
...
X~
Xl
'V(Xh ... , xn+t)=
~'f
Furthermore,
.1
X2 t n (x~)
... , Xn+t)I~ltn(Xl)I'1 V(X~, n t (X~) 1·\ V(Xh X3, . . . , Xn+t) I
I Vex!>
... ...
I
n-I Xn+1 t n (Xn+t)
~ (n
+1) mil V
m i.e.
n
-0::
Vn +1 Vn
,,;::::
-=:::
(n
IV(X 1'
0/2
of
m> 0 such that Ian -
al <
> m. Then, for n > m,
+ as +n ... + an -a I I.~ (a. =_1
m
a)
~ (a. -a)
,;:;; I _.=---=--1
+n-m~
_
n
n
2'
d2. d2.
This proves
the convergence asserted in the lemma. The above reasoning remains valid when we are considering polynomials with zeros only on E. This means that if we set ;~ ~ n that r ----> r = d as n - ..... 00.
" ' , Xn+ 1)1 = Vn+ l'
=
1;;; , we also obtain the result n
n
Summarizing what has been said, we have
VV;l ~ (n + 1) m n•
+1)
denote a positive number. Choose
But, for sufficiently large n, the first term is also less than
Theorem 1. For an arbitrary closed bounded set E of points in the complex
plane, the transfinite diameter d and the tJebysev constants
Thus, inequalities (8) are proved. They can be written in the form 't n ,,;::::.
f
The second term in the expression on the right is less .than
, Xn+l) I
Now, if Xl' " ' , X 1 are points on E such that n+ we obtain
The arithmetic mean of the first n (n +
n
+ + I tn (Xn+l) II VeX!> ... , xn)l.
Vn +t
for n
XII+l
n-l
T.
follows from
Proof. Let
d2
1
If we expand the determinant in terms of elements of the last row, we obtain
+1
T
ai
n --I XI in (Xt)
to log
Lemma 1. If a sequence of real numbers al' a 2, ••• converges to a finite
ceed Vn+ II V. n For x E E, the modulus of the left-hand member is no less than -
1)
limit a, the n the sequence of their arithmetic means (a l + a 2 + .•• + a)1 n also converges to a as n _ ""'.
Vn, we 'conclude that the modulus of the right-hand member does not ex'
m n • ·Consequently, mn .
297
these logarithms is the logarithm of the left-hand member of (9). That it converges
.. e) •
itt ... ,
v
TRANSFINITE DIAMETER AND CEBYSEV'S CONSTANT
as n ----> 00. Since Tn ---+ T as n - ""', the sequence T l' T 2' T 2' T 3' T 3' T 3' " ' , in which T·n is repeated n times, also converges to T and the sequence of their
(8)
In the first place, we have (x
v
§l.
VII. MEASURE PROPERTIES OF CLOSED SETS
T
and ;- coincide:
d = T = ;-.
't n
II'
From this theorem follows the simple fact that, if the entire boundary of some bounded domain belongs to the set E, then the transfinite diameter does not
Now for n let us set the successive numbers 2, 3, .. " n and multiply the reo sulting inequalities. We obtain 2
change if we adjoin to E the interior points of that domain. Resting on the identity of the transfinite diameter and Cebysev's constant,
2
('t; • ~. 't~)n (n+ I). (v~)n (n+ I) ~
~
we shall give two simple examples of the calculation of the transfinite diameter.
dn+1
~ [en
+ 1) !]n(n
2 + I)
Bu' (V ,)2/0'0,1) 1 uud [(n + 1),]2/0'0'" _ show that d = T it remains only to show that
('t~
2
...
1. Suppose that E is the disk
2
't~)II(II+I) (v~)n(n+ I). /
1 as n
~~. C~Y'
1zl < R.
Since
2",
1 "2 $
,u
~
\Pn (z) I~ dB =R~Ii+ ...
+Ien I~ ~ R~II,
2
2-( 't 1't2 ... 't~n(n+I)_'t
(9)
1)
:J
o.
It is easy to show that (9) is also valid when T =
0.)
298
§ 1.
VII. MEASURE PROPERTIES OF CLOSED SETS
where z = Re i8, for an arbitrary polynomial p (z) = zn + ... + C , it follows that n n m n ? Rn. But for Pn (z) = zn, we have maxz€ E 1Pn (z)/ = R n• Consequently, m = n Rn and d = T = R. Since the transfinite diameter does not vary under parallel
Theorem 2.
v
1)
polynomial zn + ... + cn' Let E • denote the set of all points z such that w p(z) €
d(E*) =Vd(E). n
/
max \Pn(z) 1= max IPn (a cos x) I
E and E·.
inequalities
O:::;X:::;l<
(e
iX
+ e-i~) I=
m~. =
max Iqil (x) I,
max 1t~. (z) I~ max It. (j:I(z)) I ~ max It y (z) I= m. zEE* . zEEo zEE·
O~X~ft
where q (x) is the trigonometric polynomial
n;~
cos x+ ... +an cos nx,
d(E·) $
an
ait,=2n - J
'
Since k
~ ~ I qn (x) I~ dx = 21 a o I~ + b
n
!
m~. ~
r
and consequently,
n
qn (x) =ao +aJ
n
Define m n = max z €Elt (z)/, m·=max €E*lt*(z)I. Then,wehavetheobvious n n z n
n
= O~~~ft max jPn (~
(10)
Proof. Let t (z) and t· (z) denote the ~eby~ev polynomials for
2. Suppose that E is the interval - a $ x $ a. For an arbitrary polynomial p (z) = zn + ..• + c , we have
=
E. Then
disk is equal to its radius.
zEE
299
Let E denote a closed bounded set and let p(z) denote a
translation of the set E, it follows that the trans fini te diame ter of an arbitrary
n
v
TRANSFINITE DIAMETER AND CEBYSEV'S CONSTANT
Vn .;;-m•. . If v _
00, we have the inequality
vd\E'i. On the other hand, if, for arbitrary fixed
W
o € E,
we denote by
z l' •• " Z n the roots of the equation p (z) - W 0 = 0 and if we denote by z~, " . • . " z· the roots of the equation t· (z) = 0, then it follows from the equation v . v n
.,
In II (Z, -
Iak I~ ~ 2Ba(~-11
k=l
/=1 k=1
..
n
III II (4 -
z~) I=
z 1)\
k=II=1
ill
that
it follows that
In t: (Z,) I= III (p (zZ) n
a2n 2 max I qn (x) I~ ~ 22 (n-!) , O:::::::.x~2Tl:
1=1
so that
an
max 1Pn (z) I~ --------r- .
Izl=1
This shows that m > a n /2 n - 11 and r> a/2. n maxz€ E lpn(z)! = a n n - 1 for the polynomial
•
k=1
Since y
n-"2
q.(w)=·II (w-p(z~))
2
k=\
On the other hand, since
/2
is of vth degree in w, we have from (11) m.~
( z ) Pn(z)= 2an n - 1 cos narc cos a =Zn+",+ Cn
max
wEE
Iq.(w)I~(maxlt:(z)lt=m:n, zEEo
and consequently, it follows that d =
T
wo) \ .
= a/2. ~~
V -vm.~
From this we conclude that the transfinite diameter of an arbitrary interval
is equal to one- fourth its length. 1) Fekete [1930].
m:.
-y/
V
(11)
-:::
==zzz
300
_~
.~._
_~~!!ii'ir~~~,
---->
00,
we have
V d(E)
::; d(E*). Therefore, equation (10) must hold. This
(that is, m::; n) because inside each of them there is at least one root of the poly
nomial p (z). This last fact follows from the fact that otherwise we could conclude,
completes the proof of the theorem.
on the basis of the maximum principle as applied to
If we apply Theorem 2 to the case in which E is a disk, we obtain the Corollary. The set of points z defined by the inequality Ip(z)! ::; R, where p (z) = zn
+ ... + C n has transfinite diameter d =
n.
= const.
:,'of the Kk' that p(z)
,K
the point
,
k
(=
1(1 = pl/m k•
f(~mk)
of this nature, dealing with outer Jordan 'content. Let us review the definition of
E denote a bounded set in the z·plane. Let us cover the plane
with a grid of squares of side lin and let us denote by s n the sum of the areas of those closed squares that contain at least one point belonging to E. The limit as s
---->
00
Therefore, on
1(\
=
00
set E does not exceed TTd 2, where d is the transfinite diameter of E. Proof. Let us first prove the theorem for the special case when the set E is
VR.
(k),m k '0
(2)
•
Equation (2), with (= pe'
Therefore, the area of the interior of the curv, K k is
•
2flmk
~ ~ x dy K
y dx = 2flmk
= ~
areas of the domains covered by the set E, we need to prove the inequality
~ ~ ~ (z' dz) = ~ \ m(z ~ ~ d~ ) Kk ~
'
k
Therefore, if we denote by S R the sum o!/dJe'
ffi (
00
~
it,!,
~
00
it,!,
~ ~~oo a~k)pmke- mk ~=~oo~ a~k)pmkemk dep) co
!
2
SR.~TtRn.
ep
equal to
defined by the inequality !p(z)! ::; R, where p(z) = zn + ... + cn' In this case, in =
~
co
varies from 0 to 2TTm
The two-dimensional outer Jordan content of a closed bounded
accordance with §I, d(E)
co
'~-QO
turn now to the theorems mentioned above. 1)
and has on
~ a<:)~~.
=
z=!(Q= ~ a.
is called the one-dimensional outer Jordan conte.nt of the set E. Let us
Theorem I.
k
p, we have for one of the branches of the function f«(}
is called the outer Jordan content (two-dimensional) of the set
the closed intervals that contain at least one point in E. The limit of l n as ---->
Consequently, one
= pl/m
'I==-():)
E. Analogously, if E is a bounded subset of a straight line, we partition this line into intervals of length lin and we denote by l the sum of the lengths of n n
\(t
m k times and the
that circle ,the Laurent expansion
and the usual measures that are employed in analysis. Polya proved two theorems
n
and to the interiors
\(1 = p exactly
mkr-;-;
of the branches of the function z = f«(m k ) is regular on
The question arises as to the relationship between the transfinite diameter
of s
111 p (z) I
Then, as the point z moves once around the curve
p(z) moves around the circle
point (= V p(z) moves once around the circle §2. -Bounds for the transfinite diameter
this content. Let
301
§2. BOUNDS FOR THE TRANSFINITE DIAMETER
VII. MEASURE PROPERTIES OF CLOSED SETS
As v
----:·
, . " , ~ ! : < ~ ~ @ ' ~ ~ ~ ~ ~ ~ ~ ~ ~ ' ' ! ! \ " ~ ~ " ; ! ! ! . ! ' ~ " , " c . : . . " " , = ~ ' ' ' ' ' ' ' - o ' ' ' _ = = - - ' ' ' ' ~ = - . . : c ' ' ; ' ' = = ' = ' ' ' ' ' - ' ' " = - = " , ~-;,=,·'=~==.=.o.-. c,-=
=Tt
(1)
2,
'II a~k) \9 pmk.
.,=~oo
For this, along with (= p (z), let us consider the inverse algebraic function :;, Therefore, if we denote by S p the sum of the areas of the interiors of the curves
z = f«(). This function has no more than n - I branch points (that is, points
( = p (z)
such that p / (z)
= 0).
K l'
Let us suppose that the circle I( I = p does not
., "
Km , we obtain
pass through any of these points. Then, the set of points z defined by the equa tion Ip(z)l = p consists of disjoint closed analytic Jordan curves K
1
, "',
Km •
These must lie outside each other because inside each of them we have I p (z)1 by virtue of the maximum principle. There are no more than n of these curves
Sp
7tp2/11
m 00 ~~
=.'-
k=l
2,2
.'- 'II a~k) 19 pm k
-
n
(3)
'=-00
Here, the right-hand member is a nondecreasing function of p since each term in
the summation is a nondecreasing function of p. This last is obvious for v > O.
I)Polya [1928].
The term with v
=
0 is equal to O. A term with v
< 0 is negative and the exponent
---:
_
VII. MEASURE PROPERTIES OF CLOSED SETS
302
§2. BOUNDS FOR THE TRANSFINITE DIAMETER
:s
crease with increasing p and is obviously a continuous function of p, including
If P (z) = 11~"'1 (z - a k ), we obviously have jP(x)! Ip (x)I.Consequently, if z € E, then IP (x)1 R. Let us denote by P the set of all real numbers x at
values of p equal to the moduli of the branch points of the function f(!,;).
which
2v/m k - 2/n in such a term is also negative. Therefore, S/17p2/n does not de
But for p sufficiently large, we have m
f{D
on
11,;\
=
1 and m 1
=
which P (x)
± R.
Let us show that each of these intervals contains at least one
:s R. This set consists of a finite number of do of which IP(z)j = R. Each of these domains contains no
that satisfy the inequality IP(z)1 00.
mains on the boundaries
This means that,
more than one of the segments in question. To see this, suppose that one of the
for large p,
~L = 1 -I ~1 \9 p'lt 2/n
2/n -
p
... 0;::;;;
(4)
1.
Consequently we have Sp/17p 2/n:s 1, that is, Sp:S 17p2/ n for all p. In particular,
S R :s 17R 2/n, which is what we had only when a 1
=
root of the polynomial P (x). Consider the set of points z in the complex plane
+ au +al~-I/n + ...,
which is the inverse of the polynomial!,; = p(z) about!,; =
:s
Then, the projection of the set E onto the real axis is con
tained in P. This set P consists of a finite number of segments at the ends of
n because the function
= p is then defined by the expansion
f~)=~I/n
IP (x)1 :s R.
303
=
a2
= ••• =
to prove. (We note that equality holds here
0, that is, only for the polynomial p (z)
=
(z - a) n.)
Let us turn now to the general case of the theorem. For this, we use the co incidence of the transfinite diameter with Ceby~ev's constant. Let t (z) denote n
the Ceby~ev polynomial and let r denote <=eby~ev's constant for E. Then, for
(> 0, if n is sufficiently great we have It n (z)1 :s (r + () n on E. Consequently, the set E lies entirely in the interior of finite domains bounded by the curves de
any
(r + ()n. Therefore, the two-dimensional outer Jor dan content of the set E does not exceed the sum of the areas of these domains and by virtue of what was shown above, this sum in turn does not exceed 17(r + ().2. / Letting (---> 0, we obtain the assertion of the theorem. This completes the proof of the theorem. fined by the equation It n (z)1
=
Theorem 2. The one-dimensional outer Jordan content of the orthogonal pro jection of a closed bounded set E onto an arbitrary straight line does not exceed 4d, where d is the transfinite diameter of E. Proof. Obviously, it will be sufficient to prove the theorem for the projection onto the real axis. Again, we shall consider first the case in which E is the set defined by the inequality
Ip(z)l:s R,
where p(z) = zn + ..• + c ' n
Suppose that
the real axis, it must be a multiply connected domain. Since IP (z)! = R on the exterior boundary curve of this domain, it follows on the basis of the maximum principle that IP (z)1 is less than rather equal to R on the interior boundary curves. Furthermore, each of these domains contains one of the segments because the interior of anyone of them that did not would not contain roots of the polyno mial P (z) (since all the roots of P (z) are real), so that, again by virtue of the maximum principle, we conclude that P (z)= const. Now, if there were no roots of
P (z) on one of these segments, there would also be none anywhere in the entire domain containing' it and, consequently, we would again conclude that P (z) == const. This proves the assertion made. Let I denote the farthest right of all the segments containing P and suppose that I contains m roots of the polynomial P (z) (without consideration of their multiplicity). We have 1:S m
:s n
with m = n only in the case in which P consists of a
single segment. In addition to P (x), let us construct polynomials PI (x),
P 2(x), ., ·of degree n with real coefficients and in particular with the coefficient of x n equal to 1 in all cases.
By P k' I k' and m k' we mean the same thing for Pk(x) as we mean by P, I, and m for P (x). Let us show that, if m < n, there ex . ists a polynomial PI (x) such that m 1 > m and the sum of the lengths of the seg ments constituting PI is greater than the corresponding sum for P. To see this, suppose that P(x) = Q(x)R(x), where Q(x) is a polynomial with zeros on I and R (x) is a polynomial with zeros outside I. Furthermore, let d denote the dis tance from I to P -I. Let us define PI (x)
n
p(z)=
domains contains at least two segments. Then, by virtue of its symm/try about
n (Z-ik),
"=1
Z=x+iy,
i"=ak+ib,,.
IQ(x + d)!
< IQ(x)1
=
Q(x + J)R(x). If x € P -I then
because each linear factor in the expression for Q(x + d) is
less in absolute value than the corresponding factor in Q(x) by an amount d. Consequently, if x € P - I, then
IP 1 (x) I < IP (x)1 :s R.
This shows that
=,=:",,,,,=====,~,,~=:"";''------~=--~-~=--'-=-----'-'-:---':;;;:;';~;;;o7::':;;:--'-~---::--.---
304
--.-:-;;:;:.;;--;;;--_-;;;..'?----..- -.. -;;.---;~--.-.-:__~;-:;.~~.~-
._;;;:;~"._
- ;....::.-
"'--.::'~,-.~.-~'""='''',,;:-_-:::.::__.5:::::;,~·;;.,:_:.-
.._ "".".. ,.,.,~c'".~"._
", _"',:..","_"_'':;''''''''':~::~''''''''''~,~'''''~--:'"''-'
;'~;~",.;;,:,~-;,;::;;:~-;~
--
P -I €
Pl' On the other hand, if x E I, we have analogously IP 1 (x - d)1 = d)! < I Q(x)R(x)! = IP(x)1 < R and, consequently, the inlerval [' ob tained from I by a displacement to the left by an amount d belongs to Pl' Thus,
I(z -
~) I.:;;; 1,
Cl) (z -
IQ(x)R(x-
PI contains P -I and ['.Consequently, the sum of the lengths of the segments composing it is not less than the corresponding sum in the case of P. Since P (x) 1 has all its zeros on P - I.and on 1', the number of segments constituting P 1 is less by at least unity than the number of segments constituting P. Reasoning
k::;
Cl> 1.
Obviously, the transfinite diameter of this set is equal to 1. The boundary of the domain B complementary to E and containing
I(z -
00
is defined by the equation
a) (z - 1/ a)! = 1, that is, by the equation
- !(z -l+exl)1 2ex
(ex-l - l 2ex
)11_ -1
n - 1 of steps at a polyno
> 2a,
mial P k (x) such that P k consists of a single segment and has length no less
and, for a 2 - 1
than the sum of the lengths of the segments constituting P. But if the segment
boundary of the domain B lie the roots of the equation (z - a) (z - l/a)
P k has length L, then, by §1, its transfinite diameter is equal to L/ 4. On the other hand, P is contained in the set of points defined by the inequality IPk(z)1 ::;
R and the transfinite diameter of that set is, in accordance with §1, equal n,-
n~
to V R. Consequently, L/4 ::;~~-~--matls, L::; 4
n
VR
=
4d, which proves the
"
n
v
the Lebysev polynomial and rthe Cebysev constant for the set E. Then, for given
(> 0
and sufficiently great n, we have Itn(z)
I::; (r + d n
1, that is, the points 0 and a + 1/ a. The difference between these two roots, which is , equal to a + 1/ a, can be made arbitrarily great bya suitable choice of a. 'In this case, the ordinary diameter of the set E can also be made arbitrarily great. =
We mention some consequences of Theorems 1 and 2 for a set E with zero by virtue of Theorem 1, equal to zero. It remains equal to 0 in the case in which
Now, let us turn to the general case of the theorem. Suppose that t (z) is v
consists of two ovals without points in common. On the
transfinite diameter. In this case, the two-dimensional Jordan content for E is,
theorem as regards the special case in question. )(
305
§2. BOUNDS FOR THE TRANSFINITE DIAMETER
VII. MEASURE PROPERTIES OF CLOSED SETS
analogously with PI (x), we arrive after some number
~c-~ ... _" ..,.,,,.. ';"'';;'-''',,"'. ""-",-,:.,,-~,".~:;;:;--:,c;;:;;:;.;:.~,,,~-~_,;;:,"~"_:;;;:_:;;::;;::;:,,,::~:::-,~ -.-£-~-",,::. "":,,~~£;:::'"'- ~"';':;;'::C?,~;;~-;;_';;':';~'
on E. But, by virtue of
all complementary domains not containing
00
are adjoined to E because this does
not change the transfinite diameter . .It follows that E has no complementary do Thus, if dee) = 0, the complement to the set E is the
what was said above, the projection of the set of points defined by this inequality
mains not containing
onto an arbitrary straight line does not exceed 4 (r + d.Consequendy, the projec
unique domain containing
tion of the set E onto the same straight line has one-dimensional outer Jordan
a boundary point of that domain. Theorem 2 proves more than this, namely, that
content not exceeding 4 (r + (), that is, by virtue of the arbitrariness of
(> 0,
not
exceeding 4r. This completes the proof of the theorem.
If the set E is a continuum, we easily obtain from Theorem 2 the result that the ordinary diameter of the set E does not exceed 4d. As we shall see in §3, this result is equivalent to a bound on the diameter of the boundary of the image of the domain
1'1 > R
under a univalent mapping by the function z
= ,
+
ao +
00.
00
and every point of the plane is either an interior or
E does not contain any continuum because if it did there would be a straight line such that the projection of the set E onto it would have a positive one-dimensional outer Jordan content. These are necessary conditions for d (E) to be equal to O. They are not, however, sufficient. As Nevanlinna
1)
has shown, there exist sets
that do not contain a continuum but that have a positive transfinite diameter. We shall not stop here to construct such sets.
a / ' + . " and it follows directly from §4 of Chapter II. Thus, Theorem 2 is a
In' conclusion, we present another interesting theorem.
generalization of this property of simply connected domains to multiply connected
Theorem 3.
domains if by the sets E we mean the boundaries of domains. The question arises, is it possible in the case of an arbitrary closed bounded set to conclude on the basis of its transfinite diameter whether there exists an upper bound for its ordi
1)
Let E denote a closed bounded set in the z-plane and let B
denote the domain complementary to E that contains 00. If fez) is regular in B and ifit has the expansion fez) = a z- 1 + a 2 z- 2 + .•. in a neighborhood of 00, 1 then, by setting
nary diameter, as is the case for a continuum? The following example shows that the answer to this question is negative. Let E denote the closed bounded set of points defined by the inequality
_/
1)
........
.~~..,....~~
306
~~~~-~-,--~~
.....
~~~~
-
........
...
.
VII. MEASURE PROPERTIES OF CLOSED SETS
at
I
a3
An = a: an
we have
lim
Tt~oo
an+l
2
(5)
n! An =
(2;i)n
~}
... }
!(Zl) ...
n-I
}! I An I~ d,
(6)
> 0, there exists a closed
with boundary consisting of a finite number of closed analytic Jordan curves. We denote this boundary by K. Suppose that E* has transfinite diameter d*
where L is the length of K. Therefore,
. lim n.;t, An I.~ d
(Zk -
ZZ)
k,Z=1 k
I
(7)
an
=
1 -2• \ 'itt
~
continuum E
! (Z') Z' n-l dz''
n=l, 2, ... ,
1
Zl
x
n-l
z-l
Zl
1 1 »+'"+-n+''' n z 00.
Consequently,
Theorem 3 can be applied to E and f(z). The determinant (5) in this case is
1 ... 1
Z2 ... Zn
dz ldz 2 ... dz n ,
1
(8)
n-l
1
Zn
If in the integral in (8) we make all the possible n! permutations of the n
1 1
is regular in the domain B complementary to E that contains
An
denotes the variable of integration in the integrals in the kth column.
1
2 1
=12. .. 3
1
Z l' Z 2' •••
n
...
1 n+l 1
n n+l ... 2n-l
which does not change the value of the integral, and if we then sum over
all these integrals, we obtain
1
!(z)=-log-=-+-2 z z z
!(Zn)Z~Z~z:" .Z~-I
n-l ZI ••.
the real axis is an interval of length no less than 1, Theorem
obtained from Theorem 3. The function
and substitute this expression into (5), we obtain
A n =(2:i)n)} ... } !(Zl)!(Z2) .•.
on~o
2 leads to the inequality d 2: 1/,4. Let us show that this inequality can also be
K
Z ,
Theorem 3, like Theorems 1 and 2, gives a lower bound for the transfinite diameter of a given set E. As an example of its application, let us look at a bounded continuum E containing the points 0 and 1. Since the projection of this
If we now keep in mind the fact that
•• "
Since ( is an arbitrary positive number, this yields (6). This completes the proof of the theorem.
In
Zk
+ 2e.
n-ro
n(n-I)
n
where
n-I Zn
Mn
Furthermore, it follows from §lthat, for the same (>0, there exists an N> 0
> N and for arbitrary points z k' where k = 1, ... , n, in the set
(9)
M= maxzt:K If(z)l, we have
< d + (.
E, we have
dZ t .,. dz n·
I An I ~ (2n)n . nr (d + 2et (n-t!. Ln ~ MnL n (d + 2e)n (nc-t), 1
bounded set E* containing E and having in its complement the unique domain
Zn
But we have inequality (7) giving a bound for the Vandermonde determinant in (9). Consequently, if we set
where d is the transfinite diameter of E.
such that, for n
Zl
!(Zn) Zl
f
307
•
• •• I2<jn-l
Proof. It follows from § 1 that, for arbitrary
.
§2. BOUNDS FOR THE TRANSFINITE DIAMETER
'" an ... an+1
I2<j
·__''''''''__----."..-. · . . . . . . . ..-.-----------
--,Oiii>"'~i~_~_~_- ~~~ "";iKih_'_"'''''''''''''~~,-~-''''_'
To evaluate this determinant, we subtract the last column from the first, second, .,. 1 (n - l)st columns. Then, after we remove common factors from the individual
rows and columns, we subtract the last row in the resulting determinant from the first, second, .• "
(n - I)st rows. When we again remove common factors from the
f~":=;;;;::;;,.;...,~~,~w",~;S;t'::::~:;';::~;-;;";:;;$"'~~~,&,,~~~~,?";;',,£;:'T_;i';':i;.t:;;/Z;=:::;;~Si':-~-::;~~~'.tt;'~"~-;'''':;~;;~·;;<=;'';;':;;::;;:'::,:'::':·---~;';:.:;,::::::::.:::;:'~:;:;:;;~'
308
n-
1,2 (n - 1) n(n+l) (2n-l)
]1 A
a z- 2 + ... are integers and if the function fez) is regular in the complement of 2
a closed bounded set E such that dee)
n-l'
. . nr-x;:
r
n~~V ~=n~moo
that, for n
1 ·2 '" (n - 1) ]2/n 1 n(n+l) ... (2n-l) =16'
(10)
f,
determinants, beginning with some number n, are all equal to zero. This last is a necessary and sufficient condition for fez) to be rational (Kronecker's theorem). We shall not stop to prove this assertion.
there exists a positive nUQlber N such
§3. ·The capacity of a closed bounded set
> N, I { 16 - e
)n :::;; AAn_ n
1
(1
The importance of the constants d and
)n
~ 16+ e .
ities obtained, we get
2
p(2N+p+1l
1
1
__ 1_
A~'HP)2 :::;;AAr-+1pP)2 ~(16+e)
Let us again look at a closed bounded set E of points in the z-plane. The complement to this set in the z-plane consists of a finite or countable set of dis us show that one can exhaust this domain by means of domains B(n), where n=1,
2
2, ' , .; each ·of which is bounded by a finite number of closed Jordan curves. Here, exhaustion is understood in the sense that [j(n)C B(n+l), B(n)C B, and every
or
16- e) 2(N+p)2
defined in the preceding section
joint domains. We denote by B that one of these domains that includes 00. Let
AN + p (116+ e)P(2N~P+1l 1 e)P(2N+P+I) ~-A--~ (16N
(
T
follow from their association with the Green's function.
If we now give to n the values N + 1, N + 2, •• " N + p and multiply the inequal
1
To
gers, it follows that (6) with d < 1 can hold only under the condition that tbese
[
This last limit can be calculated with the aid of Stirling's formula. We conclude from (10) that, for arbitrary positive
< 1, then it is a rational function.
see this, note that since, in the present case, all the determinants (5) are inte
From this formula we get
r
309
the theorem of P6lya: If two coefficients in the expansion fez) = at z-t +
rows and columns, we finally obtain the recursion formula
[
;;:';;;;;.,u;;::;;::-...."~_;;;;;:~;;;:';;,;-;,,:,.,,.:,..,;i:t-~t'i1;.':,:;.:;;;;.::;:~.~,~_._--:;':';;:;..:;;
§3. CAPACITY OF A CLOSED BOUNDED SE,l
VII. MEASURE PROPERTIES OF CLOSED SETS
A -
-
p(2N+p+1l 2lN+p)2
_ _I
A~N+P)2
point z € B beginning with some n lies in B(n). Let us cover the domain B with a grid of squa~es of side
El2
and let us consider the set consisting of points of
the closed squares in the grid that contain at least one point belonging to E. The By virtue of the arbitrariness of
f
> 0, when we let p approach 00, we get
complement of this set consists of domains one of which is the domain B con f
n2
-
1
taining 00. This domain, together with its boundary, lies in B. The distance from
lim yA n =4'
its boundary points to the set E does not exceed
n-+oo
f.
By extending the domain B, f
Consequendy, in accordance with Theorem 3, we have d ~ 1/4. This is.a sharp
we can arrange for its boundary to consist of disjoint closed Jordan curves. To
bound for d because, if E is the segment connecting the points
do this, we need only adjoin to B
°
and 1, we
have d = 1/4 in accordance with §L As will be pointed out in §5 of Cbapter XI, inequality (6) remains a sharp inequality for an arbitrary set E consisting of a finite number of disjoint closed analytic Jordan curves. replace d in (6) with an arbitrary number d'
< d,
1)
This means that, if we
this inequality will fail to hold
for some function fez) that is regular in the domain B. In that section, we shall
f
sufficiently small neighborhoods of the mul
tiple points of the boundary of Bf • By giving
°
f
suitable values
f
n
such that
f n as n -.. 00, we obtain a sequence of domains B(n), where n = 1, 2, .•• , that exhaust B. One can easily show that if the domains B(n) exhaust B, then,
for an arbitrary given
f> 0, when n is sufficiently great the distance between all
points of the boundary of the domain B(n) to the set E is less than
f.
Suppose now that the domain B defined above is exhausted by domains B(n)
present a theorem on the distribution of values of polynomials in the complex plane expressed in terms of the transfinite diameter. Furthermore, we mention
with boundaries K(n). The Green's function g (z, 00) for the domain B(n) is, as
briefly an actual application of Theorem 3 to power series. Specifically, we have
we know, a harmonic function in B(n) except at the point z
n
in B (n), it assumes the value
z l)Goluzin [1946b),
=
lXJ
°
= 00,
it is continuous
on K(n), and it behaves in a neighborhood of
in such a way that the limit
310
S3.
VII. MEASURE PROPERTIES OF CLOSED SETS
it is necessary and sufficient that the capacity of its boundary be positive:
exists and is finite. The quantity y n is called Robin's constant for the domain Consequently, in a neighborhood of z
=
00, we have
the plane is equal to its transfinite diameter; that is C(E)
where u n (z) is a harmonic function in B(n) including the point z = 00, and ap • _ . proaches 0 as z ----> 00. Since B(n) C B(n+1), it follows that the function n+
l(z, 00) - g (z, 00) is harmonic in n
negative on the boundary
including the point 00, and it is non-
B(n),
In accordance with the maximum principle, this
K(n).
last statement also holds everywhere in the domain
B(n);
With regard to capacity, we shall now prove Theorem 2.1) The capacity of an arbitrary closed bounded set E of points in
g" (z, 00) = log I z I+i" +u" (z),
g
•
Theorem 1. For a domain B containing 00 to have a Green's function g(z,oo),
lim (g,,(z, oo)-loglz\)=i" 18-+(;0
B(n).
that is, for all z E
Green's function and that h(z, 00) is the conjugate function to g(z, (). For arbi trary finite z E B, the function II
(~, z) = g(~,
z) - g(~, 00)+ log I~ - zl,
as a function of (, is harmonic in B. If we apply formula (5) of to it, keeping in mind the fact that u«(, z) = log
Yn + U n (z) either converges everywhere to + 00 in B or converges to a harmonic function y + u(z) such that u(oo) = O. In the latter case, the quantity y is called
(, z E B, II
Robin's constant for the domain B, the quantity C = e - Y is called the capacity =
log Iz 1+ y
+ u(z)
is called the Green's
function of the domain B. Obviously, C and G(z, 00) are independent of the B(n).
Green's function g(z, 00) does not assume
B, all its limiting values are nonnegative. However, we cannot assert that these limiting values are always equal to zero. For example, at isolated boundary points of the domain B, the value of g (z, 00) will indeed be positive because, in a neighborhood of any point, the function g( z, 00) is a harmonic function thatis'/ bounded below, so that it is also harmonic at that point itself. We note that, in the case in which y n + u n (z) -
00 for z in B, it is also true that y n ----> 00 be cause u n (00) = O. . In this case, the capacity of the set E is taken equal to zero.
g(z, oo)-i=
B(n)
that exhaust the domain
tively by the inequalities functions for the domains
g(z, 00) B(n)
> i n,
B
let us take domains defined respec
where
i
n
---->
0 as n
---->
g(z, 00) - y, where y is Robin's
2~ ~ logl~'-zldh(~', 00).
(1)
T
= e- Y , where
T
is C=ebyS'ev's con
stant for K. Let A denote a positive number sufficiently small that the inequality g(z, 00) > A defines in B a domain BA with boundary K A consisting of closed Jordan curves that approximate the corresponding curves constituting parts of K.
Since
2~ l'.~
dh (C'. 00) = 1
K
'
let us partition the boundary K by points (", for k =1, . , " n (numbered in order of occurrence as one moves around
n
they converge in B to g(z, 00). The result that we obtained above can be formulated somewhat differently as
=
Making use of this formula, let us show that
00. The Green's i , and
are respectively the functions g(z, 00) -
(~', Q.
K
finite number of closed Jordan curves, with the Green's function defined in Chap· the domains
z Idh
zion K, we obtain, for
constant for the d,omain B, we have
have given here coincides, when the boundary of the domain B consists of a
is a Green's function in the previous sense, for
-2~ .~ log I~' -
If we now set (= 00 and note that u(oo, z)
It should be noted that the concept of the Green's function g(z, 00) that we
Specifically, if g(z, 00)
(C, z) =
I( -
S3 of Chapter VI
K
negative values anywhere in B. Consequently, as we let z approach the boundary
ter VI.
deE).
of a finite number of disjoint closed Jordan curves. Suppose that g(z, () is its
It follows in accordance with Harnack's theorem that the sequence of the functions
of the set E, and the function g(z, 00)
=
B containing 00 with boundary K consisting
Proof. Consider first a domain
B(n),
gn+1 (z, 00) ~g" (z, 00), in+l ~i".
choice of exhausting domains
311
CAPACITY OF A CLOSED BOUNDED SET
1) Szego [1924].
K, into subsets 1" such that
....
, __ ._.....,-,
-T_~_~~'
312
---------- ------ -
---_.,
;.••-'-
-
-".~,,-~
-iii...';;'>;-;>';""
;'~k.",,,,"-;""",,,,,.~,,_,,,,,,,,·,=_-,,
VII. MEASURE PROPERTIES OF CLOSED SETS
2~ ~
dh (C', co)
= ~,
...,,,,;_ _.
§3.
·,"'''''''"";.=.,='''·>:'%''''''''"'',,'''''''''''''''·'''''''.. .,<.1.'''''''''''''''''',,,;;;:=--<'_-:<;;''''''''''__,F;,-'_«>·~_'''''''_''''''''" ....'''·O;.,'w,~~-.,.,.",~''',.,,_,0!<1«''''''''''~''''''''',.,--.;'"''
is harmonic in B and assumes nonnegative limiting values as we let z approach K. Con
k=l, ..., n.
sequently, vn(z)~O everywhere in B. In particular vn(oo)=y+(I/n)log~n~O; that is.
III
Let us compare the integral (1) with the sum
~ > e-'Y.
~ !lOg \Ck - z1.
Now as n
Il=l
sult
KA, the oscillation of log I( - zion each of the subsets lk consisting
of only one arc will, for sufficiently large n, be less than prenamed
f
> 0 inde
T =
--+
00, we obtain
T
~ e-'Y and this, together with (3) yields the re
e-'Y.
Proceeding now to proof of the theorem for the set E, consider the domain B complementary to E and containing 00. Suppose that B is exhausted by domains
pently of z. On the other hand, the number of subsets lk consisting of several
B(n) bounded by closed Jordan curves. Let y
arcs does not exceed the connectedness of the domain B. Therefore, the difference
the domain B(n). We denote by E
l 8n (z)=-2 1t
~
n
k=l
~ (log I C' - z I-log ICk -
Z [)
dh (C'. co)
Consequently,
k=llk
approaches zero uniformly with respect to z €
KA and
KA as
n --+
00. Consequently, in
T
< e -'Y n.
e-1n
~ !logICk -
Ian (z)\ < Ai
z l+8n (z)
where
(Ck
-
z) I ~ e 2A -
(2)
1•
'k)[ attains its maximum in the complement to BA
with respect to the z·plane somewhere on KA, it follows that (2) also holds at n
all points of this complement and, in particular, on K. This means that ~ $
2A -'Y for n sufficiently great, where m (see §1) is the quantity corresponding n to the set K, and consequently, T $ e 2A -'Y. By virtue of the arbitrariness of A>·0, it follows that e
T
On the other hand, if
and ~n
=
maxz€ K
tn (z)
Itn (z)1
t v (z)
Vlltv (z n)1.
n
Since the quantity III~=l (z -
~ max
111 1. (z) I,
'oJ
= 1, 2, ...
is the Cebysev polynomial for E. This means that, for every n,
there exists a point zn on the boundary of the domain B(n) such that e-'Y n $
that is,
jfI}]
$ e-'Y. On the other
z E En
k=l
where
T
hand, it follows from (5) that ~ > e-'Y n • This shows that V,n
n
).-r=
By letting n approach 00, we see that
is sufficiently great, we have
n
denote the Robin's constant for
En are also Cebysev for the boundary of the domain B(n). Let us de v _polynomials _ f.ine T = lim V~ V.n . In accordance with what we proved above, ;." = e -'Y n. n V-+oo But since E c E n , it follows that T -< T n , where T is Cebysev's constant for E.
n
=~
n
the complement of B(n) with respect to the n " " z-plane. We note that the Cebysev polynomials ""' t N () z of degree 1/ for the set v.
co)-~ 'Ylog\Ck - zl
n .I8J
log I C'-zldh(C',
accordance with (1), if z €
(5)
n-
n
For z €
313
CAPAOTY OF A CLOSED BOUNDED SET.
$ e-'Y.
n
1, 2, . ", lie on
E. Let Zo denote one of these points. Then, this last inequality implies e-'Y $
VI",v~ tv (z 0)[ and consequently, e-'Y $ V m ' tain e-'Y $
T.
By letting 1/ approach "", we obv This, together with what we showed above, gives the result T= e-'Y
or, what amounts to the same thing, dee)
=
G(E). This completes the proof of the
theorem. An interesting result is obtained when E is a continuum. Then, the domain
B is simply connected and the Green's function g(z, 00) for B coincides with the logarithm of the modulus of the function'
l'l
=
ttz) which maps the domain
B univalently onto the domain > 1 in such a way that the point z = 00 is mapped into 00. Therefore, Robin's constant for the domain B is equal to log 1['(00)1, and the capacity G(E) = 1/1['(00)1 = R. Here, R is the conformal
(4)
domain B is mapped univalently onto the domain
then the function
~ log I iff, (z) I
=
(3)
is the Cebysev polynomial for K (with zeros on K)
v n (z) = g(z, (0) -
All the cluster points of the sequence {z n I, for n
,=
radius of the domain B (with respect to 00), that is, that number R such that the
1(1
> R by means of a normalized
'"f:
•. --.
• . •n_..
.... ::-..
'::~~_::-
314
'::::"::: __ ,g":':'ii_;
"j;,;,;;"fflL
•••• _..
'c_.""'-;;;...
-_Oj~:.m..
..- ••.,-.--
.'~'''''''''''''-''"'''''''~;;"'";:''?''''"-'''''-~''.~-~~'
=
00
and F' (00)
n = 1, 2, " ' , denote a sequence of domains contained in B and containing
= I.
1)
curves and that the sequence exhausts the domain B. Finally, let B~), for n
The capacity, and hence the transfinite diameter, of a bounded
=
B(n) by removing the curve f3 and the
1, 2, .. " denote the domain obtained from
continuum E are equal to the confonnal radius of the domain complementary to E and containing
f3.
Suppose that their boundaries an consist of a finite number of closed Jordan
Thus, we have Theorem 3.
315
§4. HARMONIC MEASURE OF CLOSED BOUNDiD SETS
VII. MEASURE PROPERTIES OF CLOSED SETS
function' = F(z) that satisfies the conditions F(oo)
".~,",~-
domain bounded by it. We denote by w(z, an' B~)} a function that is harmonic in
00.
an' and equal to zero on 13. Obviously, this function is nonnegative and less than unity in Bt). Furthermore, the difference w(z, an' B~)~ w(z, a , B n+1 } since it is nonnegative on the boundary of the domain B(f3n), n +1 f3 is positive in the interior of B~n). This means that for fixed z € B13' the quan tity w(z, an' B~n)} is defined, beginning with some n and decreases with in B~), equal to unity on
In conclusion, we p.resent one more theorem connecting Green's function with C=eby~ev polynomials. Theorem 4. Suppose that a domain B with boundary K contains
and has
00
Green's function g(z, (0). Let t n (z) denote the ~eby¥ev polynomials for K with zeros on K. Then
creasing n. We can now apply Harnack's theorem, concluding from it that l~"
n
lim n -+
<Xl
ITn (z) I
m
= eg (z. <Xl), mn =
~1'
max I t n (z) I,
lim n ...... 00 w(z, a n , B(n)} = w(z, a, Bf3} exists everywhere in Bf3 and is a harmonic f3
function in B .Obviously, 0 ~ w (i, a, Bf3) < 1 in Bf3'
f3
zE K
We note that w (z, a, Bf3) does not depend on the choice of sequence B (n) because, if there are
this limit being unifonn in the interior of B.
twO
sequences B(n') and B(n") of this kind, then, for arbi
,
(") B(~ ' ) c B!3 ' and conversely. ·Conse ( ') quently, in the first case we have the inequality w(z, an" B < w(z, an'" B~ " )},
Proof. Consider again the functions v n (z), for n = 1, 2, .• " defined by formula (4). Since these functions are harmonic and nonnegative in B, the condensation
J }
and, in the second case, we have the opposite inequality.
principle can be applied to them. According to this principle, the sequence
The quantity w (z, a, Bf3) is called the harmonic measure of the set a with
tv n (z)\ itself and any subsequence of it contain subsequences that converge uni formly in the interior of B to harmonic functions. Here, the limit functions are always nonnegative in B and, in accordance with Theorem 1 they vanish at
"
trary n , there exists an n such that
respect to the domain B, the curve
f3,
and the point z € B13'
We note that, as we let z approach points of the set a through values in Bf3'
00.
Consequently, these functions vanish identically in B. This means that the se
"the harmonic measure w(z, a, Bf3} does not always approach unity. For example,
quence tv (z)l itself converges uniformly in the interior of B to O. This is
it will not approach unity as z approaches isolated points of the set a because
n
at each of these points w(z, a, Bf3} is a harmonic function. However, we can
equivalent to the assertion of the theorem.
show that, if w (z, a, B f3) 1= 0, then, as we let z approach points of the set a
§4. Harmonic measure of closed bounded sets
through values in Bf3' the limiting values of the harmonic measure w (z, a, Bf3)
Together with the methods considered above for measuring closed bounded subsets of a plane, R. Nevalinna
2)
are bounded below by a positive number. To see this, let
introduced another measure, which has proved
dan curve in B that surrounds the curve
very important in the theory of functions.
set a. On
Let a denote a closed bounded set of points in the z-plane and let any of the domains complementary to a.Let
13
13
13
denote a closed Jor
and does not contain any point of the
the function w (z, a, Bf3) has a positive minimum 0 such that
o < 0< 1. Let z 0 denote an arbitrary point of the domain 13 0 and let n be sufficiently great that f3 0 and Zo lie in
B denote
denote a closed Jordan curve in
the domain B that has no points in common with a. Let us remove
13 0'
f3 0
Bf3 that lies outside B~). Since
w(z, a, B~n)} > w(z, a, Bf3} in B~), it follows that w(z, an' B~)} >0 on f3 ' 0 Furthermore, w (z, a n , B(n)} = 1 on a n . Consequently, in accordance with the
and its
interior from B and denote the remaining portion of B by Bf3' Let B(n), for
f3
maximum principle, we obtain w(zo' an' B~)} ~ () for all n sufficiently great. 1) Fekete [1923]. 2) R. Nevanlinna [1936a]. All the material of the present section is taken from that article.
, l
As we let n approach 00, we see that w (z , a, Bf3)
o
-> 0 > O. It follows that all
the limiting values of w (z, a, Bf3) as z approaches a exceed ().
...
",,,,,,-;:;;u'_,,,,,,: .....',...
"J''''':;;e:'''''''''''''''''
""",,,,,,.,"';.
""""'"
~"'"
-,,"~~"""'-
317
w(z, a, Bf3) ~ w(z, a o' B o) > O.
The question arises as to ways of telling when the harmonic measure of a given set a is zero or positive. The vanishing or not of a harmonic measure
It follows from this last assertion that, if the harmonic measure of the set a
w (z, a, Bf3) holds simultaneously for all points of a domain B (3 because, if it vanishes at a single point in B 13' it must vanish everywhere in Bf3 by virtue of
with respect to some domain complementary to a is equal to zero, then a does not contain a continuum and the complement to a is the unique domain including
the maximum principle. However, the same phenomenon holds with regard to the
00.
We now point out the following simple properties of sets of zero harmonic
curves {3 chosen in B. To see this, suppose that for some curve {3, we have
measure.
{3'" denote another curve of the same ki~d. Let y and y' denote closed Jordan curves lying in B and enclosing {3 and {3 , such that y' lies inside y and there are no points of the set a between y and {3 *• The curves y and (3* together form the boundary of a domain B 0 contained in B. Set () = maxz€y' w(z, y, B )' 0 < () < 1. Now, let l denote any positive number. :For o n sufficiently large and for z E y, we have w(z, an' B~») < f because the con vergence of the sequence {w(z, an' B~»)\ to zero in B 13 implies its uniform con vergence to 0 inside B13' Let us denote by A and A' the maxima of the function w (z, a n' B~:) on y and y'. Since the difference w(z, an' B~~») - w(z, an' B~») is Ii harmonic func tion in the portion B~n) of the domain B(n) that is bounded by the curves an and y' and since it is equal to 0 on an and does not exceed A' on y', it follows that A < A' + f. Furthermore, since the difference w (z, an' B<;1) - Aw (z, y, B 0) is a harmonic function in B 0 that vanishes on (3* and is less .than 0 on y, it is less than zero in B 0 and, in particular, on y', where the maximum of the function w(z, an' B<;:) is equal to A'. Consequently, on y' we have A' ~Aw(z,y, Bo) ~ 'Af). From this inequality and the inequality A < A' + f, we obtain A' < (1 _ ()) - I ()f. Therefore, on y I we have w (z, an' B ~) ) ~ (1 - ()) -I (k . Since f is an arbitrary posi rive number, it follows that w(z, a, B13*) = 0 on y' and hence everywhere in B (3*' w (z, a, B(3)
""'''"''.;r
§4. HARMONIC MEASURE OF CLOSED BOUlWED SETS
VII. MEASURE PROPERTIES OF CLOSED SETS
316
::0;""_..
= 0 and let
Every closed subset of a closed bounded set of harmonic measure zero is also a set of harmonic measure zero. This follows immediately from examination of the differences between corresponding functions of the type w(z, an' B~n».
The sum of two closed bounded sets of zero harmonic measure is a set of harmonic measure zero. To show this, denote the two sets by a
l
and a
2
and let {3 denote a curve in
+ a 2• Let {B(;)\ and {B~n)} denote two sequences that l exhaust the complements to a l and a 2 respectively and that contain {3. Then, for a given point z in the domain B complementary to a and containing (3, we the complement to a = a
have (II)
lIl(z,at,II,Bt)
< 2'
(II)
£
(J)
(z, tx..!, /I> B2)
<£2'
for sufficiently large n.Here a 1 ,n and a 2 ,n are the boundaries of the domains B(;) and B(;). Let B(n) denote that one of the domains in the intersection of
Bin) and B(;) that contains (3. For suitable choice of the domains Bin) and B ~n) (they can, for example, be constructed from squares as at the beginning of §2), the boundary of the domain B(n) consists, for every n = 1, 2, .• " of a finite number of closed Jordan curves. Therefore, the domains B(n), for n = 1, 2, exhaust the domain B. The function
Thus, for a given domain B complementary to a, vanishing of the harmonic measure w (z, a, B (3) is independent of the choice of curve {3 in that domain.
(J)
Let us show now that if a contains a continuum, then w (z, a, B (3) >0 for
( z, all' B (II») ~ -
(J)
(Z, at. III B(II') 1
(J)
(z, tx..!, II' B(1I1) \I ,
any domain B complementary to a and for any curve {3 C B. Just as in the proof
which is harmonic in B~) is equal to zero on {3 and does not exceed zero on an'
of Riemann's theorem in §2 of Chapter II, we can, by a suitable transformation,
Consequently, in B(3 we have
arrange for the domain B to have exterior points•. Let B 0 denote the domain (J)
bounded by the curve (3 and some other closed Jordan curve lying outside B. Then, B<;) C B
that is, w (z, a, Bf3)
o' Consequently, just as above, we can show that, for z E B~n), w(z, a, B~n»)~w(z, a o' Bo»O.
By letting n approach
00,
we see that
(z, =
IX, B~) ~
(J)
(z,
ab
B1)
+
(J)
(z, tx..!, B\I),
O.
We shall now prove the theorem of R. Nevanlinna, which is fundamental in the theory of harmonic measure.
l
318
VII. MEASURE PROPERTIES OF CLOSED SETS
§4. HARMONIC MEASURE OF CLOSED BOUNDq'l SETS
Theorem 1. A closed bounded set has zero harmonic measure if and only if
m
319
> log(p - 1) + y'. Consequently, we obtain from (3) p
its capacity is equal to zero. Proof. As a preliminary, we shall make a necessary observation concerning
1';;;: (l - (lJp (Zl' a', B' p»)(log (p - 1)
Green's function.
so that
(lJp ( z, a,, B') p;;;: 1 -~'
B denote a domain containing 00 and bounded by a finite number of
Let
+ I')'
L
"(
1",~ (~_1\
log (p -
I)
~-.L 1"".. (~_1\
.;:::
> 0,
closed Jordan curves. Let.g(z, 00) denote its Green's function and let h(z, 00) denote the conjugate function. Let y denote Robin's constant. Formula (1) of §3
since y':5 y. This shows that if the capacity of the set a is positive, the har monic measure of a is also positive.
is then applicable and from it we obtain the inequalities
log d1 (z) ~ g(z, (0) -1 ~ log d ll (z),
(1)
(2) Now suppose that the capacity of the set a is equal to zero. Since the difference (1 - w p )M p - g' is a harmonic function in B'p and since it is non
where d (z) and d (z) are the minimum and maximum distances from the point z 1
2
negative on the boundary of B'p , it follows that this difference is also nonnega . dve in B 0 and this yields
to the set of boundary points of the domain B. We shall very shortly have occa sion ·to use these inequalities. Turning to the proof of the theorem, we may assume that the closed bounded set teferred to in the statement of the theorem is contained in the disk
Iz 1
<1
because, otherwise, we could arrange for this by means of a similarity transforma
r
g' :5 (l -
note a domain including
00
that is bounded by a finite number of closed Jordan
M1 ~ (1 -
P (z, a', B'p) the harmonic measure of the boundary a' of the domain B' with
respect to the subset B~ of the domain B contained in
Iz I < p,
p)m p is a harmonic function in B~ and it is nonnegative on the bound aryof B'p' Consequently, this difference is also nonnegative in B~; that is,
and, in accordance with inequality (1),
. Consequently,
+ 1) + 1'.
l' ~ (1 - (lJp (ZII, a', B~»)(log (p
so that
+ 1) + I')'
W
g' ~ (1 10
B;» M p'
Izl=1
Mp ~ log (p
is finite. .Let p denote a fixed number> 2 and let Mp and m p deno. te the maxi mum and minimum values of the function g' (z, 00) on I z I = p. The difference g' - (1 -
(lJp (Z2' a',
M 1 = max (Ft - log I Z 1);::: 1',
where p> l
(1) If a has positive capacity, then the constant y = limz-+oo(g(z, oo)-loglzl)
(4)
But
curves contained in I z I < 1 and satisfying the condition li' C B. Let us denote by g(z, 00) and g' (z, 00) the Green's functions for Band B'. Let us denote by W
p)M p'
For a point z 2 on I zl = 1 at which g' attains a maximum, which we denote by Ml' inequality (4) yields
tion without affecting the property to be proved. Let us denote this set by a and let us denote by B the domain complementary to a that includes 00. Let B' de
W
.
ffim (lJp ~ (lJp (ZII,
W
p)m p
zero.
(lJp (Zl' a', B;» mp'
:5 lim z -+ 00 (g' -
log Iz I)
'
~( I
'- _,
I
"
•
If we fix p > 1 and let B' (a sequence of domains exhausting B) approach B, then y' approaches
which we denote by m l' then inequality (2) yields
But m 1 = min I z 1=1 (g' - log I z I)
.
B p) ~ 1 - ,
(2)
B~. If zl is a point on the circle Izi = 1 at which g'(z, 00) has a minimum,
ml;;;: (1 -
, (l,
Iz I= I
(3)
W
00
and consequently min1zl=Iwp(z, a', B~) approaches
Since the function
W
p (z,
a', Bp'). on
I zl =
1 then converges uniformly to
(z, a, B p) where B p is the portion of the domain B lying in the disk I zl
< P,
it follows that QJ(z, a, B'/3) == 0 because the function wo(z, a', B'p) would =
y' and, by (1),
otherwise have a positive bound on
Iz I =
1. Therefore, the harmonic measure of
••_ -
----.- - - - - - - - .- . - - - - - - . . . - -
------
_.
,-- '-e:iiE2E".!!!
--~_.
,.'"
"..- ... _.~~'-'-"-'----_.,,'.',-
VII. MEASURE PROPERTIES OF CLOSED SETS
320
,,~,~._~
--
,-",-+
-'-""'-"--
.,,,--~~
,-_. __ .~.,-
~_.".,,-_.~,
321
§5. APPLICATION TO MEROMORPHIC FUNCTIONS
a is zero. This compl~tes the proof of the theorem.
Proof. Suppose that u (z) ~ M in B. Let us suppose that tbere exists a point
> m. It follows from the hypotheses of the theorem that there exists a point zJ € B such that u(z) < u(zo)' Let us describe a small disk (3 around z 1 in which u (z) < m l' where m < m 1 < u (z 0)' Ftom the hypothesis of z
As simple applications of the theory of harmonic measure, we shall no.. prove two theorems, one of which deals with removable singularities of harmonic func tions and the other with the maximum principle for harmonic functions.
o
in B at which u (z)
the theorem, tbe set a can be contained in tbe interior of a finite number of Jordan
Theorem 2. Let a denote a closed bounded set of points in the z-plane that is contained in a domaiT/: B. If the harmonic measure of a is zero, then every
curves lying in the complementary domain containing B. Let us denote the union of these
function u(z) that is harmonic and bounded in B except at points of the set a is
Jordan curves by an' so that, for the domain B~) with boundary an
harmonic in the entire domain B if we define it in a suitable way on a. On the
w(zo' an'
Uf3,
we bave
B';»)
other hand, if the harmonic measUre of a is positive, this property does not
which is harmonic in the intersection of the domains B and B~). As we let z
always hold.
approach boundary points of the domain B that are distinct from points of the set
Proof. Suppose that the harmonic measure of a is zero. Then we can assume
cause otherwise we could consider not B but a suitable domain of the type de
a, its limiting values do not exceed m - m - (M - m) < 0 and as we let z approach points of an' they do not exceed M - m - (M - m) = O. Finally, on (3 we have v(z) < m - m. It follows that v(z) < m 1 - m in Be;) B. In particular, for 1
scribed contained in B and containing a. Let u 0 (z) denote a function that is
z = z 0' this yields
without loss of generality that B is a bounded domain with boundary consisting of a single closed Jordan curve
f3
and that u (z) is a harmonic function on
f3
be
n
bounded and harmonic in the domain B and coinciding with u (z) on the boundary of B. Let us show that uo(z) == u(z) in B. For a point Zo € there exis ts' a domain B ~ ), contained in B including z the boundary (3 + a
n
< (.
If lu(z)
in B. Consequently, for the difference v (z)
we have \v(z)
I ~ 2M.
° but not containing
a,
I~ M
=
+ (M -
(zo) ~ m1
m) ill (zo.
(In'
B~n)) ~ ml
+ (M -
m}e.
Since ( is an arbitrary positive number, we obtain u(zo) $ m 1, which contradicts
of which consists of a finite number of Jordan curves and
which has the property that w(zo' an' B~»)
Iu 0 (z) I < M
II
Band givenc >0, the assumption that u (z 0)
in B - a, tben
§5.
u (z) - u 0 (z) in B - a,
"Now, let us consider the function v(z) + 2Mw(z, an'
B~) ))/
> m l' This completes tbe proof of the theorem.
An application to meromorphic functions of bounded fonn
A function f(z) is called a meromorphic function of bounded form in
This function is harmonic in B ~), equal to zero on (3, and nonnegative on an'
Izl < 1
if it can be represented in that disk as the ratio of two bounded regular functions
Consequently, v(z) + 2Mw(z, an' B~») ~ 0 in B';). Therefore, v(zo) ~ _ 2Mw > - 2Mc. Since c is an arbitrary positive number, v(zo) 2. O. Treating the
If'(z)
(1)
f(z)= '\J(z)'
function v(z) - 2Mw(z, an' B~»), we can show that v(zo) ~ O. Consequently, v(zo) = 0; that is, uo(z) = u(z) in B - a. It follows that if we define u(z) on a
i
We may assume that
I¢ (z)1
~ 1 and
11/1 (z)1
~ 1 in
Iz I < 1.
equal to uo(z), then u (z). becomes barmonic in B. On the other hand, if ·a has
A test of whether f(z) is a meromorphic function of bounded form is given by
positive harmonic measure, then, for example, the domain of definition of
Theorem 1.
Iz I < 1
w (z, a, B), which is harmonic and bounded in B - a cannot be extended to a in such a way as to make w (z, a, B) a harmonic function in B because this func
1)
Let f(z) denote a function that is meromorphic in the disk
such that f(O) f-
00.
tion, since it is equal to zero on (3, would be identically equal to zero in B - a
T(r)
and then the set a would be of zero harmonic measure. Theorem 3. Let u (z) denote a function that is harmonic and bounded above in a domain B. If
lim.;~ z. u(z) $ m for all points Z· on the boundary except for
a set of harmonic measure zero, then u(z) $ m everywhere in B.
For f(z) to be a function of bounded form, it is neces
sary and sufficient that the quantity
= d~
·i'·'...· .... , .'.
II""
Ii
~
I) R. Nevanlinna [1922].
\~ log+ If (re
2"
tB
)
I de +
I Ibkl
log I ;~ I
-_.---""".~_-
=-
_
:::"_,,·~,:::-c::-:c-:--::---:-.-::---:---~·::,~-~_-~~"_-'~-~
322
•
-~-_ ~~~
•r
_ _ ••
_:~:.~~~_
---
-"0'-'-
~.:~.-.-~:~--=--._-
__ --- -.-.,-
'·~'-_.--"W
""'~L_.~
;-.
VII. MEASURE PROPERTIES OF CLOSED SETS
=
!
log Ij (0) I=
lakl
if 0
0
I~l
r(z-_ak): r 2 - akz
we conclude that the function
f, (z)
II _
1~I
for arbitrary a such that
lal < r,
log Ij (re",O)
k
1 on the circle Izi
az)! = 1
2
we have Ir(z - a)/(r -
=
Iz I ~ r
disk
as
log F(z) =-2' 11: in
Iz I < r,
o
reiO+z log I F(re") 1-/6re -z dO '6
j (z)
IF(z)1
=
1
~ ...:..
rpr (Z) = I
+ lC
,I,
'rr
11
_ _1
rr(z 2
(II =
-akz
r (z 2
Z)
ak)
_
b) _ k
'Y
Ibkl
+
1 211:
e
e
~ log +I
re"'0 + z
I~
1
f (re i6 ) re ,'0-·· - z dO + Ie
2" 0
21<
iO
+ I f (re) iO re +. I ~ dO
1 \ 2" J log 0
re
- z
Z
Iz I < r; also, e-T(r).
Here the functions ¢ r (z) and t/! r (z) are regular in the disk
If(z)\ on the circle Iz I = r, we obtain from the above equa
lakl
a I
(4)
~r (z)' 21<
II
.
I bkl
logr~z-_ak)_ r -akz ...
= 'fr (z)
where
z I = r, then
I¢r(z)! ~ 1 and It/!,(z)/ ~ 1 in that disk and It/!r(O)1
tion the formula
logj(z)=
=
Suppose now that T (r) is less than some constant M for 0 I r n I, where r n
log·r(z-b k) B b r -
(3)
+ J f(re1 IO ) I, (re",O) ! -log
on I z 1= r.
where C is a real constant. Keeping in mind the value of F(z) and
the fact that
+ Ij I= log
r because,
f(z)/fr(z) is regular and has no zeros on Iz! ~ rand log F (z) is a regular function on Iz I ~ r. If we apply the Schwarz formula to the function log F (z) and to the disk Iz I ~ r, we have 1 ~
log Jj(re/~ IdO.
Izl < r:
=
2"
2"
~
we obtain from formula (2) the following representation of the function f(z) in the
Therefore, if r is such that there are no zeros or poles of f(z) on the function F(z)
Ibkl
Keeping in mind the fact that
O
r (z-b k) r2 _ b z '
=
323
Jensen-Schwarz formula.
has the same zeros and poles in
does the function f(z). Furthermore, If,(z)!
'.··-':~·"'"?l~~1~1'."?
Formula (2), which is a generalization of Jensen's formula, is called the
~ a ~ 1.
b k' then, by setting
II
.-.
_~,.,:;i;;;,;:~
log , ~k I
+ 2~
Proof. If the zeros of the function f(z) are denoted by a k and the poles by
f r (z)=
!
log I ~ I_
{lOg a if a;:: 1
..
._-__:~.. ~"',.".-.v.
§5. APPLICATION TO MEROMORPHIC FUNCilONS
be bounded above for 0 < r < 1. Here, the b k are the poles of the function f(z) other than the point z = o. These zeros are indexed in such a way that a pole of multiplicity n is counted n times. The summation is over all poles bk in the disk Iz I < r. The symbol lrig a is defined by + log a
,"n--~_"">_~:~':::~~::'::::::";"::'C;:;C:,":'~.,~
n approaches
kZ
for n
=
(2)
Let
1, 2, .. " denote a sequence that converges to 1 as
Then, the sequences It/! n (z)1 and 1..1.. (z)j contain subse~ 'f' n
quences It/! nk (z)1 and 1..1.. (z)1 that converge uniformly in the disk 'f'nk
2"
~ log Ij(rei~ I ~~;:~~ dO + iC.
< 1, OQ.
< r < 1.
regular functions t/! (z) and
¢ (z)
such that It/! (z)! ~ 1 and
I¢ (z)1
Iii
<1
to
Izl < l-
~ 1 in
As a result, we obtain the representation (1). This formula remains valid when there are zeros and poles of the function
Conversely, suppose that the representation (1) holds also for t/!(O)
f(z) on the circle Iz I = r. That this is true can be proved by a passage to the limit with respect to r. In the particular case in which f(O) I- 0, if we set z = 0 in (2) and equate the real parts of the two sides of the equation, we obtain the familiar formula of Jensen
which we can easily reduce t he representation (1)). Then since ~!
-'-(z) I log Ij (z) [ -log II ~'f (z) I ~ log I ~ 1(z) r--
wt
\·,;~~.1. '~
1·
t.:••,\. . •. •. . ..•••••. :\
in the disk
Izl < 1,
we have
-
log I ~ () z I,
I-
0 (to
324
VII. MEASURE PROPERTIES OF CLOSED SETS i8 ltg rI (re )
I ~ -log I ~ (re i8 ) I,
Here the a
0< r < 1,
and b k are the zeros and poles respectively of the function f( z) - c
< Iz I ~
r. Since the b k are also the poles of f( z), they are inde
pendent of c. For every r in the interval 0 ~
I-
1 \ i8 T(r)~- '211 ~ loglHre )ldB+
where the b'k' for k
=
(5)
let d r denote the distance
Izj
~ r and the set E.
Let K denote the set of Jordan curves lying one outside the other that enclose r
the set E and have the property that the distance between K r and E is less than
1, 2, "', constitute all the zeros of the function rjJ(z) in
d. For c € K, the first summation of the right side of (7) does not contain a r
On the other hand, in accordance with Jensen's formula (3) as applied to
r
c in Iz I ~ r.
single term because fez) Ie-
rjJ(z), we have
If we denote by grew, 00) the Green's
function for a domain Br containing 00 and bounded by the curves Kr and if we
logIHO)[=
)1 -'
I b'
I
r
< r < 1,
(6)
m ~
Robin's constant for Br' we obtain from (7)
0
On the basis of (6), the right-hand member of (5) is equal to -log
Consequently, for 0
denote by -h/w, 00) and Yr respectively the harmonic conjugate function and
2"
log:.....!!.... + 2~ \ log I ~ (rt>.i8) j dB.
I bkl
2~ ~ log 1/(0) - e \dh r (e, co)
10/ (0)1.
Kr
we have T(r) ~ -log IrjJ(O)\; that is, T(r) is bounded.
2"
This completes the proof of the theorem. H we replace the requirement that
=
T (r) be bounded for 0 < r < 1 with the re
2~ ~
o
quirement that its limiting values be bounded as r --> 1, Theorem 1 will remain
Ix,
valid without the assumption that f(O) Ie- 00. One can show this by modifying the
---- ...---r
proof given above.
dh (c, 00) r
[ill \ log I/(re
=
Suppose that a function fez) is meromorphic in the disk and that it does not assume in that disk a set of values w that is a closed bounded set E of positive capacity. Then, fez) is of bounded form. 1)
Izi < 1
logl;kl'
2IT. If we now apply formula (1) of §3, we obtain
2~
2"
\ gr(/(re
~
i8
),
!
co) dB +
set K is contained in the disk jw I < R. Then, for I; €
Iwi>
r
2R, we have
I l~
log w w C
We may assume that f(O) -f- "" because in the opposite
(8)
IOg';kl'
I bkl
Now let R > 1 denote a number such that, for all r in the interval 0
Proof. It will be sufficient to prove the theorem for a function fez) that is
Iz I < 1.
!
)-e \dhr(e, co)]dt+
Ibkl
a meromorphic function fez) to be bounded form. Theorem 2.
i8
kr
g;(/(O), co)=
Now, by using the material of §3, we shall prove a sufficient condition for
not bounded in
< r < 1,
between the set of values assumed by the function fez) in
r
loglbkl'
I bkl
I z I < 1.
k
in the region 0
and, consequently, 2"
325
§s. APPLICAnON TO MEROMORPHIC FUNanONS
log (1
-I ~ !) ~
< r < 1,
K r and w such that
log -} .
case, we could consider not f( z) but the function fez + a) /1 + az) for suitable a such that
Ial < 1.
For arbitrary c -f- f(O) and 0
< r < 1,
Consequently, from the formula (again see (1) of §3)
if we apply Jensen's
formula (3) to the function fez) - c, we have
log II (0) -
e
I =
~
"""
lakl
grew, co)-lr=
2~) logl~-wldhr(~' co)
'2"
log I ak I_ r
)1 log I bk I +
...
Ibkl
r
_1 211
r
~ log I/(re i8 ) - e I dB.,
= log I w I +
(7)
;',~i, !V'
, (;' j
:~;'if)'
iI ..< 1) With regard to Theorems 2 and 3, see R. Nevanlinna [1936a].
,i
i j i4
I
as applied to w such that
Iw I ~ 2R,
2~
>log I
W
wei dhr (~,
r
we obtain
+
grew, co)~lr+log Iwl-log 2 ~ log I w I+lr-log 2R.
co),
the
326
327
§5. APPLICATION TO MEROMORPHIC FUNCJIONS
VII. MEASURE PROPER TIES OF CLOSED SETS
Furthermore, since K r lies entirely in the disk Iw I ~ R, we have d(K r) = e -'Y r '5. R and consequently, y r ~ -log R. Therefore, the above inequality yields
number of zeros in Iz \ < 1. Let us denote these zeros by a for k = 1, 2, "', k indexing them in order of increasing absolute value. These zeros are the poles of the function 1/(f(z) - C), which is also a function of bounded form. If we apply
+
.gAw, 00) ~ log I wI-log 3R.
(9)
Inequality (9) also holds for all points of the domain B r lying in the disk
Theorem 1 to this last function, we conclude that the quantity
Iw \ <
2"
2R because the left-hand member is positive at these points whereas the right
I
i11: \ 16g!f(Z)I_C de+
~
hand member is negative ••
!
logl;kl'
I ak I
Using inequality (9), which is valid in the entire domain B, we obtain from r formula (8)
and, in particular, the second term in it is bounded above. Suppose that 2"
1 \
+'0
gr (/(0), 00) ~ -211: ~ log \f(re ' ) I de -log 3R
+
o
'\1
II akl < r log (r/lakl) < A for 0 < r < 1, where A is independent of r. Since the
r
~ 1~I
log I bk I
terms in this summation are positive, it follows that, for arbitrary n and for r sufficiently close to 1,
or 1
211: \
n
+
7(r)=-211: ~ loglf(re
'\1 r ~ log I ak I 0;;;;; A.
)14e
l'0
+ !
1<=1
IOg';klo;;;;;gr(/(O), co)+log3R.
(10)
I bk\
Furthermore, since the domain to
Br is contained in the domain B complementary
E that includes 00, we have gr(f(O), 00)
~
If we now take the limit first as r
~~ality -'-"-,-------
--+
1 and then as n
-->
00, we arrive at the in
00
!
g(f(O), 00), where g(w,oo) is the
log,;
k=1
k
I~A,
Green's function for B. Therefore, we obtain from (10) which means that the series on the left converges. Therefore, keeping in mind the fact that
T(r) '5. g(f(O), 00) + log 3R. This shows that T(r) is bounded for 0
< r < 1;
that is, the function fez) must be
lim
of bounded form. This completes the proof of the theorem.
x-+I
In conclusion, we present the following theorem of Nevanlinna with regard to mappings of multiply connected domains onto a disk. Theorem 3. Let w = fez) denote a function that maps the disk I z I < 1 con formally onto a multiply connected domain B containing 00 (see §1 of Chapter VI). For this function to be a meromorphic function of bounded form, it. is neces
Iz I < 1
values constituting
. __x I-x =1,
we conclude that the series I;=l(I-la k l) also converges. Now, for we have
log I ~
sary and sufficient that the capacity of the boundary of the domain B be positive. Proof. The function w = fez) does not assume in
log~
0;;;;;
a:: II = Im(log (1 -
~ -'/a~zv - ~ ~
I~
.=1
.=1
ak z )
I
-log (1 -
z· '1a' -log Iak I= k
Izi
< 1,
:J) -log I II ak
I L.. ~ z·-:; ak)ak -IV I I
2 1
[
-log I ak I
.=1
the boundary K of the domain B. Therefore, if this boundary has positive capac ity, then fez) is, in accordance with Theorem 2, a function of bounded form. Now, suppose, conversely, that fez) is a function of bounded form. Let c be a member of
B other than 0 or 00. The function fez) - c has an infinite
00
~ 2 (1 - I ak \)
'\1 ~ v_I
Iz l' -.log I ak
ak I
=
_I I - log I
2 (J -I~ak I) 1 ~
a"
I·
#4
VII. MEASURE PROPERTIES OF CLOSED SETS
328
Therefore it follows from what was said above that the series 0::>
s(z)=
~ ~
IOg\l--;-iLkZ\ Z-ak
k=1
converges absolutely in the disk
Iz I < 1
is harmonic except at the points a
k
,
CHAPTER VIII and defines in that disk a function that
where it has a logarithmic singular term
MAJORIZATION PRINCIPLES AND THEIR APPLICATIONS
_ log Iz - akl. The function s (z) is invariant under fractional-linear transforma tions in the group G (see §1 of Chapter VI) corresponding to the domain B be cause these transformations map the set of zeros a k into itself. Therefore, if we
z=
shift to the w-plaoe by means of the function
f{ -l)(w), then s (z) is trans
s (f -1 (w», which is single-valued, nonnegative, and harmonic in B except at the point w = c, where it has a singular term of the
formed into the function s 1 (w) form -log \ w -
§l.An invariant form of the Schwarz lemma
=
cl.
In §2 of Chapter II, we established a simple but extremely important property of bounded functions, known as the Schwarz lemma. A very simple and widely
Suppose now that the disk
Iw - cl ::; f
m. =
min
used form of this lemma is as follows:
is contained in B. Define
I 'llI-C [=.
8J
Theorem (' the Schwarz lemma). Let f(z) denote a function that is regular in
(w).
If the capacity of the boundary of the domain B were equal to zero, then Theorem
3 of §4 would be applicable to the portion B 1 of the domain B in which Iw -
c\ > f
and to the function - s 1 (w). According to that theorem, - s 1 (w) ::; - mf in B 1; that is, s 1 (w) ;:::m in B 1. But m f
f
---'+
+ 00 as
I z I < 1. Suppose that f(O) = 0 and that !f(z)! ::; 1 in I z I < 1. Then I z I throughout the disk I z I < 1 and 1f' (D)! ::; 1. Also, in the inequality " If(zW~-t~ I, the equality holds, for points z f- 0, only if fez) = fZ with I f I = 1.
the disk
f ---+
O.Consequendy, s 1 (w) == + 00
in B, which contradicts the finiteness, proved above, of the function s 1 (w). This
'rRz) : ;
f (0) = 0, that regular in the disk I z I < 1
If we now discard the requirement that
arbitrary function fez) that is
is, if we consider an and satisfies the ine
quality Iftz)l< 1 in that disk, then, if we set
contradiction proves the theorem. .
In connection with other results dealing with the application of the theorem
f
w=g(~)=
on harmonic measure to the study of analytic functions, we refer the reader to the monographs of R. Nevanlinna [1936a1 and Privalov [19501.
I--f(z)f .1 +zC
1-1 zlS
Cl=
for fixed z in the disk
C+Z) - fez) (T+"iC _ (C+Z)=C1C+ ... , ,
l-lf(z)l s !(z),
I z I < 1,
we see that the function g (,) .satisfies the condi
tions of the Schwarz lemma. Therefore, the inequalities mentioned in the conclu sion of that lemma hold. From this we obtain a further generalization of the Schwarz lemma.
Theorem 2. If a function fez) is regular in the disk
in that disk, then, for arbitrary , and z in the disk
329
and if If(z)
I<1
I z I < 1,
1.<::1 C-z r f (z) f (C) -= 1 - zC
f(t)-f(z)
I1-
Iz I < 1
(1)
.. '-'''" ',..
•.
''''-='===-=--~~'= '''',",,-'''"''-'"''~'~'-'''''.'~'.'-'-~='",,-='''-''-~='~-~=-
330
§ I.
VIII. MAJORIZATION PRlNOPLES AND APPLICATIONS
and, for arbitrary z in the disk
I
z 1 < 1,
lNVARIANT FORM OF THE SCHWARZ LEMM#
a z),
i
[1- I(O)/(C)
(za, Zl,
1([ = 1 and \ aI < 1. result that, in I(I < I,
where
I
1 (C) - 1 (0) ~ l CI.
Zg,
that is,
_I.
(3)
(Zs, zl>
Zg,
zJ
-=
1 z. _--!-1=-_
1+\
and (4
= e icP •
Then, by
ZI 2 12 1
I (8)
ZI~I' 1-2 Z 1 1
Formula (8) gives a convenient expression for the anharmonic ratio in question.
inequality to the form
1- ICIi I r I!(C)-!(O)I_I/(O)j'ICI' ~I",I.
In particular, the assertion made above follows from it.
1-I CI~
(4)
'.-A" •.• , •.
We now define the hyperbolic distance between two points z 1 and disk
From this in turn we get the inequality
I C1+1/(0) I
(5)
I!(c)l~ 1+I/(O)IICI Here, for (J, 0, equality holds only when f«()
= (( +
a)/(1 + a 0, where
1( I =
as the quantity 1
)
- 2 log (zs,
Zl> Zg, Z,
I
=2
log 1
-1
1-21ZI ZI
--::,ZI
1-
O~usly, if z 1 ~2.'
D(Zh
z2)=D(Z2, ZI) and D(Zh
Z2)~0.
Z2
I
(9)
Z12.
Also, D(ZI' z2)=0 only
Furthermore, for arbitrary points z h z 2, and z 3 in the disk
~--
Let z 1 and z 2 denote two points of the disk
and let z 3 and z 4 denote the points of intersection of the circle
I z I=
D (Zl' Zg)
1
+ D (Zg,
of the
I
21-_2 i
1+.
1,
and arg (= arg a if a J, O.
I z \ < 1.
Iz I < 1
D (Zl' Zg) =
We can give Theorem 2 a geometrical interpretation. To do this, we define a
Iz I < 1
331
eitp . reitp+e itp l-r z,)= -el
After some simple manipulations, as in §4 of Chapter IV, we can reduce this
special metric in the disk
= _e icP
points z3 and Z4 are mapped into the points (3 'using (7), we obtain
(2)
We note that, if we set z = 0 in I, we obtain the
I a I < I,
.. ,,, .. ,
through ZI and Z2 into a straight line passing through (= O. It follows that the
If' (z) I ~ 1 ~~~;?il with equality only for f(z) = (z + a)/(1 +
,~"""-,
IZ I < I,
Za)"~ D (ZI> zs)
(9')
z 2, and
with the circle orthogonal to it passing through the points z 1 and z 2' Let us
(the triangle inequality), .with equality holding only when the points
assume that we have numbered the points in such a way that the point z 3 is
Z
closer to z 1 than to 22'
from (9) if we note that, by virtue of the invariance of the hyperbolic distance
Let us form the anharmonic ratio of these four points:
(zs,
2 1 - 2• .
(6)
4
2. -
I1-
mapping of the z-plane; that is, if such a mapping maps the points z 1> z 2, z 3 Z4
into points (1) (2' (3 and (4, we have
(C a,
CI>
l:.J, CJ = (za,
Zb Zg,
ZI
< r < I,
and it maps the circle orthogonal to the circle
Iz I =
1 and passing
at
I 2. I+ I 2 1 I + I 21 II 2. I
~1
Z 2'
Loba~evskir, but point out that the hyperbolic distance is invariant under frac
tional-linear mappings of the disk
Iz I < I
into itself in such a way that the points icP and Z2 are mapped into the points (1=0 and (2=(Z-ZI)/(l-zl z 2)=re ,
where 0
maps the disk
I~
z 1()
metry that gives a Euclidean interpretation to the non-Euclidean geometry of
trary z 1 and Z2 in the disk \ Z I < 1. To see this, note that the function (=
z 1z)
0 and that, by
We shall not stop to construct a plane geometry based on this metric, a geo
Furthermore, the anharmonic ratio (6) is positive and less than unity for arbi
(z - z 1)/( I -
21 21Z.
with equality holding only if arg z 1 = arg (7)
z,).
z2 =
virtue of inequality (5) as applied to the function f«() = «( - Z1)/(1 the point (= z3, we have '
As one can easily verify, this ratio is invariant under an arbitrary fractional-linear and
I z 1 = 1. This property follows
under fractional-linear transformations, we can always have
z. -Z.
ZI> Z2, Z')=2 --Z ·Z.-24'
1
3 lie on a single circle orthogonal to the circle
Z 1>
Iz I < 1
into itself and that the role of straight
lines in this geometry is played by circles orthogonal to
J." 1
Iz I =
1. Furthermore,
for any fixed z l' the distance D of z 1 and 22 approaches + "" as the point Z2
I
- -------_.
VIII. MAJORIZATION PRINCIPLES AND APPLICATIONS
332
approaches the circle disk
.'=,
Iz I < 1
Iz I
::0
§l.
1. This follows from (9). The set of points z in the
whose distance from a given point Zo in that disk does not exceed
a given quantity p, where p> 1, that is, points for which D(z, zo)
S p, will be
-
INVARIANT FORM OF TIlE SCHWARZ LEMMA
Iz I < 1
Inequality (11) shows that, when the disk
333
is mapped by a function
w = fez) that is regular in the disk and satisfies there the inequality If (z)1 . the hyperbolic distance between the images of two points z 1 and
does not exceed the hyperbolic distance between the points z 1 and
p. It follows from (9) that this hyperbolic disk is defined by the inequality
and is equal to that distance only in the case of a mapping of the disk
.
I
I I-zoz
Zo _.~
+e
11
e!P !p
= tan p
--+
We now present some extensions of the Schwarz lemma.
quality
O. This shows that the l~ear differential element do z in
If(z)1 < 1
in the disk
Suppose that this function has the repre
00
I(z)= ~ cgz k , where n ~ 1. k=n.
I dz I 1-lzI9'
(10)
=
Then, !f(z)1 S \ z In in I z I
I(I
where Idz \ is the Euclidean differential element. The hyperbolic length L of the curve z
I zI < 1.
sentation
the hyperbolic metric is given by the formula
daz =
Iz I < 1
Theorem 4. Let fez) denote a function that is regular and satisfies the ine
14z/ D (Z, z+~Z)=1_I~r9(I+E), 0 as ~z
themselves
piecewise-smooth curve l into a curve of no greater hyperbolic length.
follows from (9) that
->
Z2
into itself. From inequality (12) it follows that such a mapping maps an arbitrary
and, consequently is a Euclidean disk but with center distinct from zoo It also
where (
s ~
in the disk
Z2
referred to as the hyperbolic disk with hyperbolic center z 0 and hyperbolic radius
z-
'-"'.~;;;;;;;;;;;;;;;;;;;5
z (t) for a
StSb
=
< 1 with equality holding only when fez)
::0
(Zn,
where
1.
The proof of this theorem is the same as the proof of the Schwarz lemma if
in the disk
I z I < 1 is defined .as the least upper bound of the sums I ~:~ D (z k' z k +1) for all possible systems of points z k = z (tk), where a = to < t 1 < ... < t n = b. In the·
we consider the function ¢(z) = f(z)/zn, ¢(O) = c n •
case of a piecewise-smooth curve, one can easily show that this length is finite
and satisfies there the inequality If(z) < 1 and has the representation fez) '"
and given by.the formula
Co + 'i'k:nCkZk, where n
Theorem 5. 1) Suppose that. in the disk
I z I < 1.
a function fez) is regular
~ 1. Then, in the disk I z! < 1, we have
Ii
L=~
,
I z' (t) I l
~ ••
dt.
II• (z) I ~
, ..
a
Theorem 3(the invariant form of the Schwarz lemma~ 1) If a function fez) is
Iz \ < 1
and if If(z)1
s1
in that disk, then. for arbitrary points
dar,;;;;, da zm
(14)
with equality holding only when
z 1 and z 2 in it,
l(z)=E (11)
D(j(Zl). l(z~»~D(zlt z~).
a - . 161=1. lal<1 + az
Zll+
1
ll
Proof. By virtue of the invariance of. d (7 z under a fractional-linear trans
formation of the disk I z I < 1 into itself, we have
Furthermore, in that disk,
daJ ~ daJe
(12)
(the differential form of the Schwarz lemma). Equality holds in (11) and (12) only
if fez) is a fractional-linear function mapping the disk 1) Pick [19161.
(13)
or
In view of these definitions, we can reword Theorem 2 in the following form:
regular in the disk
n I z Ill- I . . ~ I ~ 19n. (I -1/(z) I)
1 _
I
z 1 < 1 into itself·
If' (z) I
where
1) Goluzin [1945].
l-If(z) I!
If~ (z) I l-Ifdz)r'
(15)
334
00
'Y
Idz)= I(z~-co
k~n
I - col (z)
Since the function ¢(z) disk
§ 1.
VIII. MAJORIZATION PRINCIPLES AND APPLICATIONS
1z I < 1,
=
~
the initial coefficients of the function fez). Thus, in the proof that we have given,
fl (z)/zn does not exceed unity in absolute value in the
nil (z) zn+l
zn
I
r 2n -I II (z) /2 r 2n (1 _ r 2)
& --=::
r+llnl &,n P=If:1 (z)\&,n --=:: l+rllnl--=:: I,where
(6)
Therefore,
IYnl
~quantity
+ r (l-r n
II; (z) I
I
2)
r (r + IYn
+
Consequently, we can replace p in (7) with the
rlYnl),
which is at least as great. If we do this, we
_I IdzHi ~
n ..L
+
r
Iz I < 1 ep' (r) jd d dGf~ 1-lep(r)I S zl= Gtp(r)'
- pSS
r (I - r ) I-p2 nr
(17)
r = I z I and ¢(r) = rn(r + IYnl)/(1 + r \YnP, where in turn Yn . . Equality holds only when
p has the same sign as the expression
n
r
l-r 2n n P -2 rn (1-r')P+ 2
r, < P < rn •
O. Let us show that it is positive also at p
But (18) is posi
r
co+z n z+!n 1z+ +lnz I( z)_ 1+ cozn In 1+lnz
sn
I
Since x + x-I decreases in the interval 0 r<1 ,
n+ nI r
< x < 1,
::>, k+ 1 =-' r
-
B
_
I
If' (z) I ~ -1- I
]
<
II' (z) I ~
follows from what was said above that equality holds in ( 13) only when f(z) =
(20)
I(I
=
n IzIn-I 'M '2n,
1
(21)
with equality holding at a point z = zo, IZo I < 1, only if fez) = (zn - io)/(I- zozn), where 1(1 = 1.
=
1
can show that equality does indeed hold
in (13) for this last function. This completes the proof of the theorem.
12'
in I z I < 1 with equality holding at a point z = zo, Izol < 1, only if fez) = E (z ~ z 0)/(1 - Zo z). If, in addition fez) = Co + Ik=nC kZ k, where n;:: 1, then, in
I z I < 1,
with the quantity r n , which is at least as great, we obtain (13). Furthermore, it
fez)
M
and if If(z)1 < 1 there then
••••
Consequently (18) is positive. This shows that the right-hand member of (17) is an increasing function of p in the interval 0 .:S P':s r n • If we replace p in (17)
where 1(1 = 1, that is, only when Ia I < 1. By direct verification, we
\col 2).
+ cnzn+ ....
Iz I < 1
Theorem 7. If fez) is regular in the disk
we have in the interval 0
(zn + a)/(1 + az n), with
c nAI -
We *ote in conclusion that, by virtue of what was said above, we immediately
k=O, 1, ... , n-I.
R'
=
get a bo~nd for the derivative of a bounded function fez). Specifically, from Theo
I-r'
-
Co
(19)
rems 2 an'd,5, we have the following theorem.
rn • We have
+ ~ r
(n + rI) 2(rn-3 + r nI) , n-l [ n, ,n-1 + r nI) n -2(
!!.... ,in _ 2 I r
=
\.
(18)
Since the product of the roots of the polynomial (18) (in p) is equal to 1, this polynomial has no more than one root in the interval 0
and
P/(I
2n
.:S r n • Let us show that the function on the right, treated as a function of p, increases in the interval 0 .:S p .:S r n • Its derivative with respect to
an,
\cnl/(1 - \col 2).
Theorem 6.1) Under the conditions of Theorem 5, we have in the disk
p=)/.(z)l,
,
IfI (z)1
=
=
r'"
rsn_ps
and, consequently,
tive at p
n
oobtain
1/;(z)l~n:
=
335
Theorem 5 can be strengthened further if we take into account the values of ik Zk•
inequality (2) can be applied to it. This yields
II~ (z) _
Here p
INVARIANT FORM OF ruE SCHWARZ LEMMA
1) Goluzin [1945].
336
§2. THE HYPERBOLIC METRIC PRINCIPLJ!'
I"
VIII. MAJORIZATION PRINCIPLES AND APPLICATIONS
§2.
function w = w(~) maps the disk < 1 conformally onto the domain G (both mappings in the sense of § 1 of Chapter VI). Furthermore, if da z is the differen
The hyperbolic metric principle
Let us turn now to the generalization of the invariant form of tqe Schwarz
lemma to the case of multiply connected domains.
Let B denote a multiply connected domain with more than two boundary points in the z-plane. This domain can be mapped conformally (as indicated in
337
'tial element at the point z € Band da w is the corresponding differential element at the image w € G, then daw .:S da z with the same condition for equality. Proof. Suppose that functions ~ = ~(z) and t
= t (w)
map the domains Band
and it is not single-valued in B. However, we note that every branch of a single
G respectively onto the disks 1'1 < 1 and It I < 1 in the sense of § 1 of Chapter ifYI. Let us denote by z = z (~) the function inverse to ~ = ~(z) and let us define
mapping function is connected with any branch of the same function or any other
t(the composite function t = t([(z
§1
I~I < 1.
of Chapter VI) onto the disk
Here, the mapping function is not unique
mapping function by means of a fractional-linear transformation. This circumstance will be extremely important in the present case.
'.'in I~I < 1; also, '1
Ix (~) < 1
= 1 _I I dC t:I _ I c' (z) II dz I 12 - ,
,
y
,
,
(1)
on
Since the last fraction is invariant under fractional-linear transformations of the function ~(z), the hyperbolic metric does not depend either on the choice of
For any piecewise-smooth curve l: z = z (t), where a:S B, we define the length L of l by the formula L
= \' do = \' I-C' (z) II dz 1 _ j Z lJ I -) C(z) 19-
{
.) a
t.:S b,
I c' (z (tn II z' (t) I dt I -I C(z (t» II
contained in
Hyperbolic metric principle. Let B denote a domain in the z-plane and let G denote a domain in the w-plane. Suppose that each of these two domains has at least three boundary points. Let w = f(z) denote an analytic function defined by any of its elements and continued into B along an arbitrary path. Suppose that this function has the property that all the values that it assumes in B lie in the domain G. Then, an arbitrary piecewise-smooth curve l in the domain B is mapped by w = f(z) into a curve A the hyperbolic length of which-refcuive to the domain G does not exceed the hyperbolic length of the curve l relative to the domain B. These two lengths will be equal only when [(z) = w(~(z». Here, the
1"
we have
~1
:~If we now shift from the disk ;.quality
"
I
i; that is, \ da w :s da z ,
I" < 1
1 ... 1 .
to the domain B, we obtain in B the ine
It' (f (z» II f' (z) II dz I I c' (z) II dz I I -I t (f (z» II ~ I -I C(z) II ,
(3)
where w= f(z). Furthermore, if we integrate (3) with respect
1'to l, we\see that the length of the curve A does not exceed the length of the !'curve l. Equality holds in (3) only when x(~) maps the disk 1'\ < 1 univalently . into itself, so that the function f(z) = t-l(X(~(z))) has the form indicated in the wording, of the principle. This proves the hyperbolic metric principle.
(2)
Under these conditions, we now have the
function ~ = ~(z) maps the domain B conformally onto the disk
§ 1,
I t' if (z (C) II f' (z (Cn II z' (C) I I~I~ IdCI I -I t if (z (Cm 19
I
mapping function or on the choice of branch of it, and it is completely defined by the domain B and the position of the points in B.
Obviously, this function is regular
From Theorem 3 of
or
metric of the disk I~I < 1 carried over into B by this mapping. In other words, its differential element is given by the formula
c
(~») = x(~).
I" < 1.
I X' (C) II dC I I dC I t -I x, (C) [I ~ 1 - [ CII
The hyperbolic metric in the domain B is now defined as the hyperbolic
da z = da
in
< 1 and the
Corollary. Let G denote a domain iT! the z-plane and let B denote a proper
sub domain of G. Then, throughout the entire domain B, we have dca z < dBa z; that is, broadening of the domain decreases the hyperbolic distance between two points. This result is obtained from the hyperbolic metric principle by applying the latter to the function f(z) = z. We can use it to obtain a bound for dBa z by replacing the domain B with simpler domains that we know how to map onto a disk. As an example of the application of the hyperbolic metric principle, let us now prove a theorem.
Theorem. I ) Suppose that a function w
= f(z) = Co + c lZ + ... is regular in
1) For the case q '" 2, see Landau [1904} and Schottky [19041.
.
-"====-~~~''''"''''"'~-_ ~--
338
VITI. MAJORIZATION PRINCIPLES AND APPLICATIONS
Iz I
••• , a q , 2. Then, if CIt. 0, the number R is bounded by a finite quantity that only on aI," " a q , Co, and CI (Landau's theorem). Furthermore, the If(z)! is bounded in the disk I z 1 ~ OR, where 0 < 0 < 1, by a finite depending only on a I , ••• , a q , Co (Schottky's theorem).
I z I < R is given by the differential z 12 ). In particular, at the point z = 0, we have
Proof. The hyperbolic metric in the disk
= R I dz II(R 2
I dz IfR.
- I
al"
•• , a q
On the other hand, let us denote by t
onto the disk
It I < 1
Co
is
ai' ... , a q )
II dw I·
R~I CJ 11'( Cf co,
Here, equality holds only in the case in which f(z) Furthermore, for any Zo in the disk
I z I <0
-"
"w'''''
".
339
Lindelor s principle,!) Suppose that Band G are two finitely-connected
z and Zo in B, we have
go(/(z), l(zO))~gB(z, zo).
(1)
where gB(Z, zo) and gc(w, wo) are Green's functions for the domains Band G.
arrange for this by
From the hyperbolic metric principle, dow ~ du", which, for z = 0, yields
Icp'(cO,ab ... ,aq)II/'(O)I~Rl, i.e.
;'----
domains bounded by Jordan curves lying respectively in the z- and w-planes. Suppose that w = f(z) is a function that is regular in the domain B and that the values that it assumes in that domain lie in the domain G. Then, for any two pmnts
Proof. We may assume that Zo
I cp' (co,
"
If equality holds in (1) for one pair z, Zo € B, it holds for all such pairs.
mapped into t = O. Then, for w = co, we have
dot'w =
o·,;.;:'i....,
Schwarz lemma.
=
in such a way that the point w =
,-;:.."".."''''''''':;.;-
With the aid of Green's function, we can give another generalization of the
¢(w, aI" ", a q ) a function that maps the domain G constituting the entire w-plane exclusive of the points =
- - - •••,:,
§ 3. Lindelors principle
and that it fails to assume in that disk q given finite values a 1>
~
element do z
"''" _ ~¥
§3. LINDELOF'S PRINCIPLE
We shall prove this theorem and at the same time find an exact bound for R.
dol'
',,,, .._,-, ""' ,,,;,,' ,,, ,,-:.;,;,.''<,.:"'' ,,'' =;;.0·;>;' ---,,;.r;;. . ~T:',;:;'.ifu",,,;';,;:,...m,~,;;,,,,,,,,,""",.,,;~.·,~;,;,:;;;;;;.;;,;;,,;o;i"W"';'i"''''';·'''·''';;.w;,,,',,,dO'w
=
mean~
1 ..
-.,
¢-I(z/R, aI,"', a q ).
R, where 0 < e < 1. the hyperbolic
oe
because, in the opposite case, we can
of a fractional-linear transformation of the z-plane.
Consider the two functions
I go U (z), I (zo)) ll.. ... , Dq
t.
+ log \1 (z) -
I (zo) I and gB (z, zo)
I . Both t~ese functions are harmonic in the domain
+ log I z -
Zo
I.
B punctured by the point z = zoo
Theref?re, the ir difference, that is, the function
\
go (J(z), l(zO))-gB(z, zo)+log
I f(Z;=~o(Zo}1
distance between the points f(O) and f(zo) does not exceed the hyperbolic dis tance between the points 0 and Zo in the disk
Iz I < R;
that is, it does not
is also harmonic in B. This means that the difference gc ([(z), f(zo» - gB (z, zo) is a h~rmonic function in the domain B after the zeros of the function f(z) - f(z 0)
exceed the quantity
are deleted. (This difference may approach
OR
L= \ Rdr J RI_rl o
+
~log 1 6 2 1 . -6
between them and the point Co does not exceed L, then this set is closed and bounded and is completely determined by the values of co, a I, •• " a q • Conse
with a property of harmonic functions, we have everywhere in B ga(/(z), I(zo)) - gB(z, zo) ~ u, with equality holding only when
quently, if M denotes the maximum of the ordinary distance from the set E to the =
0, then If(zo):S M = M(co, aI,"',
a q ).
Here, equality holds for the
same function as before. This completes the proof of the theorem. Also, the first part of the theorem can be obtained directly from the Schwarz lemma if we apply that lemma to the function ¢ (f(z), a I,
as z approaches any of these
the limiting values of this difference are nonnegative. Therefore, in accordance
If E is the set of all points of the domain G such that the hyperbolic distance
point w
00
zeros.) On the other hand, as z approaches the boundary of the domain B, all
... ,
go (J (z), I (zo)) - gB (z, zo) -
O.
This proves Lindelof's principle. We recall that a Green's function g (z, z o) of a simply connected domain is equal to the negative of the logarithm of the modulus of a function that
a q ). 1) Linde10f [1908].
§4. HARMONIC MEASURE. SIMPLEST APPLICARONS
Vlli. MAJORIZATION PRINCIPLES AND APPLICATIONS
340
1(' < 1
z 0 is mapped into O. Therefore, in the case of simply connected domains Band G, we can give Lindelof's principle a different formulation. Specifically, if Br and
maps this domain univalently onto the disk
,=
s
in such a wa y that z
=
-1 < jR (f(z)} < 1 in
Iz I < 1
and f(O) = 0, we have the following inequalities in
I z \ < r: I
4
at (/(z»
1 ~ -;
arc tan r,
Gr are the images of the disk 1'1 r under mappings of the disk 1" < 1 onto the domains Band G respectively such that 0 is mapped respectively into z 0
I'" (/(z» I ~ -;;Iog l-r'
and Wo = f(zo), then Li.ndelof's principle asserts that the values assumed in the domain B r by the function w = f(z) all lie in the domain Gr. Furthermore, if one
I/(z)I~-log -1'It - r.
,=
of these values lies on the boundary of Gr , then the function w = f(z) maps B univaIentlyonto G. Let us give some examples of the application of Lindelof's principle. 1. In the case in which Band G are the disks
1z I < 1
and
I w I < 1,
ine
quality (1) coincides with inequality (1) of §1. 2. Suppose that B is the disk I z I < 1 and that G is the halfplane !R(w» o. Suppose also that Zo = 0 and Wo > O. In this case, a function mapping the disk
I'I < 1
univalently onto the domain B so that the origin is preserved is z
and a function mapping the disk
,= 0 is mapped into Wo =f(O)
I"
= "
number Wo at the origin lie in the disk I (w - wo}/(w + wo)l < r. This disk is symmetric about the real axis and its boundary circle intersects the real axis at the points Wo (1 + r)/{1 - r) and Wo (l - r)/(1 + r). This means that the radius of disk
I z I < r,
l+r f (0) 1l-r + r ~ at (/(z» ~f(O) l-r' ~
it in a particular case. Suppose that the function f(z) is regular in a finitely
function f(z) has zeros n l " " , nfJ- in B (these mayor may not be all the zeros of f(z) in B). Finally, suppose that all the limiting values of the quantity If(z)! as z approaches the bo~ndary of the domain B are bounded by a number M. Then, the function P.
If(z) I ~ log ~+ gB (z, nk) ,
I"
-1
< !R(w) < 1.
< 1 into G
w=~ log 1+C
1- C
Iz I < 1
~ gB (z, Pk) k=l
As z a~proaches any of these zeros, this function may approach this
" function
-00.
Furthermore,
has nonpositive limiting values as z approaches the boundary of the
domain B. Consequently, in accordance with the maximum principle, P.
- 1:
9
g(z, n k
p.
k= \
,= 0 is mapped into w =0 is of the form
Therefore, for functions f(z) regular in
£..
k=\
!/(Z)I~III t~;k:
< 1 and that G is the strip
Suppose that Zo = 0 and Wo = O. A function that maps the disk
9
~
is harJonic in the domain B except possibly at the zeros of the function f(z).
inequality takes the form
2r
'Itt
l+r
>+ 1:
g(z,Pk)
k=l
(2)
everywhere in B. In the particular case in which B is the disk \ z I < 1, this
I'" (f(z» I ~f(O) l-rs, 01-r 1 l+r 1 ()I+f ~lf(z)l~ (O)I_r'
univalently in such a way that
2
I/(z)I~Me k=l
r < 1, we obtain the inequalities
1z I
1 +r
connected domain B bounded by Jordan curves except for poles PI"'" Pv each of which is numbered as many times as its multiplicity. Suppose also that the
< r is equal to 2wor/{1 - r 2 ). In particular, in the
3. Suppose that B is the disk
2
We can also apply the idea of the proof of Lindelof's principle to generalize
=Wo (1 + ')/(1 - ,). Consequently,
all the values assumed ili the disk 1z I < r, where r < 1, by a function f(z) that is regular in B, has positive real part in I z I < 1, and is equal to a positive
the disk l(w - wo)/(w + wo)l
~
< 1 univalently onto the domain G so that
has the form w
341
and having the property that
•
I:ITI ;~:~I'
(3)
k= 1
§4. Hannonic measure. The simplest applications Let B denote a domain in the z·plane bounded by Jordan curves. I) Again 1) The definitions given below and the results remain valid for domains with non Jordan boundaries, specifically, for domains obtained from domains of the type indicated by means of univalent mappings that are continuous up to the boundaries of these domains.
342
VIII. MAJORIZATION PRINCIPLES AND APPLICATIONS
§4. HARMONIC MEASURE. SIMPLEST APPLICA1ltONS
343
let us assume for simplicity that B is finitely connected. Consider a finite num
the first part of the assertion. The second part of the assertion now follows from
ber of arcs (closed or open) of the boundary K of the domain B. Let us denote
the formula w(z, a, B) = 1- w(z, (3, B).
by a the set of all points belonging to these arcs and let us consider a function
The significance of the extension principle consists in the fact that it enables
that is harmonic and bounded in B, that assumes the value 1 on a, and that
us to strengthen various inequalities by replacing complicated parts of the bound
assumes the value 0 at all other points of the boundary K. Here, we exclude the
ary
endpoints of the arcs constituting a since the function is not continuous at such points. By §3 of Chapter VI, this function exists and is unique. Let us denote it by w (z, a, B) and let us call it the harmonic measure of the part a of the bound ary K of the domain B about the point z .. Obviously, 0
S w (z,
where in B. If we partition the boundary K into parts a 1> •• " indicated, we have
a, B)
S
1 every
an of the type
r).k'
B)=l,
Another important property of harmonic measure is given by the following theorem.
Theorem 1.1) Suppose that a function fez) is regular and bounded in a domain B with boundary K and suppose that K is partitioned into two parts a 1 and a 2 • Suppose that all the limiting values of If(z)1 as z approaches interior points of
..
~ w(z,
K with simple ones for which the harmonic measure is easily calculated.
( 1)
arcs constituting the a k (for k = 1,2) through values tn B do not exceed Mk • Then, in B,
k=1
log If (z) I ~ ill (Z,
everywhere in B because the function on the left is a harmonic bounded function in B that assumes the value 1 on the boundary K except at a finite number of
f3 )increases
and when B is extended by notifying only a, the measure w(z, a, B) decreases.
B by modifying {3, for example, we mean replacing the domain B with a domain B' containing B and with bound Here when we speak of extending the domain
ary containing the entire set a. Proof. Suppose that
B' was obtained from B by extending the latter set in
such a way that only {3 is modified. Then, the function II
(z)= ill (z,
cx., 8')
-ill
(z,
(1,
is a
harm~nic function at all points of the domain
which is harmonic and bounded in B, is equal to 0 on a
I
(2)
(3)
B except at the zeros of f(z).
the ak from inside B, all the limiting values of the function u (z)
are nonpo~'4ive. On the other hand, u(z) approaches zeros of f(z). Consequently u (z)
S0
-De
as z approaches the
everywhere in B; that is, inequality (2)
holds and the remark made in the theorem about equality is obvious .. This completes the proof of the theorem. From the theorem just proved, we get
Theorem 2. Suppose that a function f(z) is regular and bounded in a domain B and the boundary K of B is partitioned into two parts at and a 2 • Suppose that the limiting values of
I f(z)1
do not exceed Mk as z approaches interior =
1,2) from inside B. Then, for
every closed set E C B, there exist positive constants '\1 and ,\ 2 independent and is nonnegative on
of the function f(z) such that '\1 + '\2
=
1 and
If(z) I ~M:IM~9
(3. Consequently, this function is nonnegative everywhere in B, which proves on E. 1) Carleman [1921].
(z, ~, B) log MII~
As z apJroaches all points of the boundary K except the endpoints of the arcs constituti~g
points of the arcs constituting the ak (for k
B),
ill
U(z)=loglf(z)\-ill(Z, CXt, B)logMl-'--W(Z,~, B)logMIl
me as ure of its image in B l'
extended by changing only the part {3, the harmonic measure w (z, a,
+
Proof. The function
monic measure of an arbitrary part of the boundary K is mapped into the harmonic
The extension principle. I) If the boundary K of the domain B is partitioned into two parts a and {3 of the type indicated above, then, when the domain B is
B) log M 1
Furthermore, if equality holds in (2) at a point z E: B, it holds for all points in B.
points. Under a univalent mapping of the domain B onto a domain B 1, the har
An important property of harmonic measure is included in the following exten sion principle:
(Xt,
I) F. and R. Nevanlinna [1922].
(4)
~~~~-='-='-=-=-=-------------
344
~--,,:._----------
VIII. MAJORlZATION PRINCIPLES AND APPLICATIONS
---
§4. HARMONIC MEASURE. SIMPLEST APPLICAlIfONS
w~~m=~,
Proof. Suppose that MI and M2 are numbered in such a way that MI < M2. where
1>
is the angle subtended by the segment a at the point z.
2. Suppose that B is the half-disk
If(z) I ~ (~:r(z. «t. B) Mi'
(5)
~
~
For the function fez) in the domain B, we have inequality (2) which, by virtue of (1), we can write in the form
345
I z I < 1,
zs(z)
> 0 and that
a is the
,~ boundary semicircle. Then, one can again verify easily that
2
If we set AI=minze:E~(z, al> B) and A2=I-A 1 , then Al and A2 areobvi ously positive and we obtain from (5)
w(z, a, B) =-(1t -If),
(9)
~
where
If(Z)I~(~:t Mll=M~IM~1
1>
is the angle subtended by the diameter from the point z.
Let us now look at some applications of the theory of harmonic measure to the study of bounded functions.
on E, which completes the proof of the theorem.
Theorem 3. Suppose that a function fez) is regular and bounded in the half
Inequality (2) will find more than one application in what follows. Let us give an example. Suppose that the domain B is the annulus r I <
I z!
< r2'
Suppose that a l is the circle
Iz I =
rl and a 2 is the circle I z I = r2'
Then, as is easily seen,
plane 3 (z) > 0, and continuous except on the positive half without the point z = 0 of the real axis. Suppose that fez) -> a as z approaches the origin along this half-axis. Then, f(z) -> a uniformly for arbitrary &> 0 as z ---+ 0 in the angle
O:S arg z _log
rfr.
.
_
log
_I I
rfr
1 w(z,abB)-I--/-' ogr r. w(z,%B)-I--/-' ogr r r - z.
1
l
(6)
1
Proof. We may assume that a = 0 and that If(z)1 < 1 in ~(z)
log r/rl ~ Ml~' M lllog'I/'1
interval
log rlr.
(7)
be obtained directly. Specifically, consider the function z(J.f(z), where a is an arbitrary real constant. The maximum value of Iz(J.f(z)1 in the annulus r}
I z I = r,
If(z ) I ~ (r 11r)~ 1
on the circle
r
of the real axis from 0 to
(J..
It follows that, if rl and r2 are such
of the preceding theorems, we give two examples of the calculation of harmonic measure:
1. Suppose that B is the halfplane ~ (z)
> 0 and that
a is a segment of the
If we apply inequality (2) to the half
L~US
(10)
in B r • find a lower bound for w(r, art Br ). To do this, let us find an explicit expression for this quantity. The function (= rzl(r - z)2 maps the half disk B r into the halfplane 3=«) > 0 in such a way that the point z = 0 is mapped into the point (= 0 and the interval a r is mapped into the positive half of the real axis; We denote this last by a. If z and (are corresponding points, then III
(z, a" B,) = w (C, a, B).
Also, by virtue of (8),
that a is not an integer, then inequality (7) is not sharp. With regard to a sharp bound for M(r), see §4 of Chapter XI. With an eye to subsequent applications
r.
log If (z) I ~ w (z, a" B,) log IS
where r} < r < r2' When we eliminate a, this leads to (7). Obviously,
equality will hold only when fez) = cz -
"
\a
> O. Let us > 0 such that If(z) I :s ( on the
disk Br:l,z\
This inequality is known as Hadamard's "three-circle theorem". It can also
If we choose a so that r fM 1 = r~M 2, then
&•
choose arbitrary ( in (0, 1) and let us choose r
By setting M(r) = max,z,=r1f(z)!, rl < r < r2, we obtain from (2)
M (r)
~ TT -
w (C, a, B)
Thus,
~-argt , 0 ~
< arg C<1t.
I rz w(z, a" B,)= l--arg-( --)1= 1 -I arg [Z ~ ( ~ r-z ~ r
I 1-; r]Q
•
real axis. Then, as one can easily verify, we have (in the case of a halfplane,
It follows that all limiting values of the quantity w (rz, a r , B r) are at least equal to 1 - (17 - & )ITT = &117 as z ---+ 0 in the angle 0 ~ arg z ~ 17 - &. This
the harmonic measure on the boundary is defined as above)
means that there exists an
r'
in the interval (0, r) such that wlz, a r , B r) ~ &1217 = [)
'
~-:;n:'=iirpP9Pl!""""N"""'="'D~="'-'
346
VIII. MAJORIZATION PRINCIPLES AND APPLICATIONS
in the domain
I z I < r' ~
O:S arg z
~ TT -~. Consequently [d. (10)], If(z) ~ EO.
Since 0 does not depend here on r or z
-->
:s arg z :s
0 in the angle 0
§4. HARMONIC MEASURE. SIMPLEST
this means that fez)
E,
-->
that fez) approaches limiting values a and b as z approaches
0 uniformly as
~. This completes the proof of the theorem.
TT -
along the real axis. Then a = band f(z)
> 0 and l is a Jordan curve lying in that halfplane that terminates at
z = O. Suppose also that fez)
as z
-->
a as z
-->
0 in the angle -~::s arg z ~
Proof. We can again assume that a denote a number in the interval 0
If(z)l~ E
-->
0 along l. Then, fez)
-->
a uniformly
for arbitrary ~> O.
I}
TT -
=
<E < L
I f(z)1 < 1
0 and
in ~ (z)
> O. Let (
Let us choose r sufficiently small that
on the part of the curve l lying in the half-disk BT:
I z I < r,
~ (z)
> O.
Let us denote by a r the part of l extending from z = 0 to the first point at which
l intersects the circle domains
B:
and
I z I = r.
The arc aT partitions the half-disk B r into' two
B;'. Suppose that B:
is adjacent to the negative half of the
B: , we obtain
real axis. By applying inequality (2) to
and let us denote by
a;
-->
00
in the half
f(-1/ z) - b, we conclude that
Proof. If we apply Theorem 3 to the function this function approaches zero uniformly as z
-->
0 in the angle 0 ~ arg z
> O. Furthermore, we can show, in a manner analogous to the proof of Theorem 3, that the function f(- I/z) - a approaches zero as z -~ 0 in the angle ~:s arg z
:s TT.
For
&< TT/2, we see from this that the function fez) approaches
both a and b as z. --> fez)
-->
00
a uniformly as z
in the angle~~ arg z ~ TT -->
00
-
~•
Consequently, a
By means of a conformal mapping, we obtain from Theorem 3 the following two analogues of this theorem.
B:
comple
rays of that angle. Then, a =b and f(z)
-->
a uniformly as z
-->
00
oe
along both
in that angle.
T~eorem 5/1. Suppose that a function fez) is regular and bounded in a simply
real axis. Then, from the extension principle, we have
except
But, as follows from the proof of Theorem 3, the quantity Ill(Z, small r' in (0, r).
o
Thus If(z)!:S ( 1 in the part of
B:
a;:,
:s arg z :s TT - ~
with sufficiently
B;
lying in the sector
r" in (0, r). Suppose now that 0
/I ,
~ ~ acg z
< &< TT/2. If 0
:s
1T
I z I < rl and fi ~ arg z ~ TT -& . This proves the o as z --> 0 in the angle (t ~ arg z :s &. This TT -
<(
2
in
with sufficiently small
is the greater of the numbers
01,8 2 and if rl is the smaller of the two numbers r' , r /I, then If(z) 1
band
:s (0
for
Milloux. 2 ) It is of a somewhat different nature from the preceding ones. Theorem 6. Suppose that a function fez) is regular in a simply connected domain B that is contained in the disk of an arc a of the circle
I z I < 1 and containing
If(z)1
Iz I =
z = O. Suppose that
exceed
where 0
uniform convergence of fez) to completes the proof of the
fJ through values in B. Then, in B,
and is continuous except at finite points of the real axis. Suppose
Iz I < 1
and has boundary K consist.ing
1 and of a continuum fJ belonging to the disk
limiting values of
E,
1f(z) I :s I < E < 1, as
in B and that none of the
z approaches
1< E - (1-lzl\ " Z ". I/(z) I~E -arc51R-1+1 2
Theorem 5. Suppose that a function fez) is regular and bounded in the half
lS (z) > 0
=
a.
Theorems 3-5 are due to Lindelof 1) and they apply to bounded functions.
theorem. plane
=
We shall now give another theorem on bounded functions, this one due to
lying in this sector. Treating B:~ in
Iz I < r
B
a point z = :: on the boundary K of the domain B, and that it has finite
f(z) is c'dntinuous on B if we define f(c)
Br ) has a
the same way, we can show that there exists a 02> 0 such that If(z)! that part of
ft
limits ~ and bas' z a!!-proaches c along K from the two sides. Then, a
B;)'~(J)(z, a;" Br)~(J)(z, a;, B T )·
0
band
completes the proof of the theorem.
angle. Suppose that it has de finite limits a and b as z approaches
the portion of the boundary of the domain
I z I < r',
=
through values in the halfplane ~ (z) ~ O. This
connec~ed domain B bounded by a Jordan curve, that it is continuous on
positive lower bound olin the sector
~,
TT -
where ~
B r complementary to fJr' Finally, let us denote by 0;' the interval (0, r) of the Ol(Z, aT'
:s
Theorem 5'. Suppose that a function fez) is regular and bounded in the angle
Let us denote by fJr the portion of the boundary of the domain ,
a uniformly as z
and +00
~1 < arg z <&2 and that it is continuous except at finite boundary points of that
log If (z) 'I ~ (J) (z, aT' B;) log E. Here again, it remains only to find a lower bound for III (z, aT> B:). mentary to a r
-->
-00
347
plane ~ (z) ~ O.
Theorem 4. Suppose that a function fez) is regular and bounded in the half plane g(z)
APPLI~ATlONS
1) Lindelof [1915]. 2) Millou" [1924].
. l-Iz"
points on
1
( 1 1)
~===~~:o,,~
348
VIII. MAJORIZATION PRINCIPLES AND APPLICATIONS
For any Zo in the disk
Iz I < 1,
there exists a domain B containing Zo and in it
'0
mapped into a point ~o,
I I < 1.
half-disk B
axis. Specifically, let us draw through
into itself, we can arrange to ha ve the point
'0
'0
lie on the imaginary
a circle orthogonal to the circle
1"
= 1
and to the real axis. Denote by a the point of intersection of this circle with the
B, we
real axis. When we map the upper half of the '~plane into itself in such a way
obtain
'log If (z) 1 ~ Ol (z, ~, B) log e.
(12)
Here, the question consists in finding a bound for the harmonic measure w(z, {3, B). We shall first need to prove an inequality for univalent functions. Suppose that a function g(O is regular and univalent in the disk I~I
<1
and that it does
~..
.
,
~.
.. ,...
~+
... ''I
will, for arbitrary ~o such that 1~o I < 1, be regular and univalent in < 1 and will not assume in that disk the value -g(~o)/g'(~) (1-1~012). By Koebe's theorem it then follows that Ig(~o)/g '(~o) (1-1~012)1 ~
X;
that is,
I Ig (Co) ~ 1-I CO II' g' (Co)
= 1. If we
set ~o
= reie
I
(13) =
[(1 +
I" < 1
'0
the inequality
g(C)!! ~ log (1+IC')8 1_ I CI '
= g(~)
I~ I
+ VI z I .
,Ol
(zo,
~,.B) =00 (Co, W,
But
(zo
if
= : arc tan ICO I,
B') = : arc tan to 4
J"TZol
then \
JIT"Zol + VTZoT
11
(I5)
1-~=ljI,
1
tan~ =
1--
, Ar, B) ~ -1t arc tao 1+ ,n-::-TI r I Zo , . arc tan
+ l"'TZol
sin
2~ =~ tan .~
_1-'
1+tanl.<jJ- 1 +1
Zo Zo
I I
,I. _ 1 . r -I Zo I 'I' -2arcsln 1+1 Zo I'
w (zo,
(14)
1
~, B) ~ ;
arc sin :
+: ~:: :
(16)
where z 0 is an arbitrary point of the domain B. Inequalities (12) and (16) also
This is the desired inequality. Let us now see about finding a bound for w (z,
uoivalently
Therefore (15) can be written in the form
:> 1- viZl ;;0--
<1
and, consequently,
Iz I < 1
we have in
1'1
which, by (14), yields
with equality holding under the same condition. Finally, if g(') is such that 1, then for z
is mapped
onto a domain that does not contain the points 0 and 00. Inequality (14) is there fore applicable to it. If we denote by {3 I the boundary semicircle in B', we now
have by using (9)
\
log !g (0)
=
'0
into a point on the imaginary axis. Thus we may assume that = ito for to > O. We denote by z = g (,) the inverse function providing the mapping mentioned. We
I
710/0 - 710]2
in (13) and integrate with respect to r from 0
to r, we obtain, for arbitrary ~ in the disk
above is mapped into the imaginary axis. In particular, the point
Ol
4
Equality holds in (13) in the case of the function g(~) where \ 711
tively, the half-disk B ' is mapped into itself and the orthogonal circle mentioned
extended beyond the real axis, and it maps -the complete disk
g(C+Co) 1+(oC -g(C o) h (Q= .
that the points -1, a, and + 1 are mapped into the points -1, 0, and + 1 respec
have Zo = g(ito)· The function g(~) can, according to the symmetry principle, be
not assume the value zero in that disk. Then, the function
!g(O)!
349
By means of a supplementary mapping of the
z = zoo
I
~~~~~-_.==--:-=~~~=-
SI~PLEST APPLICATIONS
§4. HARMONIC MEASURE.
the function fez) is such that equality holds in the first of inequalities (11) at
Proof. I) When we apply inequality (2) to the function fez) for z €
===
---,~=.
f3,
B). We map the domain B
B ': I~\ < 1, 15 (~) > 0 in such a way that the arc a is mapped into the boundary diameter of the half-disk. Suppose that the point z 0 € B is
onto the half-disk
lead to the first of inequalities (11). Equality holds in it only when
B is a disk
with cut along a suitable radius and the function fez) is such that, as z assumes values on the half-disk B I, the function log' f(z) I is a harmonic function that is equal to 2 (1 - ¢/17) log ( on the boundary, where ¢ is the same as in formula (9),
1) The proof given here with the sharp inequality was given by E. Schmidt [1932].
From the inequality
§5. ON THE NUMBER OF ASYMPTOTIC VAL"ttES
VIII. MAJORIZATION PRINCIPLES AND APPLICATIONS
350
351
Consequently, for fixed z and sufficiently large r,
arc
. I-izi 1-lzl 1-lzl I + I z I?: 1 + I z I ~ 2
Slll
(J)
we get the second of inequalities (11). This completes the proof of the theorem. Now, we shall finally give an application of the theory of harmonic measure to a single question assoch ted with the principle of the maximum modulus. We
(z,
(x,
B) r--.J ~ (-y-
, Zo
=
which shows that, for suitable A
in
~ (z) > 0
I,;
or (2) there exist pos itive numbers A and R such that
for r>
I < r, 15 (z) > 0
and let us denote
by a its boundary semicircle. From inequality (2), we have
log If (z) I ~ (J) (z,
(x,
B) log M (r).
(18)
B) = ; -
['It - (; (; -
arc tan r Y
arc tan r
r
X'
M (r)
I in the entire angle or (2)
>e
Arl a /
(20)
where! M (r) is the maximum value of If(z)1 on an arc of the circle
I
z
1 =
r lying
R.
I
§5. On the number of asymptotic values of entire functions of finite order
Let us give an application of the theory of harmonic measure to a more com plex question, one dealing with entire functions, Suppose that fez) is an entire function and that l is a continuous curve in the z-plane that extends from some finite point to 00,
00.
If, as we let z move along
the function f(z) approaches a definite limit a, then a is called
the asymptotic value of the entire function fez) defined by the curve l. Suppose
1.
x )] = : (arc tan r Y x
:S
there exist positive numbers A and R such that
l towards
But by (9), if z = x + iy, (x,
r
with vertex at z = O. Suppose that the limiting values of 1f(z)1 do not exceed
\
R.
Proof. Let us denote by B the half-disk \ z
(z,
log M (r)
1m
> 0 and sufficiently large r, inequality (17)
in th1 angle in guestion, where r>
(17)
where M(r) denotes the maximum value of If(z)\ on the semicircle Iz\ = r,
(J)
TTU
and that all the limiting values of its modulus are bounded above by I
M(r»e Ar
15 (z) > 0
l'
r ..... 00
1 at finite boundary points. Then either (1) If(z)I
as z approaches any finite point on the real axis. Then, either (I) 1f(z)\ :::: I in
(19)
Theorem 7'. Suppose that a function fez) is regular in an angle of magnitude
> O. Now, we have
15 (z) > 0
4y
1tr'
theorem that will be used in the following section.
> I in the halfplane ~(z) > O. The reason for this is the fact that the limit
Theorem 7. 1) Suppose that a function fez) is regular in the halfplane
r--.J
By performing a conformal mapping, we obtain from Theorem 7 a generalizing
is regular in the upper half
plane and its modulus is equal to I at finite points of the real axis. Furthermore,
~ (z)
2
holds. This completes the proof of the theorem.
violated even at a single point of the boundary. Then, the modulus principle no
= oe
If (zo) I ~ 4Yo
different when the requirement of boundedness of the limiting values of If(z)1 is
ing values of the function are not all finite as z approaches the point z
2
Xo + iyo denote a point in B such that If(zo)1 > 1. Then, from (18) and (19), log
the domain B, then If(z)1 :::: M throughout the domain B. The situation is quite
If(z)1
+ _Y_) = ~1t ~ r+x r _x
we have
values of its modulus exceed M as z approaches any of the boundary points of
iz
r-x
Suppose now that the inequality If(z)1 :::: I does not hold everywhere in B. Let
recall that, if a function fez) is regular in a domain B and if none of the limiting
longer holds. For example, the function fez) = e-
1t
+ arc tan r +- x ) ,
that two curves II and lz of the type described define finite asymptotic values al and az. If al
t- az,
these values are considered distinct; if al = az, then let
us agree to consider the curves II and lz as defining distinct asymptotic values only in the case when there is some R > 0 such that the curves II and lz can not be connected by a continuous curve lying in th~ domain - - 1 ) Phragmen and Linde lof [19081.
Iz \ > R
and having
-------_.. _--"'-_. __ ._ ..._..'-
=..;;;,;;;~
__ __._- ._--_.
VIII. MAJORIZATION PRINCIPLES AND APPLICATIONS
352
-
,,--------
.._ _._.
--~--------~~~
__ _ _-_ -----------_._..
.. _._._ ...- .... _... _--_._-"'.,,--_ ....-.__ ...
... _-- _.'-'-
§5. ON THE NUMBER OF ASYMPTOTIC VALUES
the property that the oscillation of the function fez) on it is arbitrarily small. In
finite asymptotic values.
the theory of entire functions, this convention identifies the asymptotic values of the entire function fez) with nonalgebraic (transcendental) singularities of the inverse function.
If an entire function fez) is such that, for some finite k > 0 and sufficiently
353
Proof. We note that if two curves II and 12 that extend to infinity define two distinct asymptotic values of the function fez), then, outside some disk
Iz I s
R they have no points in common. Therefore, we can in general assume
them to have no points in common. Without changing the asymptotic values, we can
large r, we have M(r) <~rk, where M(r) = max1z1=r If(z)1 then fez) is called an
replace these curves with broken lines that extend to infinity and that consist of
entire function of finite order. The greatest lower bound p of the numbers k that possess this property is called the order of the entire function fez). It was
To see this, let us partition; for example, the curve II: z
assumed that the number of asymptotic values of an entire function of finite order is finite and is bounded by a quantity depending only on the order p. The com plete answer to the question was given by AhHors. 1) We shall give his solution, 2
using the method of strips, which we have already used in §6 of Chapter IV. ) Lemma. Suppose that there are a finite number n of nonoverlapping radial
strips Sk, for k = 1, 2,'" ,n, in the annulus ro < I z I < r that extend from the circle I z 1= ro to the circle I z I = r. Let Uk (Tk) denote the arc of the circle \ z I = ro (I z I = r) that serves as part of the boundary of the strip Sk' Suppose If we map these strips that none of these arcs degenerate to a point. conformally onto nonoverlapping domains I- k : ro < I'I < r', cPk < arg '< cPk+1 that completely fill the annulus ro < I" < r' and if this mapping maps the arcs Uk and Tk into the arcs cPk < arg' < cPk+1 on the circle 1'I = ro and on the circle
1'1
=
r', then r' ~ r.
Proof. The assertion follows from Lemma 1 of §6 of Chapter IV. Specifically the function t ~ (t) Slog
=
log' maps the domains I, k onto the rectangles T k: log ro
r', cPk < U(t) < cPk +1' k =
1," . , n, with sides of lengths ak
<
=
a finite or countable number of segments whose endpoints cluster only at infinity. where lim 4tl z I (t)
=
00,
=
z I (t) for to S t < tl>
with the points t', til"", t(n\ . .. and let us inscribe
in the part of I lying between the points z(t(n») and z(t(ri+1)) a broken line An such that the values of the function fez) on each segment of this broken line will differ from the values on the arc beside it by an amount less than f n
> 0 and
fn
-->0 as n -->
00.
Then, on the broken line
I;
f
n' where each
consisting of AI'
A2' ';', the function fez) has the same asymptotic value as on II' We can even assume that the broken line
I;
has no multiple points because, as we move along
it, we can bypass any closed polygons constituting part of it. Suppose that we have n curves II>"', In defining a distinct finite asym ptotic values of the function fez). On the basis of what has been said, we can assu1e that these issue from z
=
0,
that they have no other points in common and
t.hat ~Ch is a broken line without multiple points. These curves partition the z plan into domains B k' for k = 1, ... , n, indexed in such a way that each B k is boun ed by lk and Ik+1 (here, In+1 tion
f
=
II)' In each of the domains B k , the func
z) must be unbounded; otherwise, in accordance with Theorem 5" of §4
(applied to the function f(z-l + c)) its value would converge uniformly to a limit in one of the domains B k and then the curves bounding this domain
cPk+1 - cPk' bk"" 10g(r'lro). But the strips Sk are also mapped onto these rectangles
as z
and in the way required by the lemma of Chapter IV referred to. Therefore, in
would not define distinct asymptotic values. Therefore, the set of points in each
accordance with that lemma,
of the domains B k, where n
~ 'fk+1 -
~
k=1
that is,
r'
'fk 27t. r' ~--r-' loglag
ro
r o
~ r. This completes the proof of the lemma.
Theorem. An entire function fez)
i
const of order p has no more than 2p
---> ""
If(z)I>N=
sup
If(z)1
11' •••• In
is nonempty. This sets consists of domains that extend to infinity. This is true because, if one of them were bounded, inside it we would have If(z)1
> N, which
the maximum principle asserts is impossible. For every k, we shall consider not
B k but one of the domains lying in Bk just mentioned and we shall now denote I z I = ro contains
it by B k. Let ro denote a positive number such that the circle I)
AhHors [19301
2) Goluzin [1946dl
points belonging to all the domains B k, for k = 1,' .. , n. The maximum value of
354
VIII. MAJORIZATION PRINOPLES AND APPLICATIONS
If(z)! in the part of B k contained in the circle \ z
I=
'0
I z \ S'o
§5. ON THE NUMBER OF ASYMPTOTIC VALUES
is attained at some point Zk on
Let us find an upper bound for the harmonic measure w «( k>
which does not lie on the boundary of B k' Let us now look at
I z I > '0'
the part of the domain B k lying in the domain
It consists of domains.
We denote by B k that domain containing the point z k on its boundary. This B k must extend
to
00
because otherwise the maximum principle would again lead us
to a contradiction. If uk-is the longest arc of the circle
1
z I = '0 that contains a
point z k and constitutes part of the boundary of the domain B ing the domain B k with the domain B~ consisting of B
k'
k'
then, by replac·
uk> and that domain
respect to the sector obtained from lk by adjoining suitable points of the disk
,,= (
1('
< '0' It follows from the mapping of the sector by the function l/&k , where &k = (¢k+l - ¢k)ITT, that this last measure is equal in the half-disk 1(' l.s r* = " li/&k,g «(') > 0 to the harmonic measure of the boundary semicircle I" 1= r*, 0 < arg (' < 77 relative to some point = ~k + iTJk on ,(' I = ,J/&k; that is,
a
in accordance with (9) of §4, it is equal to
about the circle I z I ='0 which is adjacent to uk' Since z k is an interior point of the arc uk> then, if we denote by N k the maximum value of If(z)1 on the boWld· for all k
=
BL
we have N k
< I fez k)l.
I '1jk -2 (arc tan 1t r* - Ek
(0
(C k ,
rJ.k'
Ok)
~ 3.. (arc tan 1t
'0
<1
<,
z \
for arbitrary,
+ arc tan
> '0' We denote
_'lJ_k_
r* -
Ell
+ arc tan~) r* + Ek
'-"" 1t
r* -
Ek
r*+Ek
and that is adjacent to the arc Uk. We denote one of its boundary arcs on the
1z I = ,
the type mentioned in the lemma. As in the lemma, we map these strips onto strips
=
(
univalently onto some domain Gk contained in the sector
1(1
<,',
< arg
¢k
«
I
Nk
to a k
us de\lote by e B the smallest of the ratios for sufficiently large "
/11 (r)
on the boundary of Gk- On ak,
1!(Zk (q) I ~ M (r) = max \1 (z)\' and on
I
I/(zk (q) ~
= log /I(zk (CiJ) \ ~ Ul
(Ck,
ak' Ok) log M (r)
=
Ul
+
Ul
(C k ,
(Ck'
?k' Ok) log
rJ.k' Ok)
log
k
and
+ log N
Let us now assume that' is sufficiently great that M(,)/N k
o
1t
> O.
kB,I/&k,
Ifez k) II Nk
k
for k
for k
=
I, .. " n and let
= I, ... ,n.
Then,
lim c r, k=~, A> 0, B> 0.
r-C/O
4r o
k
Consequently, there exists at least one k such that
< 21n,
M (r)
> Ae
cr Brn/2,
lim Cr ~_C/O
> 0,
and this means that fez) is of order no less than n/2. Consequently, n/2
S P;
that is, n S 2p. This proves that fez) cannot have more than 2p finite asymp
totic values. This completes the proof of the theorem.
Nk
M(r) N
«(»,
-i r I/&k r -I/&kl og MNk(r) •
that is, such that 1/ I} k > n12. If we choose such a k in the last ine
quality, we obtain, for sufficiently large "
{} k
N k· If we apply inequality (2) of §4 to the domain Gk , to the function f(zk to the point (k = (k (Zk), we obtain log I/(zk) I
> Aecr.
. But l~=l {l-k = 277, {l-k
Izl=r
fh,
r'-.'
tIet us denote by A the smallest of the numbers N
¢k+l' Let ak denote the image of the arc Tk under this mapping and let f3k
denote the complement
4 I/&k I/&k
ro 10 M (r) '-"" 1tr2/&k_r2/&k g N k 0
I ) I~- r log ~
'0
I, .. " n, extendable to the entire domain Bic and maps that domain
1t
,I/&k I/&k 4 I/&k I/&k r ro :::>: r ro ~1t ,2/&k 2/&k=-~ 2/&k 2'· r -ro r - r o
and (1) yields
< \ (I <,'. We denote by (= (k (z) the functions car rying out this mapping and we denote their inverses by z = z k «(). Then, we have r' 2::', But, in accordance with the symmetry principle, the function (= (k (z) is, for each k
-
r' k'1jk r,2/&k _ E:
4
by Tk' We treat the domains S k' for k = I, 2, •.. , n, as strips of
lk filling the annulus
1/&
~3..(_'1j_k_+_'1j_k_)_ 4
by S k the connected subset of the domain B k that is contained in that annulus circle
'1jk ) . *+E r. k
Thus we obtain
These operations are carried out
I, ... , n.
Let us now consider the annulus
Gk)' By the
extension principle, it is no greater than the harmonic measure of the arc ak with
that is included in the intersection of the domain B k with the inversion of B k
ary of the domain
ak>
355
>1
k •
for all k.
(1)
The bound for the number of finite asymptotic values of entire functions given in the theorem is a sharp bound when p is an integer. This is shown by the ex
amples of the entire functions
_
i!£l£
:z:::;::;::;;::
356
'~':WZ
_ ...._
._
J.
_~
'"
~
__9
----
==_27'
..
_~_-
,
.-m:~==~m=','''f'J;
...
VIII. MAJORIZATION PRINCIPLES AND APPLICATIONS
d}
for k
sin zP dz -zP
=
1, 2,""
f
(z)
= PAnt (Z) + ~ PAn + l' h=1
An
P An
The theory of harmonic measure also finds significant applications in the hyperconvergence of a power series. 00
f(z)= ~ cnz,.
( 1)
,.=0 with radius of convergence R > O. If, for a suitable grouping of its terms without
+ ... + CntZAnt, + + ... +
h
+I'An
k+1
(Z )_ -Cn+1 k
Z
k
1
A
Cnk+IZ
series. For example, consider the series 00
fez) = ~ [z (1
= a(t::") = t;;>(1 + t::")/2, where p is a positive integer. Since a(l) = 1 and la(t::") I < 1 for I t::"1 s 1 but t::" t 1, the function F(t::") = f(a(O) is regular on I t::"1 S 1. From (3) we now obtain the transformed series 00
F(q =f(a (C))=P Ant (a (C»
+ z)]A
+ ~QpAn+l. k=1
n,
(2)
< Al < A2 < . ..
and An +1
> 2A n
for n
= 1,
2, .• '. If, in each term of this series, we expand the parentheses and write the
F (t::") \about
t::" =
p> 1, \t also converges on the set of points
I z \ > R,
this means that
the power series in the present case possesses the propeny of hyperconvergence.
(4)
the lowest and high
O. Si,nce this last series converges in some disk
neighborhood of the point z
Since E contains points in the domain
(CL
1
term in the form of a polynomial arranged in order of increasing powers, we obtain
E of points satisfying the inequality
Q are
nk+1
est po/Wers of t::" that appear in the corresponding polynomial. If we take p > l/(q 1), so that (p + l)/p < q, we have (p + I)A nk < PA nk +l' In this case, the serie~ (4) is a grouped series with respect to the power series representation of
a power series of the form (1) with radius of convergence given by the equation
R (1 + R) = 1, so that R = (y5- 1)/2. For this power series, the series (2) is a
(p+1»)'
.k
Here, the two quantities constituting the subscript to
n=O
I z (1 + z)1 < 1.
, k - l , 2, ''',
z
this property is called the hyperconvergence of the power
grouped series and it converges on the set
nh+l_
1 at which fez) is regular. 1)
changing their order, this series converges on some set of points z outside the
where the An are integers such that AO
(3)
Proof. Suppose that the function fez) is regular at the point z = 1 (if it is regular at a point z = e i '\ we could consider the function f(e-iaz». Let us set
Suppose that we have a power series
s R,
(z),
k+1
converges in a sufficiently small neighborhood of an arbitrary point on the circle
Iz I =
theory of power series. Let us now look at the question of what is known as
1z I
An
k
PAnt (z) = CoZAo
2p, which are defined by the same rays.
§6. The hyperconvergence of power series
closed disk
357
00
the rays arg z = kTT/p, for k = 1, 2, •. " 2p; the second has 2p asymptotic values =
_
1, 2," " where q> 1 and is independent of k, then the grouped series
of order p. The first of these has 2p distinct asymptotic values 0 defined by
e21Tkil P, for k
"__ ··_·
§6. THE HYFERCONVERGENCE OF POWER SERIES
z
sin zP zP an
"
.•.
~:;:;;;,:,'''''''.<:....,'''''''''''''~·~'''''·._._"D:."'~·=O ,~.~~~,·~.,,~r,
=
t::"
I t::"1 < p,
where
into which a sufficiently small
1 is mapped. But then, if we shift back to the vari
able z in (4), we conclude that the series (3) converges throughout a sufficiently small neighborhood of the point z
=
L This completes the proof of the theorem.
In a sense, the following theorem is a converse to Theorem 1. Theorem 2. Suppose that a power series fez)
=
I':=ocnz n has radius of con
We have a number of theorems due to Ostrowski 1) on hyperconvergence. Theorem 1. Suppose that a power series fez)
=
I':=ocnz
An
, where 0 S AO <
A1 < A20 ••• , has radius of convergence 1 and that it is a lacunary series, that is, there exists a sequence of positive integers nk, for k = 1,2,"', such that AMI> qA nh, k
1) Ostrowski [1922].
.
1) In particular, if the inequality An +l > qA n , where q> 1, is satisfied for all n = 1, 2,"', then the power series itself can be regarded as grouped. In this case, PAJJ'l.n+l(Z)=
c n / n + 1. In accordance with Theorem 1, it will therefore converge in a sufficiently small neighborhood of every re gular point on z = 1. But this last is impossible because a power series cannot converge outside its circle of convergence. This means that, in the present case, the function f(z) cannot be continued beyond the circle of convergence. This is a familiar theorem of Hadamard.
I I
... ."~
... "d.!,_
358
"'<"~'''''''_.'_'=''''''''_''
....
such that
Izol : :
.C
1.
m
=
on which the grouped series converges uniformly. This means that there exists a sequence of partial sums snk (z) = I~~o cnz n, k =
ICml~ In particular, for q nk I
1, 2, ... , of the original power series that converges uniformly on A. We may > 2nk for k = 1, 2,' ... Consider the function
\c
uk(z)=~log\Snk(z)l-loglzl (k=l, 2, ...), nk
< - q on y, where q is a positive number independent of z and k. Now, let ( denote a positive number. Define () = e- d2 , so that 0 < 1. Since
(z)
the maximum value of ISnk (z)lzn k \ in Iz I:::: 8 is attained for Iz 1= 0, if we set mS.k=maxlzi=slsnk(z)\, then, in lz\::::8 and, in particular, in Izl::::l, we have 1 1 e . Uk (z) ~ -log ma k -log 0 = -log ma k -2
where R
> 1,
series converges in the disk
Uk
(z) .'S ( in
Izl > ()
and y and to the closed set E representing the circle Iz
f
suffic i ently small that - Al q + A2( = - YJ
that, for k> K, we have on the circle
Iz I = R
P < 1, we obtain from
k=l P ql nk+ l , nk(z),
I z I < l/p,
IIp > 1.
where
the inequality
treated as a power
The remaining part of the
)
(Xl
j(z)= ~ cnzAn n=O
nk,
'ff;r k = 1, 2,··· such that
I=
R,
< 0, we conclude Uk
Ank+I/Ank
-->
<X
as k
-->
00.
Then the grouped series
(z) .'S -Tf, or,
j (z)
= PAl (z) + ~ P An + l' An k=l
k
(z)
I Snk (z) I ~ (Re-TJt k •
(7)
k+l
converges at all finite points of the z-ptane at which fez) is regular. The func tion fez), treated in the sense of Weierstrass, is single-valued in the z-plane. Proof. Suppose that the function fez) can be continued along some curve C to the point z 0, beginning with an initial element defined by the given powers series. Then, there exists a simply connected though not necessarily one-sheeted 1) domain B lying over the z-plane and containing the points 0 and z 0 such that
what amounts to the same thing,
(6)
(Xl
lying inside this domain. (By means of a conformal mapping, ine
If we choose
=
Theorem 3. Suppose that a power series
I
we see that, for k > K = max (Kl> K 2 ), we have Uk (z).'S - Alq + A2( on I z I = R, where Al and A2 are positive numbers depending only on R (note that K depends
d.
Re -7] IR ql
has \a radius of convergence equal to unity ;nd that there exist positive numbers
'
quality (2) of §4 can obviously be carried over to the present case.) As a result,
on
(5)
power series is obviously a lacunary series. This completes the proof of the theorem.
+
lim ma k=O
Iz I = 1
TJ k )n • mI~ (ReRq'
Inequality (6) shows that the series I
for k > K 2 • Inequality (2) of §4 can be applied to the domain bounded by the circles
this yields
ICml~pnk
Obviously, there exists a positive number K I such that, for k> K I, the function
it follows that there exists a positive number K 2 such that
I
(5)
which is harmonic in the z-plane except at the zeros of the polynomial snk(z).
nk'
m~nk'
1I 2 < q < 1,
If we choose q' sufficiently close to 1 that
.
nk'
(Re-TJ)n k Rm ,
< m.'S nk, where
assume that nk+1
k-+oo
m~nk'
bounds for the coefficients:
I> 1
Since
C snk (z) j ~dz,
359
I zl=R
Proof. It follows from the conditions of the theorem that there exists a neigh
Uk
l.-c 211:1
Then, fez) can be represented as the sum of a lacunary series and a series with radius of convergence exceeding l.
borhood y contained in ~z
__ '._" __
§6. THE HYPERCONVERGENCE OF POWER SERIES
vergence 1 and that corresponding to it is a grouped series that converges uni
Zo
_-=.'~'_"'=-=''''~
..
VIII. MAJ ORIZAnON PRINCIPLES AND APPLICA nONS
formly in a sufficiently small neighborhood of some point
""-'"'
fez) is regular on the closure of that domain.
By using this inequality, we obtain from the formulas 1) With regard to'Riemann surfaces and mappings of them onto a disk, see §2 of Chapter XI of the present book.
§7. NONANALYTIC GENERALIZATION OF SCHW.
VIII. MAJORIZATION PRINCIPLES AND APPLICATIONS
360
The domain B can be put in one-to-one correspondence with the disk 1(I
by means of an analytic function. The inverse of this function z = ifJ «( ) is regular
1(' < 1.
Suppose that Zo
= ifJ«(o),
(where A denotes the Laplacian operator) or, if we set u(z) = log A(z), the ine quality
where 1'0 I = p
< 1. The function ifJ«() can be uniformly approximated on an arbitrary closed subset of the disk 1(1 < I by in
means of polynomials. Therefore, there exists a polynomial a«() with a(O) = 0 of some degree p such that_the values assumed by it in the closed disk
I(I s Vp
/)"u~4e2U.
We note that in the case of the disk 1 z I Idz 1/(1-1 z
2
1 ),
if A(z) = 1/(1-1 z
F(q=!(a(q)=QpAI(C)+ ~ QAn+1' PAn k=l
k
(C)
(8)
k+l
(3)
1" s I,
neighborhood of a given point Zo € B. Then, we shall speak of a metric defined in thai: neighborhood. It is a remarkable fact that an analytic mapping maps a
is obviously, after we discard a finite number of initial terms, a grouped series
I" s Vp
1 ),
< I and the hyperbolic metric do = we have in I z I < I
Sometimes, a metric will be defined not in the entire domain B but only in some
00
uniformly on the disk
2
(2)
/)" Jog A=4A'1.
will be continuously distributed on the domain B. The transformed series (7)
for some power series. Since F(z) is regular on
361
the series (8) converges
and in particular it converges at the point (0'
Returning now to the variable z, we shift from the series (8) back to the series (7), which consequently converges at the point zoo Since the value of the function
f«() is defined at the point Zo as the sum of the same power series independently of the method of analytic continuation, it follows that fez) has a one-sheeted
metric satisfying condition (1) (or (3), respectively) into a metric satisfying con dition (1) (or (3». To see this, note that if a function z = z (0 is regular in a neighborhood of a point (= (0 and a metric ds = A(z) Idzl satisfying condition (I) (or (3» is defined in a neighborhood of the point Zo = z «(0) € B, then the trans formed metric ds (or
(3»
=
A(z«())\ z' «()lld(\ = Al «()I d(1 also satisfies condition (I)
because, as one can easily verify,
A Jog A1 = I z' (C) 1'1 /)"zlog A.
domain of definition. This completes the proof of the theorem.
If two metrics ds = A(zlldzl and ds* = A* (z)! dzl are defined in some neighbor
§7. A nonanalytic generalization of the Schwarz lemma.
hood(of the point zo, if A"(zo)
We pause now for a nonanalytic generalization of the Schwarz lemma with its
to
t~e first and we shall say that the metric ds = A(z)! dz \ admits at the point
Zo the support metric ds" =A*(z)ldz/.
application to covering questions. 1) Suppose that a generalized metric is defined in a domain
~ith these definitions
B in the z·plane,
where z = x + iy. Suppose that the differential element of this metric is defined
I,
at our disposal, we have the following generalization
of the Schwarz lemma, given by AhHors:
where A(z) is a nonnegative function. Here, the function A(z)
Theorem 1. Suppose that a continuous metric ds = A(z)/ dz
can vanish only at isolated points of the domain B. The metric is said to be
the disk
continuous in the domain B (or at a point Zo € B, respectively) if the function
(I) at every point at which A(z) ~
A(z) is continuous in
B (or at zo)' It is said to be regular in B (at the point
I
z
Proof. We set
u (z)= Jog A(z),
what follows, of especial importance are those regular metrics for which the func· tion A(z) satisfies at all points of B at which it is nonzero the inequality
Let us show that u(z) (1)
'V
(z) = log R2!!1 z I"
S v(z) in the disk
I z I < R.
0<
R< 1.
Let us suppose the oppo
site. Then, the difference u(z) - v(z) assumes positive values in the disk
I < R. Since circle I z 1 = R, 1z
1) AhHors [1938].
is defined in
1
B (in some neighborhood of z 0)' In
A Jog A~ 4A'l
I
< I and that it admits a regular support metric satisfying co ndition o. Then ds < do; where do = Idz \/(1 - I z 1 2 ) is the hyperbolic metric, throughout the disk I z I < 1.
z 0 € B) if A(z) is continuous and has continuous first and second partial deriva
tives with respect to x and y in the domain
A(zo), and if A* (z) S A(z) everywhere in that
neig~borhood, then we shall call the second metric the support metric with respect
A theorem on covering of disks
by ds = A(z) I dz
=
v(z)
-->
+00
and hence u(z) - v(z)
-->
-00
as z approaches the
the difference u(z) - v(z) attains a positive maximum at some
VIII. MAJORIZATION PRINCIPLES AND APPLICATIONS
362
363
A(z)1 dz I has at the point Zo a regular support metric ds* = A* (z)! ds I that satisfies condition (1) or, if we set u*(z) = log A*(z), condition (3). Since u*(zo) - v(zo) = u(zo) - v(zo) > 0, it fol
number p such that the image of the disk
lows that, in a sufficiently small neighborhood of the point zo, we have u*(z)
the type indicated completely covers some disk of radius p because it completely
point Zo in the disk
I z I < R.
§7. NONANALYTIC GENERALIZATION OF SCHWAR:'LEMMA
The metric ds
=
v(z) > o. But u*(zo) - v(zo) = u(zo) - v(zo) and, in a sufficiently small neigh borhood of zo, we have u*~z) - v(z) ~ u(z) - v(z). This means that the difference U(z) = u*(z) - v(z) has at the point Zo a positive maximum with respect to some neighborhood Iz - z 0 I < ro of the point z 0 and in that neighborhood it is positive. Let us show that this last is impossible. The relations ~u* ~ 4e 2u * ,~v = 4e 2v , and u* > v imply that, in the disk Iz - zol < ro,
A. (u* - v):;:: 4 (e'iu* - e'i'V)
an arbitrary function w
that is regular in \ z I < 1. How
= f(z) = z + C2z2 + •••
ever, it follows from Theorem 7 of §7 of Chapter II that there exists a positive
Iz I < 1
under an arbitrary function of
covers, for example, some half-disk with center at the origin and radius independ ent of the form of f(z). Bloch has proved more than this: there exists a positive number p such that, for an arbitrary function w = f(z) = z + ... that is regular in
, I z I < 1,
the w-plane contains a disk of radius p that is a one -sheeted image of
some portion of the disk
Iz \ < 1
under the mapping w
=
f(z).
Let us now introduce some new notation. We denote by L f the radius of the largest disk in the w-plane that is completely covered by the image of the disk
> 0,
\ z I < 1 under a function w
= f(z) = z + ••• that
is regular in
I z I < 1.
We denote
by B f the radius of the largest disk in the w-plane that is the one-sheeted image that is, ~U > O. On the other hand, in accordance with Green's formula we have for r
< ro ,
Iz-t=r ~~ rdB= lzJJO,
of a subdomain of the disk , z
=
which the function w
=
f(z) maps the disk
B= inf Bf ,
Zo + re;/3; that is,
f (z)
2x
iJ
\
.
7fi ~ U(zo+re'~dB>O. If we now integrate with respect to r from 0 to r, where 0 2"
U(zo)
~ 2~ ~
U(zo
same mapping. (In other words, B f
is the radius of the largest one-sheeted disk lying on the Riemann surface onto define
where we set z
I < 1 .under the
Iz I < 1
bijectively.) Furthermore, we
L= inf Lf • f
(z)
Here'lhese infima are over all functions f(z) = z + '. •. that are regular in the disk I z I < ; here B is called the Bloch constant l ) and L is the Landau constant.
< r < ro,
we obtain
The n merical values of these two ·consrants are as yet unknown: we have only upper and lower bounds for them. 2 )
+ rele) dB.
Tbe theorem proved above enables us to obtain the best results as regards lower bounds for the constants Band L. We begin with a lower bound for B.
Since the function U(z) attains its maximum at the point zo, this last ine quality is possible only when V (z) == U(zo) on Iz - zol = r for all r in the inter val 0
< r < ro; this would mean U(z)
== U(zo) = const in Iz - zo!
< roo This is
incompatible with the inequality ~U > O.
R -> 1, we see that A(z) :::; 1/(1 - \ z
= f(z) = z + .•.
is regular in
Iz I < 1
and has
r'
finite B f' Let z 0 denote an arbitrary point in the disk I z 1 < 1. If (z 0) t. 0, then the inverse function z = f- 1(w) is regular in a neighborhood of Wo = f(zo)· We denote by p(wo) the radius of the largest circle with center at Wo in which
Thus, we have shown that u(z):::; v(z) 2 1 ),
Suppose that the function w
=
log [R/(R 2-/ z \ 2)] in
I z 1< R.
As
that is, that ds :::; du. This completes
the proof of the theorem.
the function z =
f-
1
f- 1(w)
(w) is contained in
is regular and the image of which under the mapping z
I z I < 1.
=
Obviously, B f is an upper bound for the numbers
p(wo) for all possible Zo such that \zol
< 1.
Now, following Ahlfors, let us give an application of this theorem to covering questions.
In §7 of Chapter II, it was pointed out that there does not exist a disk Izi 0, that is completely covered by the image of the disk \ z 1 < 1 under
I) Bloch [1925] showed that B> O.
2) On the ba sis of elementary considerations. Landau [1929] gave some numerical
bounds for Band L.
_
--~~~~-==="---~----:::C-C:_'.=-"'"
364
---_.- -- ----
VllI. MAJORIZATION PRINCIPLES AND APPLICATIONS
If we denote by Gzo the domain in I z I < I that is mapped by w = [(z) onto < p(wo), then Gzo must contain on its boundary either points
I or points at which f' (z) = O. Furthermore, we can easily see that p (w), where w = [(z), is a continuous function of z in I z I < I and we
Iz I =
have, close to the point zo, where Izol where Wo = [(zo).
< I and
Let us now define a metric ds = A(z)1 dz
A(Z)=
I
in
f' (zo) = 0, I z I < I,
p(w) = I w - wol,
taking
function yt(A - t) increases in the interval 0
V'A 1 nCn (~ - zo)IJ-1 + ... I 2I cn(z-zo)n+ ···1 Ill(A-lcn(z-zo)lJ+ ... Zo
1%-1 2Jcn
+ ... I
'VAlncn
II
+ ···1
»(A -I cn (z - zo)n
0< t
ds
~
=
+ ... )
/
[Ca»! A,
yiel~s Bf
=
.
~ --.!!J. II ,(z) I ~~t everywhere in I z I < 1. In
fez). Furthermore, throughout the entire
particular, with z = 0, it
2: v'3/4. Since the function fez) is any function that is regular in
va
B~4=0.43
....
In an analogous manner, we can find a lower bound for L. For every z 0 such
Iz 01 < 1,
let us denote by p(wo) the radius of the largest circle with center
= f(zo) that is completely covered by the image B of the disk I z I <1
under the mapping w (5)
•••
z I < 1, we have the bound
at Wo
Jrp'" (w)(A - p* (w»'
w
replace p(w) with Bf in (6), we
I
Thi~ last inequality holds
This metric is regular in Gzo and condition (3) holds for it in G because it is zo obtained from the hyperbolic metric in the disk I'I < 1 by means of the analyti mappings' = y(w -
(6)
Therefore, as A --> 3Bf , we have
I
•
that
cal
< t < Bf , if we
zr'
VAlf'(z)I~2~(~-:Bf)
.
< I and ['(zo).j,. O. On the
w=!(z).
1
i)
VA If' (z) I
p* (w)=! w- lea)!,
< A/3. Therefore, if we take A > 3Bf' we have in I z I < 1 the inequality
obtain
f'
2
0, as was shown is regular and
do, that is,
Since 0(A - t) increases in 0
boundary of Gzo is a point a which either lies on the circle I z [ = 1 or has the properties that (a) = 0 and I a I < 1. In Gzo ' we define the metric ds" = A*(z)ldzl, where ( )
I
tions of the theorem established above are satisfied. But Vi(A - t) increases for
,I
). '" Z
r' (z) =
What has been said leads to the conclusion that if A is such that the function
=V'A=A:-,-,If,-'(,,-,z):..J.,.I_
n> 2, we have A(zo) = O. (Here, A(zo) is defined as
Now, let us consider the point Zo such that Izol
Consequently, when this
or else the metric ds = '\(z)! dz
2 Y'P(W) (A - p (w» ~ -1--1
This shows that, for n = 2, the metric in question is continuous and, in fact, regular at Zo and that, for lim z~zo ~ A(z).)
=0
n>l,
in a neighborhood of it, then, in a neighborhood of zo, we have pew) I[(z) - [(zo)1 and, consequently,
_I Z -
< t < Bf'
v't(A - t) increases in the interval 0 < t < Bf' then, for the metric (4), the condi
00
-
365
last condition is satisfied, the metric (5) is a support metric for the metric (4) and
(4)
r'
A(z) :.-
~~~~'=~
hence can, itself, serve as support metric.
at all points of the disk I z I < I at which (z) .j,. O. On the other hand, if a point z 0 such that Iz 0 I < 1 has the property that
cn:;cO,
=-
it satisfies condition (I). With regard to points at which
where A is a constant exceeding Sf. The function A(z) is obviously continuous
+ k=n ~ ck(z-zol,
-';"--
domain Gzo ' we obviously have p*(w) 2: p(w) and p* (wo) = p(wo). Therefore, in Gzo we have A* (w) ~ '\(w) and ,\* (wo) = '\(wo) though only on condition that the
above, at such points ,\(z)
V'AIf' (z) 1 ,W=!(z), 2 'JIP'(W) (A- p (w»
!(z)=!(Zo)
~-_.
§7. NONANALYTIC GENERALIZATION OF SCHWA~Z LEMMA
the disk Iw - wol of the circle
------
=
fez) and let us define on
Iz I < 1
the metric ds = A(z)ldzl,
where
}.(z)=
1/'(%)/
c .
w=!(z),
2p (w) log P (w)
C being a constant greater than
L f • I ) This metric is continuous in I z I < 1.
1) Again, it will be sufficient to consider functions fez) with finite Lf.
(7)
i
366
VIII. MAJORIZATION PRINCIPLES AND APPLICATIONS
Now, let us denote by Gzo the largest domain contained in
1
z
I
<1
Iz 0 I < 1,
and that has the property that its image is contained
< p(wo),
where Wo
Iw -
that is both on the circle
Wo
I '=
f(zo). Finally, let
. II'(z)
1
al=!;(O) =
denote a point
p (wo) and on the boundary of B.
Let us define on Gzo a metric ds"" = ,\""
)..*(z)=
a
c
(z)I dz I,
<1
lytic mappings: (= (l - (')/(1 + ('), (' = log (CI(w'A* (w)
~ pew)
S A(w)
the interval 0
(8)
zo
a»,
w
=
fez). Since obvi
everywhere in Gzo provided the function t log (Cit) increases in Under this condition, the metric (8) will be a support
then the conditions of the theorem that we have proved are not satis
fied for the metric ( 7). But t log (Cit) increases in the interval 0 Therefore, if
< t < CI e.
C 2p (w) log p (w)
Ita.+~)
r ( I + Cl
l+a_I)
1'(lta.-~)r(:)
2
'
3
=
+ -l) l' (~) 3 3 J
F(z) = fll3({l/~(z»,
I z I < R,
which is orthogonal to the circles of which
the sides of the triangle 6. 116 are arcs, and is single-valued and regular in that disk. It is easy to find the radius
R of that disk. Specifically, the chords con
necting the endpoints of the sides of the triangle 6. 116 are of length
.j3 and they
. make angles of 77/12 with the corresponding arcs of the triangle 6. 1 / 6 , On the other hand, a circle y containing one of the boundary arcs of the triangle 6. 1 / 6 that terminates at the poiot z
=
1 intersects the real axis at an angle of rr/12.
{3.
V3 + 1
F, which is the single-valued image of the disk \ z I < R under the func = F(z), is composed of an infinite set of triangles congruent to 6. 113
surfac~
tion and
1
ea~h
vertex of each triangle is a second-order branch point of that surface.
This la~t fact shows that a one-sheeted disk of radius greater than 1 cannot exist on the surface F. Now, consider the function
I
L~2'
F(Rz)
w=!(z)= RF' (O)=z+ ....
Let us now find some upper bounds for B (AhHors and Grunsky [1937]) and
This function is regular in the disk we constructed a function that maps the disk
wi
1 respectively, then this function w = f a(O = a 1 ( + .. ,
obtained in accordance with formula (19) of
§ 1 of Chapter III
Iz \ < 1
and it maps that disk onto a Riemann
surface that does not contain a one-sheeted disk of radius greater than
I(I < 1
Therefore, we have
onto a regular circular triangle 6. a inscribed in the disk I < 1 with interior angles of size fTQ. If this function maps the points (= 0 and (= 1 into the =
2
I
R 2 • Therefore R = Jy3 + 1. It follows from the pr1cess of the analytic continuation of the function F (z) that the Riemann
For z = 0, this yields L f ~ 1/2. Thus
points w = 0 and w
367
and their product must be equal to
«:_1 --= I - I Z II
If' (z) I:s;;; ~_,. .
§ 1 of Chapter III
~ __
Therefore points of intersection of the circle y with the real axis are I and
If we now replace pew) with Lf and take the limit as C --+ eLf' we obtain
in I z I < 1 the inequality
In
.
Consequently, the segment of the real axis lying inside y, also is of length
C> eLf, then in the disk 1 z 1 < 1 we have ds S da, that is,
II' (z) 1
,
which maps the triangle 6 1 / 6 univalently onto the triangle 6. 1 / 3 in such a way that the point z = 0 is mapped into w = 0 and the vertex z = I is mapped into from 6. 1 / 6 to the entire disk
by the successive ana.
metric for the metric (7) at the point zo. If t log (Cit) increases in the interval
o < t S Lf ,
.
the vertex w = 1. By the symmetry principle, the function F(z) can be extended
because it is
everywhere in Gzo and since p* (wo) = p(wo), it follows that
< t S p (wo).
..
Keeping this in mind, let us construct the function w
This metric is regular in Gzo and satisfies condition (3) in G
ously p* (w)
_ _~~
where
2p* (w) log p* (w)
1( ,
B(I-a. -2 B(I-a. 2
,p*(w)=lw-al, w=!(z).
obtained from the hyperbolic metric in the disk
",,-
z:=-~~_
-=>g-
that
includes z 0, where
'=
..
§7. NONANALYTIC GENERALIZATION OF SCHWAkz LEMMA
in the disk Iw - wol
L.
!&f.~
%!
IS
(with c = 1). It
follows that al can be expressed in terms of the beta function and hence in terms of the gamma function as follows:
B S 1/RF' (0), and, consequently,
1':"'<0) I 6 _ B:s;;; 1'1 (0) R 3"
(~) l' C~) V V3 + 1r(~) r
Thus, we finally have for B the inequalities
IIRF ' (0).
.- :-.
------
--.--.------------=====-~=--,;~-
_
""'===~....:::.-:::-..:.::::~~=::=.=:=;=:...~: _=_.~_===__.,_,_
..
.:..
.~_._.--.--.=~,:,.-",,,"=-o=.
_ _ ._._.~=.-.=,_.~__,, .. _.==
.
.,,=.~~~_~="",==~._._
.§8. MAJORIZATION OF SUBORDINATE ANALYTIC F~CTlONS
VIII. MAJ ORIZATION PRINCIPLES AND APPLlCA nONS
368
.... _''''====-=-~_===_'=-_====
Izl < 1, rjJ(O)
lemma; that is, it is regular in
'V3 =0.43 4
r(l)r(ll)
~B~
'"
By considering the function w = F (z) =
fI!3 (/0 I (z»,
(9)
nonunivalent majoranrs: a function [(z) is said to be subordinate in
we can show that the
is regular in the disk
Iz I < 1
F(O) = 0 are regular in I z \ < 1 and if f(z)- 0 and 0 < r < 1,
=z+ ...
and maps that disk onto a Riemann surface
F con
IIF' (0) but with the vertex of none of
Proof. Let
these triangles belonging to the surface F. Consequently, this sudace F cannot
2"
o
0
ro
J~
1
(0)
8
- r (~}r
1
3
'2";;;;' L ~
2r
(
such that
Let us write F(z) in the form
where c m f, O. Then, let us define
n)
3
2r{ ~) ,
b (z)
L:
Thus, we have the inequalities for
< ro < 1
+...,
r (-})
CO)
(2)
F(z)=cmz m
It follows tha t
L~ F'{O) = f~
I z\ = roo
=
IP dB.
denote an arbitrary number in the interval 0
there are no zeros of F(z) on the circle
IIF I (0).
cover any disk of radius greater than
2"
~ Ij(,ei~ IP dB~ ~ \ F(re lO)
sisting of a finite set of triangles obtained from t, I /3 by means of a similarity transformation with similarity coefficient
to a
We shall prove a number of theorems 1) regarding subordinate functions. Theorem 1. If two functions f(z) and F(z) such that f(O)
F(z) F' (0)
Iz I < 1
I z I < 1.
majorant F(z) if it can be represented in the form (1) in
function
w=
IrjJ(z)1 < 1 in Iz \ < 1.
= 0, and
In such a form, the concept of subordination is ~eneralized to the case of
"3
12 =OA7 '""'" '" V'V3 +Ir ( "41 )
""'"
369
= (
:0
rt
if F (z) has no zeros in 0
< I z I < ro,
n
I)
"3
\
= 0.56 ...
,
(10)
(z)mn'O{Z-Zk)
'0_
r~-z~z' if F(z) has zeros in
O
k=l
I\
Here, ZI,"', z" denote the zeros (all of them) of F(z) with each zero counted .
in acc~rdance with its multiplicity. Then, the function F(z)/b(z) has no zeros
§s.
Majorization of subordinate analytic functions
Suppose that a function f(z) is meromorphic in the disk function F(z) is meromorphic and univalent in
F (0). If the image of the disk
Iz I < 1
I z I < 1.
Iz I < 1 = fez)
where rjJ(O) = 0 and
jrjJ(z)1
F (Z»)P
( b (z)
is contained
.
F (z), we say that the function
I z I < 1. It follows in the disk I z I < R to a
:$ 1 in
all functions fez) that are subordinate
Consequently, the function (F(z)/b(z»P is regular in
= _1 r ( F (z') )P ffi (Z':' + Z }d6 z' = r eiO 2~ J b (z') - z . • 0
(3)
0
mula (1), we obtain in
I z I < ro 2"
~ l- r \ F (z') IP ffi (Z' + cp (z) ) dO. b (cp (z» --: 2~" b (z') z' -lfl (z) IJ.£l-IP
(4)
o
given univalent
Furthermore, since Ib(z')1 = Ion
majorant F(z) is defined by the formula
where rjJ(z) is an arbitrary function satisfying the conditions of the Schwarz
for
in the interval \ z I < ro. Therefore, if we represent f(z) in accordance with for
that the set of
j(z)=F('f (z»,
I z \ < ro
2"
=
fez) is subordinate to F(z) in the disk I z I < 1 and that the function F(z) is a univalent majorant of f(z). We denote this by writing fez) -< F (z). This obvi ously is equivalent to regularity of the function F-I(f(z» = rjJ (z) in the disk
I z I < 1,
I z I ~ roo
arbitrary p> O. By Poisson's formula, we have
and that a
Suppose that f(O)
under the mapping w
in the image of that disk under the mapping w =
in
(1)
••
1) Rogosinski [1943].
Iz'l =ro
and \b(z)1
<1
in
Izi
we have
370
VIII. MAJORIZATION PRINCIPLES AND APPLICATIONS
II
(z)
I~ P
i- Jr IF (z')
211:
{' ffi
J
where z =
Te
it
,
o we obtain
--+
TO,
z' -cp (z)
< r < ro,
21tffi
2"
211:
and keep in mind the fact
(Z'z' -+ Ifcp (0) ) = 21t (0) ,
k=n+1
n
L: I a" k=1 (6)
IF (r oeiO) IP dlJ,
\
(7)
Iz I =
ro, is obtained from this one by means of a passage to
r ~
/9 9k
n
L: I A k=1
k
r
9 tk •
1
n
Proof. Suppose that fez)
=
al = But
Ia II :5 i
F(ep(z» and ep(z) (11
Ab
~=(l.~A)
=
Ik=lakzk. Then,
+ t1gA
I•
and, from inequality (2) of 91 as applied to the function ep(z)/ z at I
a
2
1
:5
1-
I a 11 2 •
Therefore, from what was said above, we obtain
(11
(11
(11
This completes the proof of the theorem.
~ la"I'~ ~ IA"I'·
Proof. We set
and
r I~ I Al I, I a~ I ~ I It ·1 A, I+ (1 -I It) I Al I ~ max ( I Al I, I At j).
n
"=1
lall :51 All
la21 :5 max (IAII, IA 21).
z = 0, we find
Theorem 2. If the functions fez) = Ik=lakzk and F(z) = Ik=IAkz k are regular in I z I < I and if fez) -< F (z) in I z I < 1, then, for n = 1, 2, " . , (8)
"=1
Theorem4.l) If two functions fez) and F(z) such that f(O)=F(O)=O and F' (0) = 1 are defined and regular in the disk z I < 1, if F (z) is one-sheeted, and if f(z)-< F(z), t~en, J
/I
811 (z)
k=1
As r --+ 1, this yields (8), and completes the proof of the theorem.
0
the limit. This completes the proof of the theorem.
=
"=1 F rom this we obtain
this yields (2). The case in which ro is such that the function
F(z) has zeros on
If fez)
~ I a" I'r'" + ~ ~ ak 19r'" ~ ~ I AkI'r'k.
(5)
Theorem 3. Under the conditions of Theorem 2, we have
o
r
~ (Z' + cp (z) ) dB.
(Z'Z' -If + cp (z) ) dt = (z) .\ II (re it ) IP dt ~
and, as
IP
o
If we integrate (5) over the circle 1 z I = r, 0 that, by virtue of the theorem of the mean,
371
nOOn
2"
'It
<§8. MAJORIZATION OF SUBORDINATE ANALYTIC FUNCTIONS
=
/I
(Xl
k ~ a"z", SII (z) = ~ A"z", R/I (Z) = ~ Akz . k=1 k=1 k=II+1
F(ep(z», then, obviously (since ep(O)
=
,1+lzl
II
0),
Izi
(z)I~{l_.I.. ,\B' \/(z)I~{l_I?'I\I'
(10)
These inequalities are sharp with equality holding only for a function of the form '00
l(z)=S/I(cp(z»+RII(~(Z»=SII(rp(Z»+ ~ ak z" k=lI+l
I
and, consequently,
=
'L"lZ~~\1
,
\
"ll = I € 1=
1.
(11)
Proof. From the representation of fez) in the form (1), we have If' (z)\ = (Xl
~
Sn(cp(z»=SII(z)+
a;;z".
But Sn (ep (z» -< Sn (z) in 1 z I
< 1.
Therefore, by Theorem 1, we may write
211:
~ I S/I (rp (re » I' dB ~ ~ I S/I (re ) I' dlJ,
o
iO
0
IF
'(ep(z»J . 'ep' (z)l.
But inequality (8) of 94 of Chapter II holds for
lep' (z)l,
I F' «cp z» I ~
+
1 I ~ (z) I (l-jcp(Z)i)8'
, .
l-lzl"
1+ IIf (z)I)'
1
If' (z) I ~ ( 1-1 If (z) 1 . 1 -I Z 1 1) Sch iffer [1936], Lohin [1949].
2
•
If' (z)l,
so that
1-I If' (z) II
Icp (z)l~
Consequently,
2"
iO
(9)
and inequality (2) of 91 of the present chapter holds for
k=II+1
that is, by (9),
(z)
(12)
\
_
"',,_
372
_
_
_...._
'~,"""',.. ~"~~_
_A'
~....,=,._.~~ ..,",~ __,,=,
~.
I,
"_,,",_~O_~·~·O._"_
~~.~~.-~~,,"_~~'._v_~"~'~"
we have the first of inequalities (10). We obtain the second
1¢(z)1 ~ I z I only for
¢(z) = ."Z, where
1.,,1
<Xl
2~i ~
of inequalities (10) by integration of the first. Since equality holds in the relation ship
"~~.
"~""~---
= 1 and since equAlity holds in
An (C')
IC'I=I
[! k=O
C~+I
= 1, equality holds in the relations (10) only for a function of the form (11).
This completes the proof of the theorem. Theorem 5. Let f(z) ~
I.'k = lakzk
-
dC'
denote a function that is defined and regu
If the starlike domain referred to is convex, we have the sharp inequalities lanl ~I, for n = 1, 2,···.
I A~ (Q I~ 2~
S
I An (C')
1,'/=1
belongs to B for
I z I < 1,
F(z) is a convex domain. Since wk
=
=
k=O
~ 2~ IC'/=I ~ k=O I! C'~:1
n,
(w I + .. ; + wn)/n €
=
An
is subordinate in
Iz \ < 1
to the majorant F (z). But then, it follows on the basis
= 1) that Ian I ~ 1 for all n = 1, 2,···. If the image of the disk 1z I < 1 under the mapping w = F(z) is a starlike
Iz I < 1
PI (C
=
(in
1(1
z1 < 1,
we have
~ 1 we have F1«(¢(z))-< F 1 «(z)
co
~ An (C) Zk,
I
AdC)
k=l
we have on the basis of what we have already shown
= 1,
IAn«()1
(13) ~ 1 for
1(1 < 1
and
= 1, 2," '. But An «() is an nth-degree polynomial in ( such that An (0) = O. Therefore, in accordance with Cauchy's formula, we have for I(, < 1 n
A' (r) n "
=
I
2'll:i
r J IC'/=1
An (C')
(C' -C)I
dr'
,,=
J
PI (C<jl (z)) zn+I
dz, 0
< r < 1,
~
r
J
'f(z)P~(,!,(z)) dZ=_1
jzl=r
I = IA~(l)1 ~ n. l )
Jr
2nl Iz\=r
zn+I
P(,?(z))
~n+1
dz = an'
Equality holds only for a function of the
next theorem, we shall need a lemma, due to Rogosinski, which
is a sharpening of the Schwarz lemma. Lemma. Let ¢(z) denote a function that is defined, regular, and bounded
above in modulus by 1 in the disk I z 1< 1 such that ¢(O) = 0 and ¢"(O) 2: o. Let Zo denote a point in Iz I < 1. Let ~zo denote the domain contained in the
onto a convex domain (see §9 of ChapterIV). Let us
I¢(z)l ~ I z I. Therefore, for arbitrary in I z I < 1. Therefore, if we set
A'(I)=~ 2m n
T~~rove the
domain, then the function FI(z) defined by F(z) = zF~(z), F 1(0) = 0, Fi(O) = 1 maps the disk
r
1
(C) = 21ti
form ( 1). This complet es the proof of the theorem.
of Theorem 2 (with n
represent f(z) in accordance with formula (1). In the disk
I C12k•
k=O
we obtain
Cons quently,. Ian
k=l
rIdr: 1= !
I Z 1=1'
n
~ !f(1jkz l/n)=an z+ ...
n-I
Hence as (--> 1, we see that \ A ~ (1)1 ~ n. Now from
f(."k z )
e 21Til for n = 1, 2,"', it follows that B. But this means that the function where."
k=O
II! S+1 rIdC' I n-I
Proof. Let us consider first the second case, in which the image B of the under the mapping w
21ti, 1,1=1
. [~ Ck ]! An (0 "'"""f'JiiT dr:
n-I
ordinate to F(z). Suppose that the image of the disk I z I < 1 under F(z) is a starlike domain. Then, we have the sharp inequalities I an \ ~ n, for n = 1,2, ....
I zI < 1
n-I
r ,J
1
and, consequently,
lar in I z I < 1. Let F(z) denote a function that is regular and univalent in the same disk. Suppose that F (0) = 0 and F '(0) = 1 and suppose that f(z) is sub
disk
373
r
the first of inequalities (12) only for the function F(¢) = ¢/(I - "'1¢)2, where
1"'11
'''''''''-''.,~
§s. MAJORIZATION OF SUBORDINATE ANALYTIC fuNCTIONS
VIII. MAJORIZATION PRINOPLES AND APPLICA nONS
Since I¢(z)! ~ I z
~.,-"_.~_ •.;~~.~~,.,,-~
disk I z 1~ \ Zo I, containing the disk I z I ~ I Zo 1 , and bounded by an arc of the circle I z 1 = I zol 2 and the arcs of two curves passing through Zo and tangent to 2
the circle 1 z I
= I
zol 2. Then Wo
=
¢(zo) belongs to ~zo·
Proof. In accordance with inequality (3) of
f(z)
=
z-I¢(z), the point Wo
=
§1
as applied to the function
¢(zo) lies in the disk with boundary defined by
1) This inequality can also be proved if we replace the assumption that the image of the disk I 1 < 1 under the mapping w = F(z) is starlike with the assumption that it is symmetric about the real axis. For more infomlation on this, see Littlewood [1944].
z
~
=~~~U~"'~,?"-,-":~~:~~~~E:'~'0_",,",~"~"-~""~~~-~~;:·:!:!"':~~F=~~~I:-y'J'~::-~~.'~~_~~'::~
~
374
VIII. MAJ ORIZA nON PRINCIPLES AND APPLlCA nONS
the equation I(w - a 1z 0 )/(zo - a 1w)!
=
§s. MAJORIZATION OF SUBORDINATE ANALYTIC "UNCTIONS
I zol, where a 1 = ¢ '(0) and 0 sal S 1. as 1, let us find the envelppe of all
If we now take a= a 1 in the interval 0 S
the disks obtained. These disks are obviously contained in the disk Iw and their centers lie on the segment (0,
1
s
I
Zo
I
at all points z ~ z 0 in the domain ~z 0 mentioned in the lemma. This is true because the function F' (f(z))
=
a 1z + .•• satisfies the conditions of the lemma
and hence, by ,applying (16) to the point z = F -1([(zO)) € ~zo, we obtain (15).
The equation of the envelope is
Z1»)'
To prove inequality (16) with 1z 0 I
determined from the equations
=
X,
we define the number p € (0, 1) as
a root of the equation
F (w, IX) = ffi -
(lOg w- azo ) Zo-aw
F~(W, 1X)=ffi(, w-az Zo
w
_
o
= log 1Zo I, zo-aw
(l
)=0.
(14)
o
aw)
) =0
m(wzo-aw - azo ) =
or
IF (z) I ~ I F' (0) I II ~~ ~ 1\1
I z I < 1, ,
we have
IF (zo) I> IF' (0) 111 !z~ ~
S
as 1,
this determines the arcs of the
Then, on the boundary of the domain ~zo there exists a point z 1
at
Z,
spo nding to the value s a = 1, 0, and - 1) and the other pas sing through the points
arc by 1. Theiangent to the image of t he arc l under the map
S a 1 S 1,
=
cover the domain ~zo mentioned in the lemma. This completes
the proof of the lemma. Theorem 6. 1) Let fez) and F (z) denote two functions de fined and regular 1z 1
in the disk
<1
F(o) = 0, that arg
and suppose that F (z) is univalent. Suppose that f(O) =
f' (0) = arg
F ' (0) [this condition is dropped if
that f(z) -< F(z). Finally, suppose that f(z)
t
1
z
I
s X.
(The number
X is
and
F(z). Then,
If(z) I
f' (0) = 0],
I = X,
zQ
ping
"/2
= F (z) at the point
wI =
F (z 1) makes an angle w
=
w th the radial direction at that point. Let us calculate
this Ie in a different way. We first find the angle y formed by the t ngent to l at the point z 1 (directed toward the origin)
Figure 16
and the ra ial direction at the point z l ' Let us denote by R the radius of the circle containing the arc 1. Then, from the right triangle with vertices at the 2 points 0, z 0, and the center c of that circle, we obtain R2 = \z 01 2 + (R -I z 01 ) 2.
Therefore, R '" (1 + Izo 1 2)/2. Furthermore, from the triangle with vertices at the (15)
not the best constant. 2)
Proof. To prove the validity of inequality (15) at a given point Zo such that Izo
.
'Obviously, the point z 1 lies on one of the circular boundary arcs passing through Zo (see Figure 16). Let us denote this
zo, -i Izo Izo, and - zo. It follows that the circles I(w - a 1 Z 0 }/(zo - a 1w)1
l
•
which IF(z)\'attains the value m = maxzE.1zoIF(z)l.
circle s pas sing in one cas e through the points z 0, i I z 0 1 z 0, and - z 0 (corre
0
(18)
1\1
now suppose that inequality (16) is not satisfied everywhere in the domain ~z 0
that (w - azo)/(zo - aw) = ± i Izo I, i.e. w = [(a ± ilz o 1)/(1 ±ialzo I)]zo on the
I,
(17)
If \ z I < p, inequality (16) follows immediately from (18) and (17). Let us
O.
This equation together with the first of equations (14) leads to the conclusion envelope. As a varies in the interval 0
=(ltiLD'; p=0.123 ... >Izol'i= 116 ,
for z € ~zo.
conjugate, the equation takes the form
m( (w - az_)1(zo -
p p)1
Since the function F(z) is univalent in
If we replace the first fraction in the second of these equations with its complex
Izo
375
it will be sufficient to show that
I F(z) I
points 0, z 1, and c, we have
(R -I ZO l'i)'i =R~ + I Zl I~ - 2R Izll sin "(.
Therefore,
. _ I Z1 IB + I Zo II 16lzd l +l sm "( - I zd (l + I ZO IB) 171 Z1 I 1 l=arcsin 161 z11 +1 17 11zd
Now, the angle of inclination of the tangent to the arc l at the point z 1 (with 1) Biernacki [1936] .
2) .Xia Dao-xing (Hsia Tao-hsing) has shown that the best constant in this case is
the number (3 - ..;5)/2 (see §3, subsection 3° of the supplement).)
respect to the real axis) is equal to -y + arg z l' Furthermore, the angle of incli nation of the tangent to the image of the arc 1 at the point
wI
= F (z 1) is equal
_..----.::==--==::::=..-========.:=:::--==::::=-~~-===:=::::.:==_ _ _==_=.:...___=.=.:::=:__===_..:.-._=__~=~=:==__..:.......=_====::.:::::_===. __=__:._.::_:'::=.:::':=~~::=':::...--:;:;---====-':::':=-==~_:=:==="'===-""=====,",,--O==-~:==~=~:::::'=="_==::=--=--=-_:=:::::~.:
_______=.--;,:--=====;=.=:;---
376
VIII. MAJ ORIZATION PRINCIPLES AND APPLICA nONS
§s. MAJORIZATION OF SUBORDINATE ANALYTIC FUNCTIONS
to -y + arg z 1 + arg F' (z 1)' Finally, the angle between this last tangent and the radial direction at the point
wI =
Proof. Since the function F (z)
F (z 1) is equal to y - arg z 1 - arg F'(z 1) +
= T-
arg zF'(z) 1 1 F (ZI)
arc sin l61z11' +1 _ arg z1 F'(z)1 . 17 I ZI
I
I z 1< 1,
the same is true of the function
arg F (z 1)' As a consequence of all this, we have 0)
+. ; . is regular and univalent in
= C lZ
377
o(C) =
F (ZI)
But inequality (12) of §1 of Chapter IV holds for this last argument. Therefore,
for arbitrary z such that
1z I
F
C+Z) -F(z) (l+iC F' (z) (I _I z I I)
< 1.
= C+...,
1'1 < 1,
Therefore, in
(19)
we have
we obtain
z 1 1+l zll . 161 11 +1 +log Iz I 1·1";·«810 171'01 1- , ",;;;; arc
°
la(QI",;;;;{,I+,lrC'~8 (~+l~:r-._I,.,•.
l+lzol< 1t 16p'+1+10g'_"1 2
510
• ..,
If we substitute equation (19) into this inequality and set 0
I x I < 1,
where However, this contradicts our previous conclusion that
(i)
=
F' (x)
I F' (z),
tion proves the validity of inequality (16) and, consequently, inequality (15) on the circle
I
z1=
XI.
we obtain
11/2. This contradic
z I .:s XI is attained on the circle I z I = XI, it tollows that (15) holds also in 1 z I < XI. This completes the proof of the theorem.
1
and suppose that [(z)
that under the additional assumption that F(z) is starlike in
p = (3 -
/5)/2.
=
Le~\us set
If we also assume that [(z) is univalent the number p is defined
+ p + 2 atc tan
I-p
(i)
1-
I
,
that is, p = 0.39· ••.
where I
In conclusion, we shall prove one more theorem:
zI <1 0, that
Iz \ = r
Y
(I-arl )I+lool'(a-r t )s+2(I-arl ).(a-r' »'"
and ~(w(z» ='\, and also the fact that the numerator of the deriv
ative of the radicand with respect to ,\ can be reduced to the form _
1
and that F (z) is univalent in that disk. Suppose that [(0) = F(O) = arg r' (0) = arg F' (0), and that [(z) -< F (z). Then, in the disk I z I < 0.12 ... , we have 1['(z)l.:::: IF'(z)! with equality holding only [or [(z) == F(z).4)
(z) satisfies the conditions of the Schwarz lemma.
+
-
p = 1 't , 2,
Theorem 7. 3) Suppose that [(z) and F (z) are defined and regular in
a+oo (z)
x = ¢(z) in (20). Keeping in mind the fact that \
have I)
as a root of the transcendental equation 2 ) log 1
F(¢(z). If we represent the function ¢(z) in the form
=
we ea~ly conclude tha t
0.382' .. and
I z I < 1, we
\
cp (z) =z· I +aoo (z)' 11= cp (0) ~ 0,
that the radius p of the largest disk in which inequality (15) holds, under the - .j5) /2
I",;;;; ( II +_II~ t z: I)\ I I -I-1 xz I'I"
.
Remarks. We note that, using more complicated inequalities, we can show
< P .:::: (3
and hence is negative for
I
(I-r' ) (I+a) I-ar'
Ia I < 1
and r
(1 + I < 1,
0)
l~ a-r
Ix I < I
1 00
r (I-a)
I+r'-r(l+a)
+
(i) \
< r.
Consequently, (20) yields
1) Goluzin lI951b]. (See editor's footnote on p. 374).)
va
+-
(see
§ 3,
)
we obtain
S \ I-z
for
l
l-arS
r(l-a)(I+r) ",;;;; I - ars - ar r3
2) Biernacki ll93G and Goluzin l1939cl. 3) Goluzin lI951b]. 4) (Xia Dao-xing has shown that the best constant here is the number 3 subsection 3° of the supplement).)
(x z)/(1 - zx),
(20) I-zx Now, let [(z) and F(z} denote the pair of functions mentioned in the theorem
But since the maximum value of I[(z)/ F(z)1 in the disk
conditions of Theorem 6, lies in the interval 0.35
,=
r (I-a)
\
I
F' (
",;;;; ( 1r(l-a) ) I-I
1 F'(z) ~I--'+'----;rl~-r--"(If-+-'---a~)
_
WWM?
::mrr::w"1!i!M!!lW!4
378
nm
"!fl!W1!!!!l!Q'"
'"
~
'''!Wi'i'4W!Ji;~
'i"t"'?'t'ft''ii¥il!W3.."%'rA~'?W'"t'iW'''''E%''fID'''iW'''F{j.nFt¥i!;\,h·§'';>Wrl~'0i,.;~tJ@fJ1ii;'ft:Pi'''?!o/f!!''%W®i''r7.'R'tj~·'"m
§s. MAJORIZATION OF SUBORDINATE ANALYTIC FUNCTIONS
VIII. MAJORIZA nON PRINCIPLES AND APPLICAnONS
that is
I
F'(,(Z»! F' (z)
2
j'(r) =l-r aa+ I
(l+r2-2ar)2 l-r2 ~ l+rl-2r . ',n.
4r 2
(21)
The function On the other hand, we have the familiar inequalities
at a= 1. If a
a -r- r I' ( ) 1 1 -I 'ill 12
I 'Pt () z 1~ 1+ ra ' 'Pt z ~ 1 _ r2
for
I z I < 1,
we have, in
where ¢ l(Z)
IF'(z)l.
= z-l¢(z) = a .... On the basis of these inequalities,
< 3, that is, if r> 3 -..,(8, this derivative is negative for a= 1.
From this we conclude that the inequality
+1 rr
2
I 'Pt I~) = (r + I
(1 -
,! !)
..,(8 = (l
-=- r I 'ill/).
Theorem 7 be univalent in in the disk
+
a+r
,!
Iz I < 3 - ..,(8
l+ ar
'()I a(l+r l )+2r 1-1,11 1'P Z ~ 1 +rl+2ar l-r 2 ' If we multiply (21) and (22) and set a
I I' (z) I F' (z)
[_
~
(22)
= (l
+ r 2 )/2r, a ~ 1, we obtain
(
0 -
{1.)2
0-
1
+I +
011
0
11
(23) •
Here, the sign- of the derivative with respect to a of the right-hand member coin cides with the sign of the polynomial
p (a) =a3 - 3a - (3a~
+ 1) a. -
2aa.~.
1,
p(a.) ~p(I)=a3 - 3a~ - 5a -1 =(a+ 1) (a~ - 4a -1), -
4a - 1 ~ 0, that is, if a ~ 2 + ,/'5, we have p ( a) ~ 0 in 0 S
a
<1
and, consequently, the right-hand member of (23) is an increasing function of a. But it is equal to unity for a= 1. This shows that If'(z)\ that is, if r
S IF'(z)1
if a 2: 2 +
< 0.12. Equality holds only when a= 1, that is, only when ¢(z)
=
V5, z,
so that fez) == F(z). This completes the proof of the theorem. Remark. Consider the two functions
F(z)= (l +Z)S' f(z)=F
is not
Iz I > 3
0.17· . '. If we make the additional requirement that the function f(z) in
that is,
so that, if a 2
1f'(z)1 S IF'(z)1
always satisfied under the conditions of Theorem 7 if z is such that
+
S a<
2r
as a function of a has a derivative equal to (a - 3)/(a + 1)4
r I,t! 1 -I If II r ~ 1 -I 'P 12 l 1+ r I I 1- r ~ 1 + r a + r 1 - r2 ,
For 0
a= 1 +r2
Consequently, for a less than but sufficiently close to unity, we have If' (r) I >
I z I < 1,
~ I 'Pt I
f' (r)
(a+a)8'
379
(z tt:z), 0 ~ a. ~ 1,
which satisfy the conditions of Theorem 7. We have I) Goluzin [1951rl.
I z I < 1,
then the inequality
If' (z) I s IF' (z) I holds
though not always in a larger disk. 1 )
§l.
LIMITING VALUES OF POISSON'S INTEGttAL
381
Theorem 1. If the function fee) is continuous at e = eo' then u(r, e) - .... ieo by any mode of approach through
f(e ) as (= reie approaches the point e o points in the open disk 1(/ < L
(>
Proof. For given CHAPTER IX
If(t) - f(e o)! since
BOUNDARY VALUE PROBLEMS
FOR ANALYTIC FUNCTIONS DEFINED ON A DISK
< d2
0, we take 0 in the interval 0
whenever
It - eol < O.
< 0 < 17
Suppose that Ie -
such that
eol < 0/2.
Then,
2"
;~ ~
1)
P(r, t-B)dt=l,
(3)
o we have
§1. Limiting values of Poisson's integral
U
As we know, an integral of the form
;1t
(r, B) -- f (B o) = .
2"
~ (f (t) -
f (B o» P (r, t - B) dt.
b
Let uS partition the integral on the right into two integrals II and 1 , the fiest 2
·2"
u(r,
B)=2~ ~f(t)p(r,
over ~he arc (eo - 0, eo + 0) and the second over the remaining arc 11 of the
(1)
t-B)dt,
.circle\\(1 = 1. Noting that Per, e)
where
\
! rcos 0+ r1I=1+2
00
l-r
?
(2)
n
r cosnB
Since It -
e\ > 0/2
is called Poisson's integral. t and has period 217, the integral (1) yields the solution of the classical Dirichlet
in the open disk \(1
< 1,
where (= reie. This solution is a harmonic function
it is continuous in the closed disk \(\ :5 1, and it as·
sumes the values f(e) on the circle
1(1
= L In this section, we shall study Pois
son's integral under less stringent conditions imposed on the function f(t). Let f(t) denote an arbitrary real function that is summable in (0, 217) and tbat has period 217. Obviously, the function u (r, e) defined is again a harmonic function in the disk
1(\
< 1.
2)
by the integral (1)
Then we have
on 11 and hence, P (r, t -
Il~l~
In the case in which a real function f(t} is continuous for all real values of
I( I < 1,
j
2"
E~ P(r, t-B)dt=z. E P(r, t-B)dt~~
°0- 0
,
11=1
problem for the disk
°0+0
E\
l
P(r, B)=l -
\1 ~hl~4~
> 0 in the disk 1(\ < 1, we obtain
l - rll
~
1--2rcos-+rll 2
This shows that 12
-->
0 as r
--->
e) < P (r, 1 \
·21tjlf(t)!dt. 11
L Consequently, there exists an TJ> 0 such
1121 <: d2 whenever r> 1 - TJ· Consequently, for r> 1 - TJ and Ie - e01< 0/2, we have 1u (r, e) - fee 0)1 < (, which completes the proof of the theorem.
that
As a consequence of Theorem 1, we obtain the familiar theorem that, if the function f(t) is continuous foe all t, then the function u(r, e) defined on the circle
/(1
=
1 as the limit of u(r, e) when the approach is through points inside
the disk I( I < 1 is continuous on '(I :5 1 and u (1, e)
if fee) is continuous on some arc of the circle 1(1 1) Stricdy speaking, this chapter is not a parr of the geometric theory of functions. However, the information contained in it will be used to a considerable extent in the fol lowing chapter. 2)ln this and the following chapters, a familiarity with the fundamentals of the theory of functions of a real variable is assumed. Thus the integral (1) is understood in the sense of Lebesgue. Also, because of the periodicity of !(t), we can displace the interval (0, 217) and hence we can interpret the integral as being over the circle 1~ I = 1.
380
0/2), we have
=
=
f(e). On the other hand,
1, then the function u(r, e)
defined on that arc as the limit of u (r, e) when the approach is through points inside the disk 1(1 < 1 will be continuous in the union of the open disk set of interior points of that arc.
1(1 < 1 and the
Theorem 2. If a function fee) has a finite derivative ~t e = eo' then au(r, e)/ae -+ ['(eo) as (= reie approaches the point e,e o through points in
-:--=::===-~=~~~=-~:':~,,=,~~::~i'>;.~~,;';':~~;~';'~i;='~;:":;'-~;0=~:Mi:iw~~..:i~;:~,:;:~_~))~~~"";;_,,,;i.~~'i'';;;;"_';;;~~''i;;;,.-.i'~
382
IX. BOUNDARY PROBLEMS FOR ANALYTIC FUNCTIONS
the open disk
1(1
side the disk
I(I < O.
§1. LIMITING VALUES OF POISSON'S INTEGR'AL
< 1 along any nontangential path (that is, a path that lies in h ' l y in . h some ang 1e w£t vertex at e iB o tat, sUI,(f"£c£ent 1y c 1ose to e 'iB o, l'£es enUre
linear in the interval (a, a + 217): [(()
that is
A() + B and that is extended as a periodic
=
2r (t-ll o) sin (t-6) I elf --: reID j II
Q But le it - reiBI ~ !ei(t-B)_
Proof. Let us fitst prove the theorem for the case of a function [(()
rl > Isin{t- ()j
,ou (r, 6) -1 all
2~
-
~
a6
2~
-
a
1
=-~(f(a+27t- 0) -
2r I t-O o I
I Q I~ I eU -
C I(t) oP (r, t -,ll) dt
I
~
at
l(a+O» P(r, a -a)
'0 i (Oo+~+~) re''0 =elo+se 2" we have •
It follows that, for a < () 0 < a + 217, the derivative au (r, ()/ a() --> A as re iB iBo iBo along any path leading to the point e from inside < 1. e
' . I re,91 = \ ell - ei90 - e;e
--->
1(1
= \ 2e
~~~_~
k
6)+r 9
i (
) /-9 0 --CL
2
t-ll
sin - 2_0 -
&
---> [
I
'B as reI
(() 0)
e
--->
l
membering the periodicity of [(()),
all
6) _
aUI (r, oil
ll)
e; I~ 2] sin
(ex -
t-ll --T) I
a0 I
..
= _1
sm'2
Q is bounded for It - ()ol < a o' If re iB is sufficiently close to e iBo , that is, if ( < (0' then Q is bounded on the remaining parr of the interval a < t < a + 217 because the denominator in (6) is then bounded below. Thus, for « and all t, we have 1QI
a+2" \'
2~
~ a
(I (t)t -- I6 (llo) _ 0
XP(r t-a) ,
I' (a »)
au (r,
0
I
2r(t-llo)sin(t--ll) 1- 2r cos (t - 6) r9
+
dt,
We write this factor as follows:
< a 0 < 17/2,
00
ll)
_,aul
< Q0'
(r, 6) all
with the tan
(0
Therefore, we obtain from (5)
I.,;:: Qo -= 2~
(5)
Let us show that the las t factor in the integrand in this equation is bounded iBb for t € (a, a + 217) if re iB is sufficiently close to e and lies in an angle with vertex at e iB o and sides making angles, a 0' where 0 gent to the circle 1(1 = 1 at the point e,B o.
~
that is,
iB o
Now, from (4) and the same relationship applied to u (r, ()) we obtain, re
au (r,
I
QI~~,
f' (()
())/ a()
a;o~a;-S;7t-I1o,
(again the same sort of substitution), COllsequently,
e ,B o and extended as a periodic function outside that interval. From what we
au 1 (r,
- () 0 1 ::; a o and note that
(4)
•
Let us denote by u (r, ()) the function obtained from (1) if we replace [(t) with l the linear function [(() 0) + (t - () 0) 0) in the interval (a, a + 217), where e ai l have shown,
2
1i
' -t -26-0 I~ <:to) '-; 2 I t - 2 60 I =-; 2 sm , 2' tI o It' "-' 2' sm (Cto - 2 X I sm
Let us turn now to the general case. It follows from (1) that
2~ ~
e;'>O,
. (0o+~+-")
,
l ell -
a
ou(r, 6)=_1 \/(t)p(r, t-a) 1-2rcos(t
re lO I .
Furthermore, if we assume for the moment that
a
a+2" 1 C +~ ~ 1'(t)P(r, t-a)da=-AP(r, a,-a)+A.
all
(this last is obtained by replacing
aginary parr). Consequently,
a+2"
C I(t) ap (r, t - 6) dt
(6)
the absolute value of the expression as a whole with the absolute value of the im
function outside that interv_al. Then, from (1) we obtain a+2~
383
a+2" C I I(t) - 1(°0) -I' (a) IP(r t - a) dt ~ t -llo 0, • a
But Theorem 1 is applicable to this integral because
f' (() 0)
exists and hence the
first factor in the integrand approaches 0 as t,---> () o' Thus, this integral itself approaches 0 as (---> 0, that is, as re iB --> e' Bo in the angle in question. Therefore, by passing to the limit, we have , I 1m
ou a6(r,
ll) -11'm aUI (r, ll) =1' (a) all 0,
384
IX. BOUNDARY PROBLEMS FOR ANALYTIC FUNCTIONS
which completes the proof of the theorem.
repeating the same reasoning as above, we can prove
Theorem 3. If the value of the function f(()) in (1) at (j equal to (j=(jo:(d/d())rtf(t}dt=f«(j)18
0
= (jo
is finite and
Theorem 4. If the function a (t) in (7) has a finite derivative at () = () , -8 0 where < () 0"< 2TT, then u (r, ()) - a' «(j 0) as (, = re i8 approaches e' 0 along
°
then u(r, (j)-f«(jo) as (,=re i8
approaches the point e .8 0 along an arbitrary non tangential path from inside
any nontangential path in
1(,1 < 1.
1(,1 < 1.
Since aCt) has a derivative almost everywhere in (0, 2TT), we obtain, as be
Proof. Suppose that 1) 0 interval a
385
§2. REPRESENTATION OF HARMONIC FUNCTIONS
< (j < a + 2TT
-
< a < (j o'
2TT
We define the function F «(j) in the
fore, Corollary 2. The Poisson-Stieltjes integral (7) has almost everywhere on
by the formula
I(, I = 1 limiting values equal to a' «()) along all nontangential paths.
9
F (lJ) = ~ f (a} de,
All the theorems of this section are due to Fatou,
a
I
F' «() 0)
=
f(() 0)' If we now inte §2. T~e representation of harmonic functions
Ia+2,.
by means of Poisson's integral and the Poisson-Stieltjes integral
u(r, e)=2~F(t)P(r, t-e) a
a+2" __ 1 2~
\
~
n
at
~
2~'
tegra. The answers to these questions involve special classes of harmon.ic func tions ~which we shall introduce special notation. Specifically, we denote by
•
a+~
this section, we shall take up questions of the possibility of representing
harm nic functions by means of Poisson's integral and the Poisson-Stieltjes in
F(t) ap(r, t-O) dt=~P(r a-e)
a
who initiated the in
vestigation of the boundary properties of analytic functions.
and then extend it as a periodic function. Then, grate (1) by parts, we obtain
1)
X ~fOO~+~~~FOOP~t-~~
h , where p > 0, the class of functions u (r, (j) that are harmonic in the disk p
\(,1 < 1 and have the property that the integral 2"
~ III (r, e) IP de o
and Theorem 3 is obtained on the basis of Theorem 2. Since f(()) is equal almost everywhere in (0, 2TT) to the derivative of F«(j), is bounded for 0
Theorem 3 leads to the important
which holds for
Corollary 1. Poisson's integral (1) has almost everywhere on the circle
(1)
< r < 1. It follows from the obvious inequality 0< p' < P and all x ~ 0, that, if u(r, () E h p '
xP
,
< 1 + x P,
then u(r, ()) €
h p " so that the class h p is broadened with decreasing p.
I(, I = 1 limiting values e qual to f«()) along all non tangential paths. An analogous result holds for the more general integral, known as the Poisson
We begin with the representation of harmonic functions by means of a
Poisson-Stieltjes integral. To do this, we need as a lemma the following
Stieltjes integral
Theorem 1 (Helly). Every uniformly bounded infinite family {[(x)! of real 2"
u(r,
e)=2~) P(r,
t-e)drJ.(t),
(7)
functions that is of uniformly bounded variation
2)
in the interval (a, b) contains
a sequence that converges at all except possibly countably many 3) points in (a, b) to a func tion of bounded variation.
where a(t) is a function of bounded variation in the interval (0, 2TT), which can always be extended as a periodic function outside that interval.
1)
Specifically,
Proof. Obviously, it will be sufficient to consider the case in which all the 1) Fatou [1906].
1) The integral (1) can be regarded as the integral (7) with aCt) = f~ f(t)dt. This follows from the properties of a Stieltj es integral.
2) This means that the total variatioqs of all the functions in the interval (a, b) are bounded by the same finite number. 3) Helly' s theorem still holds even without this exception.
386
§2. REPRESENTATION OF HARMONIC FUNCTIONS
IX. BOUNDARY PROBLEMS FOR ANALYTIC FUNCTIONS
functions f(x) are nOildecreasing in the interval (a, b).
1)
Consider the set of all
By using a diagonalization process, we can select from the family {f(x)1 a
f k (x),
for k = 1, 2, .. "
Let us show that this sequence converges not only at these points but everywhere I'
u(r, g) belong,to hI' For u(r, ()
\'1 <1
to have such a representation in the disk
when the function a(t) is nondecreasing, it is necessary and sufficient that
that converges at all the points r l' r 2' .••.
in (a, b) except possibly at countably many points. Let
be representable in that disk by means of the Poisson-Stieltjes integral (7) of §1, where a(t) is a function of bounded variation, it is necessary and sufficient that
rational points in (a, b) and arrange them in a sequence r l' r 2' •••.
sequence
387
u (r, ()
? 0 in
I(I < 1.
Proof. 1°. If equation (7) holds for u (r, ()) in
denote a positive num
1(\
< 1, by representing the
function a(t) as a difference of two nondecreasing functions, we also represent
ber. We assert that the set of points x in (a, b) for which
the function u (r, ()) in that disk as a difference of two nonnegative harmonic func
lim !k (x) k-+oo
(2)
lim fix) :;::: e k-+oo
tions. ,Let us denote these by u 1 (r, () and u 2 (r, (). Then, for arbitrary r in the interval 0
< r < 1,
we have
is a countable set. If this were not the case, the set of these points would have 2"
a cluster point and hence it would contain a monotonic, let us say increasing, sequence of points x , where n = 1, 2, .... Between each pair (x ,x n
.
V
v+I
),
v = 1, 2, "', there would be a rational number r at which the sequence Ifk(x)! nv converges. By setting lim f (r ) = f(r ), we obtain k ~oo k nv nv
f (r n) - f
(r n._l)
=
-+00
a r «()
:;::: lim fk (x.) - lim fk (x.):;::: e, k-+oo k-+oo v
= 1, 2, .. '. Consequently, f(r
v
-->
00.
nv
)
> -
f(r
ni
) + (v + 1) 1', that is, f(r
nv
) _
00
f
all points for which (2) holds with different
f
v ' we see that
f k (x)
(r, 8) dB
=
21t (Ul (0)
+
U2
(0)),.
(3)
I ~=o = u (0).
Then it follows that
hI'
he Consider the function
''\It u (r, t) dt which depends on the parameter
r, where 0
< r < 1.
Since
2ll
l(lr(R)i~ ~ u(r, t)ldt~M o
as
a sequence of values I'v that approach zero and choosing
U2
0
Conversely, let us suppose that u(r, () €
"
But this is impossible because of the uniform boundedness of the func
tions f(z). By giving
u(r, ~) €
2~.
-+00
2"
+~
Her~ and in what follows, we shall write u (r, ()
k lim fk (r n._l)
k llm fk (r n) -
2"
~ I u (r, B) I dB ~ ~ Ul (r, 8) dB o ~
and n
n
k=I
k=I
converges
I
~ I(lr (B k ) - (lr (B k +1) I ~ ~
everywhere in (a, b) except possibly at a countable set E of points. The limit ing function f(x) is defined and monotonic on the set (a, b) - E. Consequently,
0=B 1
at points of the set E it has left- and right-hand limits. If we define the value
<8 <" 2
°*+1
I
~ u (r, t) dt ~ 21tM,
Ok
.
of the function f(x) at points x in E as the arithmetic mean of these limits, we
where
obtain a function f(x) that is defined and monotonic throughout the interval
o < r < 1,
is uniformly bounded and of uniformly bounded variation with respect
(a, b). Also, f n (x) -
to r for 0
< r < 2rr.
f(x) in (a, b) except possibly at countably many points.
This completes the proof of the theorem.
k
Regarding the representability of harmonic functions by means of a Poisson
2)
For a function u(r,
el
that is harmonic in the disk
1(1 < 1
to
1) This is true because every function of bounded variation can be represented as the difference of two nondecreasing functions and the total variation of each of these func tions does not exceed the total variation of the given function. 2) Plessner [1923].
a r «(), for
Consequently, if we take a sequence of functions a
Pk
«(),
1, 2, . ", Pk --.1, we can, in accordance with Helly's theorem, take a subse
quence a ,
P k
«()
that converges everywhere in (0, 2rr) except possibly at count-
ably many points to a function a«() of bounded variation. Let us now fix the
Stieltjes integral, we have Theorem 2.
=
M is finite and independent of rand (), the set of functions ,
point (= reif) and let us suppose that r
< P~ < 1
for k> K. Then, in accordance
with Poisson's ordinary formula we have, for k> K, 2"
1 \" u(r, B)=2 ~ U(Pk, t) 1t
0
pi/- r2
'9
h -
Pk cos (t - 6)+ r odt.
2' r
388
IX. BOUNDARY PROBLEMS FOR ANALYTIC FUNCTIONS
If we integrate by parts, we obtain
21CU (r, 6) = a pk (21C)
pi/ _
aQ.b f3 ~ aa + {3b.
Ph"-r'
2pi/ cos 6 -f..-. r·
~
-
t a
Pil ( )
iJt - '.
'2
<x
val 0
l
Ph - r '" ---"
=
(5)
b f3 xQ. - ax - (3b increases in the inter
since
-1»0.
~'(X)=~((~)~
Mdt.
o 00
1> (x)
To see this, note that the function
21t
--+
and {3 are all positive and a + {3 = 1, then
Proof. We note that, if a, b, a
\' a '
If we now take the limit as k we have
389
§2. REPRESENTATION OF HARMONIC FUNC'tIONS
and remember that ap'k (217) = 217U(0) = a(217),
Since 1>(b)
=
0, it follows that
1> (x)
~ 0 for 0
~ b; this proves (5) for a ~ b.
<x
By reversing the roles of a and b, we can prove (5) in an analogous manner for 2"
a ~ b. Let us replace a, (3, a and b in (5) with lip, l/q, a P and b q • Then, for
21CU (r, 6) = a (21C) P (r, 6) _ ~ a (t) aP (r'"t - 6) dt. Inverse integration by parts now yields formula (7) of
§ 1.
a, b, p and q all positive with lip + II q
=
1, we have the inequality
I I q .ab,,;::::-aP+-b -='p q •
This completes the
proof of" the first part of the theorem. The second pact is proved analogously.
(6)
, Now, since the functions Ig (x)! P anJ Ih (x)! q are summable on E, it follows Corollary 1. For a function u(r, e) to belong to the class hI' it is neces from''fM that g(x) h(x) is summable. Furthermore, from (6) we have sary and sufficient that it have a representation in I" < 1 in the form of the dif .
\
ference of two nonnegative harmonic functions. Proof. Suppose that u (r, h) E hI
Then, u (r, h) can be represented in
"1 <: 1
in accordance with formula (7) of §1. Consequently, as was shown above, it can
I g(x) I g(x)/Pdx)IIP
(" .,
1±
(
.
B
I h (x) I
~ I h (X) jq dx)llq
!'I
1.
=
use the generally accepted notation. Specifically, we shall write f(x) E LP on E and E is a linear space) to indicate that If(x)\P is summable on E.
Holder's inequal Hy. If g (x) E LP and h (x) E L q on E, where p > 1, q> 1, and IIp,+ llq = 1, then g(x)h(x) E L on E and
E
B
E
q
everywhere on E for some constant ,\.
E
B
t
P
+ (~ I h (x) IP dx ) liP. E
Proof. By using (4), we have
+ h (x) IP-l dx + ~ Ih (x) 1·1 g(x) + h (x) dx ~ ( ~ Ig(x) lP dxt ~ Ig(x) + h (x) jq(P-l) dx y;q B B + (~ I h (x) I dxY;p (~ i g(x) + h (x) Iq(P-t) dx y;q .
~ 1 g(x) +h (x) \P dx~ ~ I g(x) ,.\ g(x)
For p = 1, we shall write simply L instead of L 1.
P
A Ih (x)1
(S Ig (x) + h (x) IP dX) liP ~ (~ i g(x) lPdx
we need to have some auxiliary inequalities, which we shall now prove. We shall
I~ g(x) h (x) dx I~ (~I g(x) I d't; t P(~I h (x) jq dx )'Iq.
=
we have
of Poisson's integral and for many other questions that will concern us later ,on,
>0
x,
B
. Minkowski's inequality. If g(x) E LP and h (x) E LP on E, where p> 1,
To establish conditions for representability of harmonic functions by means
(where p
I h (x) Iq ) d
I
that is, we obtain inequality (4). It is easy to show that equality holds in (4) only when \g (x) IP
This follows from the theorem just proved and Theorem 4 of §1.
/P
B
Corollary 2. If a function u(r, e) belongs to the class hI' it has definit~
circle
I g (x)
\ ( I
~; PSI g (x) IP dx + q SI h (x) /q dx
be represented in the form of the difference required. The converse follows from (3).
limiting values constituting a boundary function u (e) almost everywhere on the
dx
E
B
I
B
P
(4)
(
P
B
l!
P- 1
(7)
390
§2. REPRESENTATION OF HARMONIC FUNOfIONS
IX. BOUNDARY PROBLEMS FOR ANALYTIC FUNCn-ONS
Since q(p - 1) == p, this leads after some ~mplification to inequality (7). In
U(T, e) =ffi
accordance with the remark concerning inequality (4), it follows that equality holds in (7) only when g (x)
=
(1l-C +C) '
Ah (x) on E for some constant A.
Using Holder's and Minkowski's inequalities, let us establish a test for the If a sequence of functions f n (x), for n = 1, 2, .. " •
defined on E approache~
a limit function f(x) almost everywhere on E and if
1'1 < 1, by Theorem 2 can be represented in
which is harmonic and nonnegative in the form (7) of
possibility of taking the limit under the Lebesgue integral sign. Specifically,
§ 1.
,=
1'1
But its limiting values on the circle
everywhere except at the point
form (1) of §1, we would have f(t)
=
1 are equal to zero
1. Therefore, if it could be represented in the 0 in (0, '27T) in accordance with Theorem 1
=
of §1 and hence we would have u (r, e) == 0 in do have 1)
~ IIn (x) IP dx ~ M,
1'1 < 1,
which is impossible. We
E
where p
>1
and
M is
391
Theorem 3. For a function u (r, 0) that is harmonic in the disk
finite and independent of n, then
lim ~ In (x) dx
n-+co E
1'1
have a representation in the disk
= EV(x) dx.
(8)
<1
I"
<1
to
by means of a Poisson integral (1) of
§1, where f(t} E LP, where p> 1, in (0, 27T), it is necessary and sufficient that u(r, e) E h . P
Proof of the sufficiency. If u (r, 0) E h , where p> 1, then u(r, 0) E h and, in
Proof. In accordance with a familiar lemma of Fatou, we have the inequality
P
accord¥ce with Corollary 2 to Theorem 1, we conclude that u (r, e) has almost
~ I/(x) IPdx ~ lim ~ IIn.(x) IP dx,
E
everywhe\e on the circle
n .... coE
I"
=
1 limiting values along all nonrangential paths, and
partic~ar along the radii. Furthermore, these limiting values define a function u (0) E LP 'because, by Fatou's lemma,
in from which it follows that f(x) E LP on E. Furthermore, by applying the inequal ities that we have proved, we see that, for an arbitrary set e C E,
I~ (/n (x) e
I
2~
....
I(x))dx ~ ~ lin (x) - I (x) I dx
~ I u (e) I o
~ (~l/n(x) - I (X) IP dXytP (mes e)llq
2",
1
e
U (T,
_
e) -
IP dxytP
+ (~I/(x) I dx),IP] ~(mes P
e)llq.
I ~ 211
p2_r 2
u (p, t) P2 - 2pr cos (t - 6) + r 2 dt,
Since the fraction in the integrand is bounded as p
2M 1IP,
---->
T
<. p < 1.
(9)
1 and since
2",
e
)~ I U (p, e) IP de ~ M o
By making mes e sufficiently small, we can make the right-hand member of this inequality arbitrarily small independently of n. And this, in accordance with a
for 0
familiar test for the possibility of taking the limit under the Lebesgue integral
a result, we obtain
sign, yields (8).
< P < 1,
we may take the limit in (9) under the integral sign as p
---->
1. As
2",
Let us now turn to the representation of functions that are harmonic in 1 by means of Poisson's integral (1) of
§1.
For example, the function
I~ <
§1. We note first of all that not every
function u (r, e) that can be represented in the form (7) of sented in the form (1) of
~ lim ~ I U(T, e) IP de. ' .... 1 0
Consider Poisson's formula
e
~ (mes e)llq [(~ IIn (X)
2~
P de
§1
can also be repre
u~~=~~UOOP~t-~~ 1) The question of the representability of u(r, (J) by means of Poisson's integral (1) of §1 with f(t) € L has also been solved, but the complete result is complicated to for mulate.
:::~~'~::':'_."_"~ •• ~.•
-_.
1(1
< 1; that is, we have the representation (1) of §1 with f(t)
=
u(t) €
=
... ~="'.o=w.,",,,,,,,,,,,,,,·_~k~'~~~~""'''-
393
2"
~ I v (r, B) 19 dB = ~ I u (r, B) 19 dB - 21t (ag - b~). o 0
with f(t) E LP, where p> L Then, by applying Holder's inequality (again,
lip + l/q
~~~~.". ,,,,,,-';~-_,w,,,,,,,,,
2"
LP.
Proof of the necessity. Suppose that equation (1) of §1 holds in the disk
1(1 < 1
,.
§3. LIMITING VALUES OF ANALYTIC FUNCTIONS
IX. BOUNDARY PROBLEMS FOR ANALYTIC FUNCTIONS
392 in
,<.__
In conclusion, we note that the representation (1) of §1 is simply Poisson's formula
1), we have
2"
211:
~
21'
lu(r,
B)IPdB~ ~ {2~ ~ 2"
=
2"
(10)
t-B)dt,
1(1 < 1
which gives the value of a harmonic function u(r, 0) in 1/(t)lpl/p(r, t-B)·pl/q(r, t-B)dtY dB 2"
~ ~ 2~ ~
in accordance with Theorem 3 of
2"
1/(t)IPP(r, t-B)dtdB=
~
in terms of the
limiting values of u (t) along nontangential paths. This follows from the fact that, the circle
1/(t)IPdt,
1(\
=
§ 1,
we have f(O)
=
u (0) almost everywhere on
L Therefore, Theorem 2 can also be regarded as giving a suffi
cient condition for applicability of Poisson's formula (10) to u(r, 0). This condi
I( I < L Consequently, u (r, 0) € hp • This completes the proof of the theorem.
tion is expressed in teems of the membership of u (r, 0) in the class hp , where >1.
Finally, let us pose the question: What conditions must a function u (r, 0)
1(\ < 1 satisfy for its harmonic conjugate v(r, 0) to have a representation in accordance with Poisson's formula in 1(1 < I?
§ 3.
that is harmonic in the disk
The limitin!5 values of analytic functions
Using the preceding results regarding harmonic functions, we can easily de rive a very simple result regarding the existence of limiting values of analytic
A sufficient condition can be obtained from Theorem 2 and the following
functions.
theorem. Theorem 4.
B)=2~~· u(tjP(r, o
2"
~ (2~)P[~
2"
in
u(r,
1/(t)IP(r, t-B)dtr dB
1)
If u(r, 0) €
h p , where p> 1, then v(r, 0) €
hp .
"
The proof of this theorem is rather complicated if p 1= 2. Therefore, we shall not present the proof in the general case.
2)
In the case p = 2, it follows easily
= ao+ ~ 00
ex:>
+~
(r, B) = bo
is regular and bounded in
Then, its real and imaginary parts represent bounded harmonic functions
n bnr sin nB),
n
(bnr cos nB
+ anr
sin nB),
n=1
f«()
I(I
=
1 as (approaches
also has definite limiting
values along nontangential paths almost everywhere on that circle. These limit ing values define a function on
n
I(I < 1. in 1(1 < 1
which belong, in particular, to the class hi' In accordance with §2, these func this circle along nontangential paths. Therefore,
(anr n cos nB -
n=1
V
f«()
tions have definite limiting values almost everywhere on
from Parseval's formula applied to the expansions u (r, B)
Specifically, suppose that a function
1(\
= 1, which we shall denote by
f«()
or by
f(e i8 ) where (= e i8 . This very simple result can now be generalized to the class N (introduced
which constitute the real and imaginary parts of the Taylor expansion ex:>
by R. Nevanlinna) of functions
f«()
u+lv= ~ (an+lbn)~n.
Cf'
Specifically, in this case, we have We can always assume that ¢ unity in the disk 2) A proof can be found, for example, in the book by Zygmund [1959).
1(\
< 1 and that
(C) •
/(q=~ (C)
Ii=O
I)M. Riesz (19271.
that are regular in the disk
can be represented in that disk in the form of the ratio of two bounded functions:
«()
and
lfr«()
(1)
are bounded in absolute value by
1(\ < 1.
Another way of characterizing the class N is on the basis of Nevanlinna's theorem:
394
§3, LIMITING VALUES OF ANALYTIC FUN\jTIONS
IX. BOUNDARY PROBLEMS FOR ANALYTIC FUNCTIONS
Theorem 1. For a function f«()
i
0 to belong to the class N, it is necessary
Cf'r(Q=
and sufficient that the integral +
2"
~ log If (re i8 ) I dB o
be bounded by some constant M for 0 Proof.
1)
If the function f«()
i
~r (C) = e
< r < 1.
+
rk
2"
o
cm(m + c
(3)
0
2"
mt1
(m t 1
1: d1t ~ log I ~ (re ) I dB= log I em I+ o iO
log I ~k I
is over all the zeros of 0/ «() that lie in 0
0/(0
in 0
< I" < 1
(4)
< I" < r. Since the right-hand mem < r < 1, the right-hand member....
1 in that disk. If we take a sequence of numbers r k in the interval 0
that approaches 1 as k ---+
00,
1'1< 1
to a regular function 0/ «(\Also,
I"
¢ «()'t~!_~s!~gular in proof of the theorem.
I"
< 1;
also
11> «()I
From the representation (1) for the functions f«() r
< 1,
k
logf(Q=
1:
log
ICkl
0 in the class N for 0
2"
0 is such that the integral
= -
Then, in accordance with the Jensen-Schwarz for"
2"
~ log I ~ (re iO )[ dB,
~ log I Cf' (re iO ) 1 dB o
0
and, as in the first part of the proof of Theorem 1, the last two integrals do not
rr~C~r::~)
+ 2~
2"
~
iO log If(re ) \
decrease in the interval 0
'0
~~;o~; dB +lC,
~r
Therefore, not only the integral (2) but also
2"
0
f(Q= OPr (C)
the integral
~ I log If (re iO ) II dB o
where C is a real constant. This can be represented in the form (5)
(C) ,
where we set
for 0
(6)
is bounded by a finite quantity independent of r. This property of
functions belonging to the class N will now be of use to us. Theorem 2. If a function f«()
i
0 belongs to the class N, it has definite
limiting values f(e i8 ) almost everywhere on the circle
''I
gential paths. Furthermore, [logl f(e i8 )11 is summable on 1) The proof gi ven here differs only slightly from the proof of the more general Theo rem 1 of §S of Chaprer VII.
<
+ ~ I log I ~ (re iO) 11 dB
0 0 0
mula, we have in
i
k
o < r < 1.
to some function
we have k
< r < 1.
<1
$ 1 in I" $ L This completes the
of (3) is a nonincreasing function ofr and, consequently, is bounded above in
(2) is bounded for 0
<
we can, on the basis of the condensation
~ [log If (re iO ) II dB ~ ~ [log I Cf' (re iO) II dB
i
also I¢r«()! $ 1 and
Io/«()I $1 in < LSince the values of Io/ rk (0)1, for k = 1, 2, " ' , are bounded below by a positive quantity independent of k, it follows that o/«() i o. It fol
ber of (4) is a nondecreasing function of r in 0
Conversely, suppose now that the function f«()
< r;
principle, choose a subsequence 10/ rk' «(H from the sequence. 10/ r «(H that con
and the sum
This shows that the integral (2) is bounded in the interval 0
I"
lQ,~ from (5) that the sequence I¢rk' «(H converges in '"
+ m log r.
O
Here, the (k denote the zeros of the function
<1
verges uniformly inside the disk
+"', where m 2: 0, then in accordance with Jensen's formula as applied to the function o/«()/(m, we have, for 0 < r < 1,
=
reiO_C dC
0
The functions ¢/() and o/r«() are regular in
10/ r «() $
~ log\f(reio)ldB~-- ~ logl~(rei8)ldB.
Now, if o/«()
l'0 1 r + -2it ) log It(re iO )(e +C
0 belongs to the class N, that is, if it can be
then, since If«()1 $ 1/10/«()\ in that disk, we have 2"
2"
represented in the form of the ratio (1) with I¢«()I $ 1 and Io/«()I $ 1 in the disk
1'1 < 1,
1 11"
-2it~IOg+1 _IiO ) I~d8+iC reiO+C 0 f(re
r(C-Ck) I _ e
ICkl
.
(2)
n
395
=
1'1
1 along all nontan = 1.
Proof. If a function f«() E N, not identically zero, is bounded in
1'1 < 1,
396
IX. BOUNDARY PROBLEMS FOR ANALYTIC FUNCTIONS
then, as was noted above, it has almost everywhere on
It::" 1;0
§3. LIMITING VALUES OF ANALYTIC FUNCTIt>NS
1 definite limiting
values f(e i8 ) along all nontangential paths, and, in particular, along radial paths. In accordance with Fatou's lemma, we have ~3
23
~ I log II (e I6 )
o·
In accordance with what
W!lS
II dO ~ lim ~ I log II (fe i6 ) II de.
It: "1 ;0 I. 1>(t::")
and
Suppose that the sequence
r/J(O
~ 1 definite limiting values f(eiB).If we again apply Fatou's
"1;0
The finite limiting values of the function f(t::") €
I.
log
If we apply this inequality to the functions
N along nontangential
As a consequence of Theorem 2, we have the following uniqueness theorem.
< r,
f n (t::")
(9)
and use inequality (8), we ob-
loglln(~)I:o;;;;:; r+!:! M. If we fix
t: "
in the disk
It::" 1< I
\t::"1 < 1. Obviously f(O = [, (t::") - [/t::")
equal in the disk
€ N. If [(01=0 in It::"\ < I, then, by Theorem 2, Iloglf(e i8 )11 is finite almost everywhere on I" = 1; that is, f(e iB ) is finite and nonzero almost everywhere on I" ;0 1. This contradicts the condi tion according to which [(e i8 );o 0 almost everywhere on E (except at points at 00).
In connection with the boundary values of functions in the class N, we also have the following convergence theorem:
and let r approach I, we obtain the inequality '+1 CI M
Theorem 3. ') If two functions [, (t::") and [/0 € N have the same boundary values on a set E o[ positive measure on the ei~cle 1t::"1 ;0 I, they are identically
;0
I"
tain the result that, in such a disk
values.
[2(t::")
we obtain in any disk
I/(~) I ~ 2~ ~ log I/(fe i6) I ffi (~:~:~~ ) dO.
paths, which exist almost everywhere on It: " I ;0 I, will now be called its boundary
;0
< r,
23
\\.""..
This completes the proof of the theorem.
which [, (t::")
converges at all points belonging to a subset
terms o~ the right are nonpositive for where 0 \. r < 1,
have almost everywhere on
lemma to the integral (6), we conclude that jloglf(e i8 )11 is summable on
Proof.
I" I"
(8)
= 1. Then this sequence converges uni formly in the disk 1t::"1 < 1 to a function belonging to the class N.
I definite limiting values along nontangential paths and these limiting
It: "1
I[n (01
E of positive measure of the circle
values are nonzero almost everywhere. But then the function f(t::") has almost everywhere on
+
~ log lin (fe I6 ) IdO ~ M. o
Proof. If we take the real parts in formula (5) and note that the first twO
is an arbitrary function belonging to the class N, then in
its representation (1) the functions
"I =
23
said regarding the integral (6), the right-hand member
are almost everywhere nonzero on
f(O 1= 0
the interval I" < 1. Suppose that there exists a single finite constant M such that for every n and for 0 < r < I,
(7)
Consequently, /logl[(e i8 )!1 is summable on I" = 1. But then, the values of log If(e iB ) 1are finite almost everywhere on "I = I; that is, the values of [(e i8 ) Now, if
Theorem 4.1) Let I[n (t::")1 denote a sequence of functions that are regular in
r-1o
of this inequality is bounded.
397
lin (Q 1:0;;;;:; e'-ICI This inequality shows that the functions disk
I" < I.
f n (t::")
are uniformly bounded inside the
\t::"\ < 1.
Let us suppose now that they do not converge throughout
I"
Then, just as in the proof of Vitali's theorem (§2 of Chapter I), they contain two subsequences I [n'k(t::")l and Ifn"k(t::")!, for k;o 1,2"" that converge in < I to distinct functions that are regular in that disk. By hypothesis, the sequence of
"I
the differences [k (0 converges in
;0 fn' k (0 - [nil k (0 .converges on the set E to 1>(0 -1= O. Furthermore, since 2)
zero and it
to a function
1) This theorem was proved by Hin~n [19241 for uniformly bounded functions and later generalized by Ostrowski. 2)This is true because, for arbitrary positive numbers a, and a ,
'{-
1) This theorem is a special case of a general theorem of Privalov, which will be
treated in §2 of Chapter IX.
log (a 1
+a
+
f)
+
+
2
~ log (2 max IJ.k) = max log (2ak) ~ max (log a.k k='.2
k=I.2
+ ~Iog 12 1
k=I.2
+ 199 a. + log+. 2. +
D
+ log+ 2)'
398
§3. LIMITING VALUES OF ANALYTIC FUNc"hONS
IX. BOUNDARY PROBLEMS FOR ANALYTIC FUNCTIONS
+
+ Jog+ lin. (Q I+ log+ 2,
+
log Ilfk (Q 10;;;;; log lin'k(Q I we obtain from condition (8), for k
The complete answer to this question' is given by the following theorem of
k
1, 2, .,. and 0
=
Ostrowski.
< r < 1,
Theorem 5. For a function f(I;,) that is regular in the disk 1I;, \ Blaschke function, it is necessary and sufficient that the integral
2"
i", ~ l~g ICfln (re/a) \ dB 0;;;;; 2M + log 2.
(10)
On the basis of Fatou's l~mma, we conclude from this, in particular, that ¢(I;,) €, we have a
+
=
2a - 1a I, it follows that, for II;, I < r, 2"
I \
+
(re
/8
l8
n
(1;,). Since for arbitrary real a
~ log 1I (re i8) IdB
+ C)
2"
-
i8
(re/
8
be bounded for 0
Then, for any fixed I;, in
1(,\ <::
[We point out that, in accordance with Jensen's formula,
1(,1 < 1
~ the form
\
+ C)
2"
+ log 2) - r-ICll \ r + I CI 2", ~ \log 1Cfln (re
18
.
f(I;,)
) II dB.
the disk I(, I
<1
where m = 0 if "[(0) 2"
b (I;,)h (1;,),
t-
(Q = ( ;
r
Cflr (~) hr (Q,
0 and m = multiplicity of the zero z = 0 of the function
f(I;,) if f(O) = 0 and where
. ICfln (e) lidOu. + log 2) - 1I-ICII + ICI ~ \J \log o /8
II«rrr~C-- 5ek
k
Cflr (Q =
00
in this inequality, the integral on the right approaches
00
2" lr
by virtue of Fatou's lemma because ¢(e/e) = 0 on E and mes E > O. This shows that ¢
n
(I;,) -
0 in the entire disk
earlier about ¢
(1;,). Thus the sequence Ifn (I;,)J converges uniformly in the disk
1(,1 < 1. This completes the proof of the theorem.
We now consider another very important question associated with the zeros
1(,\ < 1. The
question arises whether there exists a function b (I;,) that is regular in the disk
1(,\ < 1, equation disk
Ib (1;,)\ ~ 1 in that disk, that satisfies the 1 almost everywhere on the circle 1(,1 = 1, and that has in the
that satisfies the inequality
I b (1;,)1
=
1(,1 < 1 the same zeros and of the same multipliCity as f(I;,). If such a func·
tion b(I;,) exists, it is called the Blaschke function for f(I;,)·
III
'8
re'
+l
re
-c
211: J logl!''''''';lI ~d8
1(,1 < 1. But this contradicts what was said
of analytic functions. Let f(I;,) denote a function that is regular in
) ,
C
0<'1 Ck
If we let n approach
(12)
1(,1 < 1. In accordance with formula (5), we have in
I
using Fatou's lemma,
=
b(I;,)~.gular in 1(,1 < 1 and Ib(I;,)1 ~ 1 in 1(,\ < 1 ana where h(I;,) is
regular and has no zeros in
1, we obtain, letting r approach 1 and again
(')1 0;;;;; II+ICI 1og I Cfln" _I CI 2 (2M
< r < 1.]
sary and sufficient condition for the function f(I;,) to have a representation in
where
r+ CI (
< r < 1.
the integral (11) is a nondecreasing function of r for 0
Il og ICfln (re ) II ffi refS _ C dB
0;;;;; r _II CI 2 2M
(11)
o
Proof. First of all, let us show that the condition in the theorem is a neces·
log ICfln (~) 10;;;;; 2 . 2", ~ log 1Cfln (re ) 1ffi re/8 _ C dB
I \ 2", ~
< 1 to have a
2"
o Let us now apply inequality (9) to the functions ¢
399
h;(Q=e'
0
.
1(,1 < r we have: ¢r(I;,) is regular and l¢r(I;,)\ < 1, and also h,
00.
quence
I-t.'f' r k (I;,)J
contains a subsequence that converges uniformly in the disk
[1;,\ < 1 to a regular function ¢(I;,). Here, either ¢(O) = 0, in which case ¢(I;,) _ o in I(, I < 1, or ¢ (0) t- o. 'In the firs t case, the corresponding subsequence of the functions h
rk (I;,) approaches
00
in II;,I
<1
and, in particular, at I;, = 0, which
means that the integral (11) is not bounded for 0
< r < 1.
In the second case, on
the other hand, this same sequence converges to a regular function with no zeros in
1(,\ < 1, and then the in tegral (11) is bounded for 0 < r < 1.
I ,_ ~ .. _ •• _ ~ 1. ~
_t
, 11.- .- L =__ J..- _
I ~L
~
_~.
400
IX. BOUNDARY PROBLEMS FOR ANALYTIC FUNCTIONS
1('
This shows that, if the integral (11) is bounded, we have in
<1
21<
2=
0
Here, in accordance with (13), we have b(m)
< r < 1,
+~
I bn <01 iO
log I h (re ) I dB
~ 21t log 1h (0) I;
1b «)I
f«),
then we have the representation
where the function
h<0
It remains to show that the diS~"
1(' < 1
and has no zeros in
«)
I
< 1,
has a Blaschke function.
If the function ¢«) i 0 is regular in 1(1 < 1 and I¢«)I ~ 1 in '" < 1, then when ¢ <0 has only finitely many zeros in that disk, the representation (12)
in
I"
\
< 1,
1(1 < 1.
«)
other hand, if there are infinitely many zeros, (1' (2' .. "
1(1 1 ~ 1(2 1 ~
in
'I (1<
~
\
'f
iO
) I~
1 21;
S
I
I-i-
Ibn <eit)1 2~
(13)
k
o
2~
== 1, we obtain
I
ia l' b (re ) Jb (reIO) n
00
with n. We
'(I <
it
(15)
0
1(1 = r,
where 0
< r < 1,
to
2"
~ Ibn «)I ~ 1
~ I b (ell) I dt ~ 21t, o
i
contains a subsequence that converges in
0 that has the same zeros as ¢
«)
1(' < 1
2"
~(l-lb(eit)Ddt~O.
and satisfies in
o
1 the inequalities ",
'l'
•
2"
IdB:=:;; Jib r (e )ldt.
that is,
I bn «)I
to a regular function b «)
'(I <
and keeping in
b <0· Therefore, if we take the limit under the integral sign in (15) as r __ 1, we obtain
1
Now, the sequence
(14)
g
But the sequence Ib n ~)l conl'erges uniformly on
approaches 1 uniformly as (approaches the circle
I( \ =1- Considering the function ¢ «)/ b n <0, we easily arrive at the inequalities
in
'a C=re',
g (e it ) 1 P (r, t - B) dt.
By applying inequality (14) to the function mind the fact that
and we denote by bn <0 the finite product obtained by replacing
I¢«()I
is regular and bounded in
211
o
Ibn «)\
1'1 < 1.
~ fg(reiO)ld/t~~ Ig(eit)ldt. o 0
k=l
note that the function
0 in
so that
indexed in such a way
IT IC I I-fktek 00
b (C)=Cm
g«)
g(~)=~ ~ gee' )P(r, t-B)dt,
1. On the
we form the infinite product
••• ,
h«() f,
b <01 = I almost everywhere on I( 1 = 1. .
2"
that
Having the function (13), we imme
Integrating with respect to () from 0 to 2", we obtain
I-eke'
where the product is over all the zeros of the function ¢
~
§2 I
I g(re
,
Ib * «)I
2~
is immediately obtained with
b(C)=cmll r-~k
then, by
,
that disk. To prove the sufficiency of the condition in the theorem, it remains, !!os before, to prove that the bounded function ¢
''I < 1 we have the inequalities
that is, we have b* «() == b «(). This shows that the
To do this, we note first that, if the function
is regular in
If the sequence
then, in addition to b <0, there would
< 1,
diately conclude that there exists the representation (12) with
f«) = b
Ib«()! ~ 1b* «)I;
and
1(1
= II;:1 I(kl.
with these properties. Then, in accordance with what
. infinite product (13) must converge in
It follows, in particular, that the condition in the theorem itself is necessary
in
«)
was said above, it follows that in
< r < 1
because, if b «) is a Blaschke function for
itself'did not converge in
be another function b *
0
that is, the integral (11) is bounded for 0
401
I¢ «) I ~ Ib
the re
2=
~ lo~ If (re iO ) I dO = ~ log I b (re iO ) IdB
o
FUNCTI~NS
:~".
quired representation (12) <where b <0 = (m¢ (0). Conversely, if (12) holds, then, for 0
§3. LIMITING VALUES OF ANALYTIC
~
But it follows from the inequality' b(OI
::; 1
in
1(' < 1
(16)
that' b(eit)1 ~ 1 almost
402
IX. BOUNDARY PROBLEMS FOR ANALYTIC FUNCTIONS
everywhere on the circle almost everywhere on
1(1
1(1
= 1. This, together with (16) shows that \bCe
§4. BOUNDARY PROPERTIES OF FUNCTIONS it
)! = 1
= 1. This completes the proof of the theorem. In particu
lar, since the boundedness of the integral (2) in 0
implies boundedness of
this function,'we have in
By integrating this inequality with respect to
Now, let us denote by Hp , for p > 0, the class of functions fCO that are
1(1
regular in
<1
2n
and have the property that the integral
~ II ("re i6) IP dB
for arbitrary p> O. Furthermore, the inequalities
P and X? 0 and the inclusion relation fCO €
1(1 , < 1P belong to the P x < x + 1, 0 < p' <
Hp imply that fC() €
\(1 < 1,
I"
(2)
•
where bCO is a Blaschke function and hC() is a function in Hp
that is nonz ero in
< 1.
For arbitrary fixed p in the interval 0
< P < 1,
fpC() I) F. Riesz [1923].
=
bpCOhpC(),
< r < 1. If
we'now replace r with
pi I p, where pi < p, we obtain an inequality proving that the integral (1) is non decreasing in 0 < r < 1. Turning now to .!,he proof of the theorem, we note that, in accordance with the representation (2) holds in the disk regular and has no zeros in
1(1
<1
with the function
hC(),
t§3,
which is
1(1 < 1. Let us show that h C() € H .
Let us denote by b n C() the product of the first n tactors in the representa tion (13) of §3 of the Blaschke function
bC()
and, for fixed ( in the interval 0
( < 1 and fixed n, let us choose T/ > 0 such that Ibn C()I > 1 - ( for which it is possible for us to do. For 1 - 1/ < r < 1, we have 2"
~\ Ifbn(re(e i8)) [P de ~ "
the function
fCp() is regular in I( 1:::; 1 and consequently has in 1(1 < 1 the repre
sentation
0
which is now proved, for arbitrary rand p in 0
Proof. Let us show first that the integral (1) is a nondecreasing function of
< r < 1.
2"
1/(rpe i6) I P de ~ ~ 1I (pe t6 ) IP dB,
P
Hp' for p > 0 and if fC 0 ~ 0, then
fC() = bC()hC(),
~ o
Suppose thattheleast upper bound of the integrals (1) in the interval 0 is equal to M.
P
If fC() €
2"
+
With regard to functions in the class H , we have a number of theorems. 1)
(3)
accordance wIth 0), we have a fortiori
H pi; that is,
H cH, for all p and p', where 0 0 and x 2: 0, we conclude that fC() € N if fCO € Hp ; that is, HeN for all p > O. It follows from this last p fact and §3 that a function fC() € Hp , for p >0, has, almost everywhere on the circle \(1 = 1, definite limiting values along nontangential paths and these form a limit function fCe iB ). In accordance with Fatou's lemma applied to the integral (1), by letting r approach 1 we conclude that the function fCe iB ) € LP in CO, 211). Theorem 1.
2"
BU~i,~ce IbpCOj:::; 1 in \(1 < 1 and IbpC()\ == 1 on 1(1 = 1, it follows that If~C()1 :::;~COI in .1(1 < 1 and IfpC()1 == Ih/Ol on 1(1 = 1. Consequently, in
< r < 1.
Obviously, all functions that are regular and bounded in P
0 to 211, we obtain
(1)
o is bounded for 0
e from
~ I hp (re i6 ) IP dB ~ ~ I h p (e i6 ) \P dB. o 0
2n
=
Ihp(eit)IPP(r, t-e)dt.
o
§4. Boundary properties of functions in the class Hp
f pC()
1(1 < 1
\hp(reiB)IP~2~ ~
function.
r in 0
403
p
2n
Corollary 1. Every function fC() ~ 0 belonging to the class N has a Blaschke
in
H
where bpC() is its Blaschke function and h pC() .;,. 0 in 1(1 < 1. The function [hpC()]P is regular in 1(1:::; L Consequently, from Poisson's formula applied to
the integral (11), we obtain
class H
IN CLASS
1(1 > 1 -
i6
M
,n.
But since this integral is a nondecreasing function of r in the interval 0 inequality (4) also holds for 0
o < r < 1.
< r:::;
<
T/'
(4)
< r < 1,
1 - T/, that is, throughout the entire interval
If we fix r and let n approach 00, we obtain, remembering the arbitrari ness of ( > 0, the result that, in 0 < r < 1,
..,=oooo-=:--,,.
~--==-~-=-===---=-"'===:--=.~. __-=====--===_==-=~"._-=====~_~~=-==o-=-===.= ---:::~_·~=·=-~~_---:-=-~~:-:",=o::=:::~:,---::::::·:~::.:..",==.,..,.-::~-_-":",,,,,o,-o"·==~~:-::~':-_·:::;:::::_:;::=,.."c--=:~-:;",::o-":,:::,:::,:;",:;::,-~~----"::...::....,··
404
405
§4. BOUNDARY PROPERTIES OF FUNCTIONS IN eLASS H p
IX. BOUNDARY PROBLEMS FOR ANALYTIC FUNCTIONS
2"
I h(reIB)IPda~M.
o H and, in fact, that the least upper bounds of the inte P grals (1) for fC') and hC') in the interval 0< r < I are equal. This completes
·2"
~ I/(re/~ IP da ~ ~ I/(e,e) IP da. o 0
2x
~
=.:::-.;:.:::.....,.,,.,....._"""'==_=--"""~,;':;~-:~_-::=_:_~~.:::.::_-~~~-"c_=_-_="",~==,",.=,::=·
(9)
Therefore,
This proves that h C,) €
2><
2"
lim ~ I/(reiB)IPda~~ /1(eiB)\Pda. r-1o
(10)
0
the proof of the theotem.
Theorem 2.
2)
If fC') €
[The existence of the limit on the left follows from the monotonicity of the integral (1).]
H , where p> 0, we have P
2"
2"
lim ~ I/(re' ) IP da = ~ B
r -10
II
(e
iB ) \P
de
On the other hand, by applying Fatou's lemma to the integral on the left in (8),
(5)
we obtain the opposite inequality. Therefore, equation (5) holds.
0
To ~ove (6), let
and 2"
lim ~ I/(reiB)~/(e'B).lP4a=O. r-I
(6)
o
l
denote any positive number. Th~n, let us choose 0> 0,
such that, or any set e with mes e < 0 we have Ie IfCe,B)! P dO < l in the interval CO, 217). Sin e IfCreiB)1 - ..... IfCeiB)1 almost everywhere in CO, 217) as r -+ 1, by taking an arbi y sequence of numbers r < 1 that approaches 1 as n -+ we " conclude on the basis of Egorov's theorem that there exists in CO, 217) a set e 1 such that mes e 1 < 0 ac.d IfCr eiB)1 -+ IfCeiB)1 uniformly outside e as n -+ " 1 Therefore, if we deno{e the complement to e 1 with respect to CO, 217) by E, we 00,
Proof. It will be sufficient to assume that fC') ~
o.
If we represent fC') in
accordance with formula (2), we conclude from Theorem I that the function
00.
[hC,)]p/2 € H 2. Consequently, by virtue of Theorem (2) of §2, the real and imag inary parts of this function can be represented in \(1 < 1 ·itt terms of their limit ing values by means of Poisson's integral. But then, the function [h C,)]P /2 itself wilrhave such a representation in "I
= 2~ .~.
lim
< 1:
~ II (r "e iB) IP de = ~ II (e IB)
,,-00 E
2"
hP/2 (re iB)
have
hP/2 (e/ i ) P (r, t -
E
(7)
iB
~ I/(r"e ) IP da= ~ I/(e' B)
"_00 el
In accordance with the Bunjakovskil-Schwarz inequality, this yields
o
2"
per, t - a) dt,
Ia -
o
b IP ~ ( I a I
e1
+ I b I )P =
2"
~ I h (re/ B) IPda ~ ~ 1 h (e lt ) IP dt. o 0
But since
IfC')1
~
jhC')1
in
\(1 < 1
and IfC')
we obtain from (8), for 0< r < 1,
=
IhC,)!
almost everywhere on
l.
Therefore, there exists a
for
(max (21 a I. 21 b I»P
I b IP) ~ 2P (i a IP + I bill),
for arbitrary complex numbers a and b, we have (8)
2"
~ I/(r "e
iB
) -
o
I (e iB) IP da ~ ~ I/(r"e tB ) -
l(e iB )1 P da
E
+ ~ 2"( I/(r"e/BW + I/(e iB) IP) de ~ 2~e +"2 P2e = el
1) F. Riesz [1923].
l
"
= 2 max (j a Ill,
so that
1'\ ~ 1,
IfCr eiB)IP dO <
P
2"
(12)
n> N. Now, let us choose " N sufficiendy great that we also have the inequality IfCr e,B)-fCe,B)IP N. Then, since positive number N such that
t - a) dt]9
~ ~ I h (eitH'
f
jP da.
el
But the right-hand member of (12) is less than
2"
a) . pl/2 (r,
(11)
It follows from (5) and (11) that
a) dt.
lim
I h (re/B) IP ~ [~ I hPI2 (e il ) I PI/2 (r, t -
IP da.
(2~
+ 2 +I) e. p
(12')
§4. BOUNDARY PROPERTIES OF FUNCTIONS
IX. BOUNDARY PROBLEMS FOR ANALYTIC FUNCTIoNS
406 Since
I:
2"
If (r neiB) - f
B (e/ ) IP
d6
= O.
second of the integrals on the right, the integrand approaches zero almost every
Also, because of the arbitrariness of the sequence
Ir n I,
where in CO, 217) and at the same time it does not exceed the function MIfCeie)1
equation (6) follows.
for some finite M independent of p. Therefore, we can take the limit under the in
This completes the proof of. the theorem.
sentation in
tegral sign in this integral as p
that fCO E HI' If this condition holds in
1'\ < 1,
of the theorem.
(13)
f(eit)P(r, t-6)dt.
Proof. If fC') can be represented in the disk
1'\< 1
Theorem 4.
1)
If fC') €
Hp , where p > 0, then, in the representation (2), we have Hp . Consequently, Poisson's formula (13)
Proof. If fC') €
by the formula
hC/J €
2"
¢ Ct) E L for
t
2"
211:
2"
o
0
.
hP(C)=2~ ~
CO, 21/.), then it easily follows that, for 0 < r < 1, €
Hp ' where p > 0 and if IfCeie)l $ M almost every
where on 1(1 = 1,. then IfC')1 ::; M in 1'1 < 1. Furthermore, if fC') € Hp ' where p >0, and if fCe,e) € Lq, where q> p, then fC,)€ Hq•
f(C)=in~ ~(t)P(r, t-6)dt, is applicable in 1(1 1(1 < 1 and IfCeie)1 to the inequality
~ Ij(re iB ) I d6 ~ ~ I
that is fCI;) €
1. This means that it approaches zero as
dent of p, is eqdal to zero and this yields formula (13). This completes the proof
we have Poisson's formula
2"
f(C)=2~ ~
where
--->
For a function fCI;) that is regular in 1'1 < 1 to have a repre p ---> 1. We conclude from inequality (14) that its left-hand member, being indepen < 1 in terms of Poisson's integral, it is necessary and sufficient
1)
1(1
407
p
r in 0 < r < 1, bounded as p ---> 1. Therefore, in accordance with (6) with p = 1, we conclude that this integral approaches zero as p ---> 1. Furthermore, in the
n ....... ooQ
Theorem 3.
H
But in the first of the integrals on the ~ight, the Poisson kernel is, for fixed
is an arbitrary positive number, this means that
lim ~
IN CLASS
h (e it) P(r, t-6)dt P
o
< 1 to the function [hC'W € H l' Since IfC')1 ::;!hC')1 in = jhCe ie )! almost everywhere on 1'\ = 1, this formula leads 2"
If(C)IP~2~~ If(ett)\PP(r, t-6)df.
HI'
Conversely, if fCI;) €
o
H l' then, for 0 < r < p < 1, we have
2"
I \
'!h
Therefore, if IfCeie)1 ::; M almost everywhere on
pI - r S
it
f(re' }=2n ~f(pe )ps_2prcos(t_e)+rsdfo
IfC')1 ::; M in 1'1 < 1.
\'1 = 1,
it (ollows that
On the other hand, if fCe ie ) E U, where q> p, then, by
using Holder's inequality, we have
Therefore, 211
If(re/ B) -
2~ ~ f(l1
it
)P(r, t - 6)dt
If(C) IP ~ 2~
I
211
If(peit)-f(e it )
I
S n
~
pl_r ..
M'
..
•
l
~ 2~ (~
dt·
\ \ I pI - r +21t~ If(e it )IDI-2orcos(t-O)-I-rl-P(r, t-6) I dt.
1) Fihtengol'c [1929].
~
If(e it )
IP PPl9 (r, t -
q-p
6). P-q- (r,t - 6) dt
·211
211:
~~ 2n \~
~
q_p
211
If(e
it
9 )1
P (r, t - 6) dttq .
(~
per, t - 6) dt)-q 211
(14)
= (2n;p/q 1) V. I. Smirnov [1929].
(~If(eit) Iq per, t - 6) dt
r
9
•
~~-__ ~:..::-~~~
~ __ ~_~~_ -~~-:
408
~=~~~
"~~~~~_~~_~~~_-,=-_~o~~p~~
__
~~~_~~~--.-=~~~~~_~~:_~
__
__
~~"~_~~_=_~~ ~_ -=-_~~~~~" =~~ ~~~~~~ _~_=~~~~_~ ~"_"__
_=_~~_~=~~~ ",~~~~~~~~_~~_
'._'--~~--::-~=..:::=-:.-:=-~~="'=--"---~=:-;--=-:::::.:::-~~"=--~:~:~:::.::~:~,':~~~~--=,::::-=~=-,:=;==-~~~,:,,":,,~:-=:=;;:::;-::-~~--~~=~~~~="-,,"..,...~
IX. BOUNDARY PROBLEMS FOR ANALYTIC FUNCTIONS
§s.
that is,
in
2",
2",
'S
o
I(I
=
q
r, we obtain, for 0
If we rule out the case F«() == 0 (when what we are trying to prove is
> 0 in 1(1 < 1. Therefore, if F«() = Re1q" then 1C1>1 < 17/2 in 1(1 < 1. Now, suppose that 0 < P < 1. If we set u p (r, () = 3t([f«(W) = RP cos pC1>, we have u p (r, () > 0 in 1(I < 1. Therefore
If(c)lq~2~ ~ If(e/t)lqP(r, t-fJ)dt. If we integrate this over the circle
1(1 < 1.
409
FUNCTIONS CONTINUOUS ON A CLOSED DISK
obvious), we have 3t(F«(»
< r < 1,
RP = Ell. (r;
6) & up (r, 0) cos p~ --=:: It
2",
If(re/ e) I dfJ ~ ~ If(e it ) lq dt.
•
cosP2
0
If we integrate this over the circle
rI J
This proves that f«() E H q . This completes the proof ofthe theorem.
2.,
Theorem 5. 1) If the function f«() is regular in the disk '(I < 1 and if f(re ie ) - .... f(e il ) almost everywhere on the circle 1(1 = 1 as r -+ 1 and if
1(1
=
r, for 0
--k- r
< r < 1,
we have
2..
F (rete) I
P dfJ
~
cos"2
Up
~
(r, fJ) dfJ =
p
2ltU
~O) •
cosP2
f(e Ie) E L in (0, 217), then, for Cauchy's formula Consequently, F«(Y E H
.1) P
21<
f
1 \ f(e it ) de it (C) = 2'lti ~ elt C
(15)
1(' < 1,
it is necessary and sufficient that f«() E HI. Further more, for an arbitrary function f«() E HI' Cauchy's theorem to hold in the disk
4 that f«() E H1" Consequently, we have proved the necessity of the condition in the theorem. This completes the proof of the theorem. §5. Fun,ctions that are continuous on a closed disk
21<
~ f (e it ) dell = 0 o
(16)
Proo f. H f«() E HI' then the proof of form~las (15) and (16) is the same as the proof of formula (13). Suppose, conversely, that formula (15) holds for the function f«() in I( I < 1. The integral on the right can be regarded as a linear combination of four Cauchy integrals of the form 2",
2~i ~ :t~t) d~ ,
(17)
o
where ep (t) is a real nonnegative function in (0, 217). Now, if we denote by F «() the integral (17), we have _
1 \
1- r cos (t - 6)
ffi(F(C»-2lt ~ cp(t) 1_2rcos(t_6)+r2dt;:=:O,
1) F. and M. Riesz [19161.
In conclusion, we point out some properties of some very important types of functions that are regular in the open disk
holds.
2"
On the basis of this, we conclude that the Cauchy
integral appearing in formula (15) belongs to the class H for arbitrary· p in the . p interval 0 < P < 1; that is, f«() E Hp . But f(e 1e ) E L. It follows from Theorem
_
iO
disk
1(1 < 1
and continuous on the closed
1(\ ~ 1. Theorem 1.
2)
For a function f«() that is regular in the open disk
1(1 < 1
to be continuous on the closed disk I( I ~ 1 and absolutely continuous on the circle 1(\ = 1, it is necessary and sufficient that f' (z) E H 1" If f' (z) E HI' then df(eie)jd(J= ieief' (e ie ) almost everywhere on the circle 1(1 = LIn the equation above, f' (e ie ) means the limiting values of the derivative f' (z) as z approaches e ie along nontangential paths, and df(eie)jd() means the derivative of f(e ie ) with respect to (). Proof. Suppose that the function f«() is continuous on continuous
00
1(1
= 1. In
1(1 < 1,
1'1
~ 1 and absolutely
we have 2"
f (re/ e) = 2~ ~ f (ell) p'(r, t - e) dt
C=re. 1) This property of Cauchy integrals was proved by Smirnov [1929].
2) F. Riesz [1923].
(1)
.....~~
410
IX. BOUNDARY PROBLEMS FOR ANALYTIC FUNCTIONS
and, consequently,
2"
2"
o
0
f(q=~ g (t)P(r, t-a)dt+c=~(g(t)+e)P(r, t-a)dt,
2"
ire10j' (re/o) = ~ \ f(eit) iJP (r, t - 0) dt 27t~
(2)
iJt'
that is, the function f(O can be represented in the disk
If we integrate (2) by parts, we see that the function i(,f' (0 can be represented in
1(' < 1
by a Poisson integral. Consequently, i(['(O € H
411
§5. FUNCTIONS CONTINUOUS ON A CLOSED DISK
1
;
that is, f'«()EH
1(1 < 1
(5)
by means of a
Poisson integral of an absolutely continuous function. Consequently, it is continu 1
•
ous in the disk I(I
<1
I(I
and it is absolutely continuous on the circle
=
1.
Conversely, let f' «() denote a member of H l' Then, the function i(f '(0 can be represented in 1(' < 1 in terms of its limiting values ieiBf' (e iB ) bymeans
Obviously, f(e it ) = g(t) + c and equation (5) becomes Poisson's formula (1). Furthermore, it follows from (1) that lim ->1 (df(reiB) / dO) = ie iB f' (e iB ) almost
of Poisson's integral:
everywhere on k
lrj' (Q =
ix ~leltj' (e
The function
)
P (r, t - a) dt.
(3)
t
g(t) =
lei/j' (e it ) dt
~
~_____
o
(4)
~
is absolutely continuous in (0, 217) and, in accordance with
C~y,S theorem,
g (217) = O. Integrating (3) by parts, we have 2"
6) dt,
2"
1(1
=iJ~ ~
Proof. Since f(O can be represented in
< 1,
If(e 1o ) - f(e 1o ,) I ~KI on the circle 1(1
where c (r) depends only on r. But c (r), being the difference between two func
1(1 < 1,
is itself a harmonic function. Since
!i.e (r) = die (r) drs
we easily obtain the result that c (r)
=
(0
I(I < 1
by means
E H l' It follows on the
basis of Theorem 1 that f(O is absolutely continuous on the circle
1(1
=
1.
+ 1-r dedr(r) •
If'(q I ~ Proof. :For a
0< a
< 1.
=
0
< a.~ 1,
11 !:.\1 ,., r = Iq,
where M is a finite constant, be satisfied in 1(1 =
(6)
(7)
< 1.
1, the theorem is obvious. Let us consider the case when
Suppose that f(O is continuous on \(\ ~ 1 and that inequality (6) is
satisfied on 1(1 = 1. Since f(O can be represented in with Cauchy's formula
alog r + b, where a and b are constants.
Since c(r) is continuous at r = 0, it follows that a
a- a' [",
1, it is necessary and sufficient that the inequality
=
o
tions that are harmonic in
in accordance with formula
imaginary parts of the function i(f' (0 can be represented in
f(Q=~ g(t)P(r, t-a)dt+c(r),
,
1(1 < 1
(1), we find, just as at the beginning of the proof of Theorem 1, that the real and
g(t)P(r, t-a)dt.
211
Thus, in
ous on the closed disk 1(1 ~ 1 and if it is a function of bounded variation on the circle 1(1 = 1 (that is, if the real and imaginary parts of the function f(O are functions 0 f bounded variation on that circle), then f(O is absolutely continuous on the circle 1(1 = 1.
Theorem 3 (Hardy-Littlewood). For a function f(O that is regular in the open disk 1(1 < 1 to be continuous on the closed disk 1(1 ~ 1 and to satisfy a Lipschitz condition
that is,
Therefore, in
1. This completes the proof of the theorem.
of a Poisson-Stieitjes integral and, consequently, f'
l~f' (q = ~ g(t) iJP (r,,,,~ -
iJftei8)
r
=
Theorem 2. If a function f«() is regular in the open disk I( I < 1 and continu
, it
"I
211
f(q=~ \ f(elt)e lt 21t ~ elt - t dt,
0 and hence c(r) == const = c.
1(1 < 1, it follows that, for I( I < 1,
1(1 < 1
in accordance
412
IX. BOUNDARY PROBLEMS FOR ANALYTIC FUNCTIONS
I' (Q =
21t
§5. FUNCTIONS CONTINUOUS ON A CLOSED DISK I
21t
~ C f (e it ) eit dt _ ~ C f(e it ) - f(e la ) it dt 21t .~ (e lt _ C):i - 21t.~ (eft no e , o 0
If(eiB) -
h
I
+~
where'; = re it . Consequently,
, I e If(eit)-f(e ia ) I k
If(C)I~21t~ o
"" Ie If(eil
.
1-2rcos
d
S
21t
I la d
k 'P
< x < 17/2, we have
(1_ r)2 + 4r
1t 2
-1t
f.
(I _ r)2 + 4r
1t2
•
0
This shows that inequality (7) holds in 1/2:::;
disk 1(1 < 1, is continuous on the closed disk condition
on the circle
1(1
=
on the closed disk
1(\ < 1 for suitable finite M inde
converges for every 0 and hence, the limit fee/B) =
f(re,e) exists fot every O. Furthermore, if we integrate (7) with respect
< r < 1, we see that f(,;) is bounded in 1(1 < 1 and,
consequently, can be represented by means of a Poisson integral in terms of its limiting values fee/e). Let us show that (6) holds. It will then follow by virtue
f(O
is continuous on the closed disk 1(1 :::; 1. Obviously, it will be
sufficient to prove (6) for
10 - 0"
. ,
0<
and satisfies a Lipschitz
11
~ 1,
(8)
< 1. We have I
where we take for l a curve consisting of the radial segments (e ie, he· e ) and (e/ B ', he/ B') and the arcs (he/ e, he/ e ') on 1(\ = h, where h = 1 - 10 - 0'1. In
1(\
C' la
(9)
:s 1. If(re, a) - f(re i.) I ~KI re ia - rei. la, If(re ill) -f(r'e iB) I ~KI reia _ r'eiala
(10)
(1l)
B for arbitrary points re/ , re iB' ,
assume that
10- 0"
r' e /B in the disk I(I < 1. To prove (10), we may < 77. Now, the function rPB,e'(';)= (f(,;ei~ _ f(,;e ie '»/,; is
regular in '" < land continuous on the closed disk 1(\ : :; 1. Therefore, the max imum of its absolute value on the closed disk 1(\ :::; 1 does not exceed the maximum of the quantity If(ei(tte~ - f(e'(tte' »1 in [0, 277). In accordance with (8), this maximum is no greater than k 10- 0'1 CL. Consequently,
a If (re/ ) - f (re ia ') I ~ kr Ia-
f(eIS) - f(e ia') = ~f' (C) d"
this case, we have
a
Proof. It is easy to see that (9) will be proved if we can prove the two particu lar inequalities
M greater than the least upper bound of the quantity
then (7) will hold in 1';1 < 1. Conversely, suppose that inequality (7) is satisfied in the disk 1(\ < 1. Then,
of §1 that
1';1 ~ 1,
If(C) - fCC') I
1(\ < 1/2,
to r from 0 to r, where 0
(2+ h)la~6'la
1, then it satisfies the complex Lipschitz condition
ro
pendent of r. If we take
' .... 1
h)1 a
Theorem 4 (Hardy-Littlewood). If a function f(,;) that is regular in the open
o
lim
dt
k<pad
l+a
Mf' (reie)dr
(J -
a If(e, ) - f(e'B') I ~ k I a- 0' la,
--
the integral
a
that is (6) with k = M(2/ a + 1). This completes the proof of the theorem.
k1t a r 2 C <pad
(1 - r)I- CL If'(,;)! in
a
=M(2 (I-h)a+ hIO-O")=M
ro
< ~1t
B'
~ 2 ~ (I ~~~1 + ~ (I ~~1 IX
Therefore, on the basis of what was said above and also inequality (6),
II' (Q I ~ ~
I
a If' (re/ ') I dr
-1t
1-2rcos
1t
•
+ IB~ h If' (he it ) Idt I
1t
ll-re11o-t'l:i dt=21tJ
But since sin x> (2/17)X for 0
f (e ia') I ~ ~ 1/' (re 1a ) I dr
413
a'i a•
... But
I O~&~I' ~ -= 2. sm 6-0' 2
1=~leiS_e/a'l 4 '
(12)
so that (10) with K
(77""/ZGt)k follows from (lZ).
=
10
Let us now prove (11). Suppose that 0 ~ r < r' rem 3, inequality (7) holds in
1'1
< 1.
In accordance with Theo
T
r' < (1
+ r)/Z and hence-
1'1 < 1
T
r' - r < 1 - r',
< 1;
that is, f(O satisfies the condition of Theorem 3. But then, in accord
circle
we have
!(r'ei~ I ~ f1 ~'\1 " ~ dt ~ M (...r,--..~
=
u(r, e) + iv(r, e) is regular in the open disk
and u (r, e) is continuous on the closed disk
1'1
=
r' > (l
1!(re i9 ) _
+ r)/Z, so that
!(r'ei~ I ~ M
IT'T
(1- t)" a
r' - r>
1-
r',
converges, then f(') is bounded in the open disk then
< M (1- r')" ~ ~ Ir' a
a
disk
1'1 < 1
If a function f(')
=
r I".
1'1 .: ; 1
closed disk
and satis
0< ~< 1,
a' I",
where (, = re iB ). Consequently,
"\
I
I!(QI~-;- j
-11
" 2\
A (t)
A (t)
II-reitrdf+Mh
where MI is \llso a constant. If we replace the absolute value in the denominator of the integral on the right with its imaginary part and use the inequality sin t
Proof. Since f(O can be represented in the open disk
1'1 < 1
in accordance
"
..
+
I \ eit C f(~)=21t ~ u(t) eit_cdt+iC,
2"
2u (t) eit dt _ 21t ~ (e it - C)s -
1.- \ 1C
~
u (6) it dt (elf - c)~ e •
u (t) -
Consequently, just as in the proof of Theorem 3, we obtain the inequality
If' (Q l~
11
~\I--'"
f(')
Theorem 7. I)If a function f(')
2..
1.- \
""2
1 !(~)1&2 -= 1t ~\ ~dt+M"':'::1.r 810 t -== r \~ ~df+M t h
(14)
which proves the boundedness of
it follows that
>
(Z/77)(, we obtain
"2
2..
I)Privalov [1919].
""2 2 \
I!(QI~-;- ~ II_reitldt+M~-;- ~
with the Schwarz formula
f ... -
dt
.
IU(f+6)-u(6)III_relf l +M,
where M is a constant. Therefore, in accordance with (15), we have
(13)
then f(') satisfies a complex Lipschitz condition on the
' (T) _
We transform this formula to the form
1t
I u(6)- u(a') I ~Kla -
1'1 = 1, 1'1 .: ; 1.
< 1.
1\ elt+C !(q=21t ~ (u(t)-u(9)) ..If_r df+u(9)+iC
fies the Lipschitz condition
on the circle
I"
2"
u(r, e) + iv(r, e) is regular'in the open
and if u (r, e) is continuous on the clos ed disk
1'1 < 1.
Proof. Formula (14) holds in
the theorem. 1)
(15)
where the function A(t) is nondecreasing and is such that the integral fc{(A(t)!t)dt
In both cases, we have obtained (11) for suitable K. This completes the proof of Theorem 5.
and satisfies on the
lu(e) - u(e')1 .::; A(je - e'l),
" =Mjr' -r)".
T
On the other hand, if
I' I ~ 1
1 the condition
T'
1!(re'9 ) -
415
ance with Theorems 3 and 4, it follows that equation (14) holds. This completes Theorem 6. If a function f(O
~~
(1
1'1
FUNCTIONS CONTINUOUS ON A CLOSED DISK
the proof of the theorem.
<. 1. From this inequality we get r' ,.
If (re"j - f (r'e;') I,,;; ~ If' (Ie;') Idt,,;; ~ If
§s.
IX. BOUNDARY PROBLEMS FOR ANALYTIC FUNCTIONS
414
1'1 < 1,
disk
=
in
I"
< 1.
u(r, e) + iv(r, e) is regular in the open
and if u(r, e) is' continuous on
1'1.::; 1,
then e i f(1;,) € H
p
for all
p> O. Proof. Let p and
f
denote positive numbers such that p f
u (r, e) is continuous on the circle
1'1
=
< 77/Z.
Since
1, it follows that, in accordance with
the familiar theorem on the approximation of periodic functions, there exists a 1) Smirnov [1932].
416
IX. BOUNDARY PROBLEMS FOR ANALYTIC FUNCTIONS
trigonometric polynomial
'fi/ m
S
such that
+ ~ (an cos n6 + bn sin n6),
(6) = ao
Iu (0) - s «;1)1 ~ (;
n=1
'
in the interval (0, 217).
If we set
m
p(~)=ao+ ~ (an-lbng n n=1
(so that :R(p(e ie ))
=
s({m and _1_ 21t1
"
p(O, we have, for 0
i~ (e) dC _
J\
\
C -e
i~ (0)
CHAPTER X
BOUNDARY QUESTIONS FOR FUNCTIONS THAT ARE
ANALYTIC INSIDE A RECTIFIABLE CONTOUR
< r < 1,
§ 1.
.
I c I=r If we now separate the real and imaginary parts, we obtain
In the present chapter, we shall study various questions associated with the boundary values of functions that are regular in a domain B that is bounded by a
2"
1 21t } I eip'!' (C) I COS P (u (6) -
The correspondence of boundaries under conformal mapping
rectifiable closed Jordan curve. 10 these investigations, an important role is
s (6» d6 = ffi (e'P'!' (0».
played by certain boundary properties of functions that map the domain B uni valendy onto a disk. We shall begin with an exposition of these properties, which
Therefore, since COS
P (u (6) - s (6» ;;:::: cos po.
were established by N. N. Luzin and L' L Privalov. 1)
> 0,
A continuous curve C: z = z (t) = x (t) + iy (t) for a ~ t ~ b is said to be
we have 1 211: ~ 21t \ I eip'!' (e) Id6 .;;;;;; !Jl (e'P'f' (0» ~ cospe'
rectifiable if, for an arbitrary number n + 1 and an arbitrary sequence of values (16)
t l' t 2'
••• ,
I~ I :
1, we obtain the result that e if ( ~) €
the sum
n
~
Inequality (16) has been proved for all r in the interval 0 < r < 1. This means that ei¢(~) € H .If we now keep in mind the fact that the function e-ip(~) is bounded in
< t 2 < t 3 < ... < t n + 1 = b,
t n + 1 such that t 1 = a
Hp also. This completes
I Z (t k +1) -
k=1
Z
(tk )
I
remains bounded. The least upper bound of these sums out of all possible com binations of values t 2' ••• ,t n is called the length of the curve C. It is obvious
the proof of the theorem.
from the definition that a necessary and sufficient condition for a Jordan curve C to be rectifiable is that the functions x (t) and y (t) be of bounded variation in (a, b). Furthermore, it is obvious that, if the curve C is rectifiable, any arc of it
is also rectifiable. We denote the length of an arc z
=
z (t) for a ~ t ~ t', which
is a function of t', by s (t '). Let us prove the following lemma regarding s(t).
Lemma. If x(t) and y(t) are absolutely continuous in (a, b), then
t
s(t)=~Iz'('t)ld't, a
I
1) See Privalov (1919].
417
a~t~b.
(1)
__,·_'
'--.--m-.=_;=""""====-~=."="'=====
...='''''':'=,,. --,.
-., ..~•._.
.__ ,.~ __ ~_~_.
-=:..-:::----
__ .~~~r:::.._==_ __= : ~... '_:_'m.~..".._ ..
-~
'r~,±. ....
_~!!!!"!!,o::;;.
"' ."""""" _
Proof. Suppose that a
and that a
=
t1
< t 2 < ... < t n + 1 = t.
"_.. ~'M·,,,!:~
Then, the
I Z ('t) -
Z (tk) 1= V(x ('t) - X (tk))~
+ (y ('t) -
.""'.•"."...;,o::;;;g:............ _.
1)_
_ "fot'!i!', __
,J"" •.__ .t. __
;:;;;;
. . . _,!!'~'=_
419
1, .•. , n) is absolutely continuous in (a, b) because the
s (t);?; ~ S' ('t) d't.
Y (tk))~; I
a
!
absolu~e value
of the difference between its values at any two points of the interval
But at points at which all the derivatives s '(T), x '(T), and y '(T) exist, we have
(aJ b) does
S'('t)= lim .1s~ lim
not exceed the sum of the absolut e values of the corresponding differ~;ces between functions x(t) and y(t). Consequently, ¢k(T) has a derivative almost everywhere in (a, b) and
"
4t-+0
4~-+0
A'C
almost everywhere. Therefore, for tk
< T < b,
1_'.
I.
s(t);?;) I Z' ('t) I d't.
Inequalities (2) and (3) lead to equation (1). This completes the proof of the lemma. Suppose now that a finite simply connectl'd domain
".
T
=
tk
maps the domain
we have
Z (t k )
1<
t k +1
~
In accordance with Theorems 1 and 2 of
IZ' ('t) Id't.
I.:I .$ 1.
AI so,
continuous in (a, b) except possibly at countably many points. Excluding these
1) w(':) is continuous in the closed disk
tinuous on the circle
\.:1
=
t
't-7l
()d
S t
a+h
t}~
= ieitw '(e it ) almost everywhere on the circle '':1 = 1; the length s(t', t ") of the arc z = w (e it ) for t'.$ t S t" is given by the
formula t"
()d
't-7l J S 't
S t
~
a
s('C)d't] <
(t', t") = ~
I ro' (e lt ) I dt.
(3 ')
t'
a
ath s('t)d'C-
t
1\
absolutely con
2) w '(':) E HI;
4)
h> 0,
J
t~h
1.:1 .$ 1 and is
1;
a
\ s(t+h)-s(1:)d _1
=
3) dw(eit)/dt
points, we have in (a, b) by Fatou's lemma t t lim \ S (c h S (t) d't ~ \ S' ('t) d't, a
of Chapter IX and the lemma just
curve C, then thg following hold: (2)
On the other hand, since s(t) is obviously a nondecreasing function, it is
+ k-
§5
w (.:) is regular in 1.:1 < 1 and maps that disk univalently onto a finite domain B that is bounded by a rectifiable closed Jordan Theorem 1. If a function z
a
= ~[
of Chapter II, the function
proved we immediately.obtain
t
a
Let z = w (0 denote the
2.". Since C is rectifiable, the function w(e it ) is of bounded variation in (0, 2.,,).
S(t)<~lz'('t)ld't.
h
§3
the curve C can be represented parametrically in the form z = w (e it) for 0.$ t .$
Therefore
J
1.:1 < 1.
X (z) is con tinuous on B and the function w (0 is continuous on
IZ'('t)ldt.
tk
h_O
B conformally onto the disk
inverse function. In accordance with Theorem 4 of
+ l'
IZ (t k +1) -
and since
B in the z-plane is
bounded by a rectifiable closed Jordan curve C. Suppose that a function ~ = X(z)
tk
In particular, for
(3)
a
~
19,,('t)1< ~
At.
t
+ 19k ('t) I + y (x (t) - x (tk»" + (y ('t) -:; (t k»2 . yX' (t)2 + y' (t)2 -_I Z .('t, )I ~ Y (x ('t) - x (tk» + (y 2
YAX'+AyJ=V X'('t)2+y'('t)2=\Z'('t)I.
Consequently, at these points and hence at all points t E (a, b),
I (x (1:) - x (tk» x' ('r:) (y ('t) - Y (tk» y' ('t) ! Y (x (t) - X (tk»2 (y ('t) - Y (tk»"
J
._.
t
9k (t) = =
.. ~o·"t'%;E~""':~
it follows that
function
(for k
"!f'--.;:-z:;;;~~ __; .."'*~_
§l. CORRESPONDENCE OF BOUNDARIE~
X. FUNCTIONS ANALYTIC INSIDE A RECTIFIABLE CONTOUR
418
.~.-.'="~!".,,
Let us now look at various sets of points on the boundary C. A set E
t+h
~ ~ t
s(t)d'C<s(t+h),
cC
is said to be closed on C if it is closed as a set of points in the plane. A set
420
§ 1.
X. FUNCTIONS ANAL YTIC INSIDE A RECTIFIABLE CONTOUR
CORRESPONDENCE OF BOUNDARI~S
E c C is said to be open on C if its complement is closed on C. One can easily
~ 100' (e il) I dt
show that an open set on C is the union of a finite or countable set of arcs. We
E
421
< e,
define the measure of an open set on C as the sum of the lengths of the arcs of which it is composed. Now, let
E
denote an arbitrary set of points
00 I
C.),The .
then
,> ~ I w· (."1I dt ~ ene".1l1 »")
outer measure of the set E is defined as the greatest lower bound Ofihemeasures of all open sets on C containing E. We denote this outer measure by m E. We define the inner measure, ~hich we denote by miE, of the set E by t
1 00'
(e it ) Idt
~ ]Ie (me, E _ ! or.)),
formula
miE = s - me CE, where CE is the complement of E with respect to C .and s is
so that
the length of C. We say that the set E is measurable on C if me E = miE. When
mes E <"}I"e
this is the case, we denote the common value of me E and miE by mes E and
+ B("}I"e) =
Tj
(e).
call it the measure of the set E. H E is measurable on C, then it follows from
This shows that mes E can be made arbitrarily small by making ( sufficiently
the re lations
small and 7J«() depends only on (.
miE=s-meCE,
miCE=s-meE,
meE=miE
that CE is also measurable on C. In particular, measurability of open sets implies measurability of closed sets. In general, one can show that all the theorems regarding Lebesgue measure can be carried over to the study of curves and, for this reason, we shall treat them as proved.
denote the set of points on
measure zero (of positive measure) on the circle
I(I
=
1 onto a set of points of
measure zero (Of positive measure) on C, and conversely.
corresponding to it. For given (> 0, there
< (.
In accordance
with the remark just made, we have mes 0 s < 7J«()' and consequently, meE. But me E s does not depend on (and
7) «()
--+
< rf..().
0 as (-.... O. Therefore, me E s =
o.
3. To prove that corresponding to a set of positive measure on
1(1 = 1
there
is always a measurable set of positive measure on C, we note that, if E s is a measurable set on
1(' = 1 such that mes Es = 0 and let Ez
1(1 = 1
exists on C an open set Oz containing Ez such that mes Oz
mes E s =
Theorem 2. The mapping mentioned in Theorem 1 maps a set of points of
Proof. 1. Let E s denote a set on
Now, let Ez denote a set of points on C such that mes Ez = O. Let E s
1(1 = 1
and mes E s > 0, then on
'(I = 1
there are closed sets
Gt,k), for k = 1, 2, " ' , such that Gt) C E sand mes G k ) --+ mes E s' Let us
l
k
k
t
for k=l, 2,"',
denote the corresponding set on C. For any (> 0, there exists on the circle
define GS=U:=lGt ). Then, GscEsand mesGs>mesG
1(1 = 1
so that mes G s = mes Er • Consequently, the set N s = E s - G s is a set of measure
an open set 0 s such that Esc 0 sand mes Os <; (. GmespDnding to it
zero on
on C is an open set Oz that contains Ez • We have
meEz
< mes Oz =
I, and we have therefore represented E s as the union of the set
N s of measure 0 and the closed sets G~k), for k = 1, 2, .... But corresponding N s is the set Nz of measure zero on C and, obviously, corresponding
~ 100' (e it) Idt.
to the set
o~
But meE z does not depend on (and the last integral can be arbitrarily small for sufficiently small (. Consequently, me E z = mes E z = O. 2. Before proving the converse, we make the following remark: Since
UJ '«() 1= 0 almost everywhere on '(I = 1, the measure of the set E(lUJ '«()1 < () of points on 1(I = 1 at which IUJ '«()\ < ( approaches zero as (--+ 0, it follows that when we set mes E (IUJ ' «()1< () = 8 «(), we have the limit relation 8 «() -.. . 0 as ( ....... O. Therefore, if E is a measurable set on the circle
1(1 =
),
I(I
= 1 such that
to the sets G k ) are the closed sets G~k) on C. Consequently, the set sponding on
t C to the set
E. corre
E s is the set Nz U U:= 1 G;k), which is a measurable
set. Its measure is positive because otherwise, E s would, from what we have proved, be of measure zero. In an analogous way, we can show that corresponding to a set of positive measure on C is a measurable set of positive measure on
1(1 = 1.
This completes the proof of the theorem.
Remark. It follows from part 2 of the proof of Th eorem 2 that a function (= X(z) that maps the domain B onto the disk 1(1 < 1 is absolutely continuous on
~;:;;:-..::;;~it\.'t#~.,,~-.'!ft'_"E'.i&-g;3~';":;::-:'~~~~"'~~""~~':;:';;,"~;;;;:;-:-~a:iE:£~~~=.B'_E'l':"'_'T.>-~-_~;¥$.~E:'f'~.-:::;~~~~~~~~*,_~~:E'~-:,~;;..:.~3=""'~"i'E5-3:-~SES?:-::5-~ft~~~.EiE"1~ffi,"':\Sf2~'!',ftES£-£."ti:.'£f'r<"]Si'-:S::2'":~E"~. :::~.~,:--:;?~~ ~~""-::::';:=;'·::·::·-·;-:;:7:~h-::-7;'·,ct.,*trE';).:tiL;'~::;;'
§ I.
X. FUNCTIONS ANALYTIC INSIDE A RECTIFIABLE CONTOUR
422
C if we regard it as a function of arclength on C because the inequality k~I' k -
s
s
"1 < ( imp ' 1 'ie s
point z
-
j!
x. (z (S.m 1< 1/ (e).
=
w«() in Theorem 1 is conformal\al"!.9 t every
lor, respec-tively, conformal almost everywhere on C.
1(1
=
1 in the one case and on C in the other,
\
Since the derivative dw (e it)! dt exists and is equal to zero almost everywhere on the circl e
1(I
°and vice versa. Also, if these curves have tangents at the points indicated,
l, this is equivalent to conservation of angles at the point (0 in the shift from the
I(I ::; 1
to the closed domain
11
or the other way around. This completes the
proof of the theorem. Let us now look at the que stion of the correspondence of boundaries under
Proof. Obviously, we can consider the conformality of the mapping only at points on
423
the curve l. Since this angle of rotation is the same for all pairs of curve s ,\ and
disk Theorem 3. The mapping z
1(I =
CORRESPONDENCE. OF BOUNDARIH'S
a will be the angle of rotation of the tangent in the transition from the curve ,\ to
k
~ I x. (z (s;'» where on
:i-;';iFE_Si~;;;:eti2;t~i:?~~-!'.;~~:;S:2:.,£;q2jti~:;";:;'$J?:~~~F;i.!~~
= 1, there exists almost everywhere on C a unique tangent such
conformal mapping of domains bounded by smooth Jordan curves. By a smooth Jordan curve C: z
=
z(t), where a::; t ::; b, we mean a curve possessing at every
point a tangent that rotates continuously as z moves along
C, that is, a curve
for which the angle 0 (z) of inclination of its tangent to the real axis is a con
that in a neighborhood of a point of tangency, the curve C lies on both sides of
tinuous function of the point of tangency. In the case of a closed curve, we must
the normal to C at that point. Let us show that angles are preserved at all points
also have O(z(b)) = O(z(a)) + 217' Obviously, a smooth curve is a rectifiable curve.
on
1(1 =
1 that are mapped into points of the curve C with this property. Let (0
denote one of these points on
1(I =
1 and let z
° denote the point on
C corre
sponding to it. Then, every branch of thl'; function
,
II
Therefore, 0 (z) can also be regarded as a function of the arclength s on C:
0= O(s). We now have Theorem 4. 1) Let z = w (~ denote a func tion that is re gular in the dis k
(1:, ~o) = arg w (~~ - ~ (~o)
\(1 < 1
and that maps that disk univalently onto a domain B bounded by a smooth
closed Jordan ,curve C. Let (= x(z) denote its inverse. Then argw'«() is a is continuous in the closed disk
1(1::; 1
except at the point (0' But since the
curve C has a tangent with the property described at the point z 0' the function
continuous function in the closed disk tion in the closed domain
u «(, (0) approaches a definite limit a as (approaches (0 along the circle
1(1 =
function u «(, (0) in the portion of the disk
1(
1
<1
contained in a sufficiently
small neighborhood of the point (0 is less than 317. This shows that the function
1(1 < 1 and it can be represented in circle 1(I = 1 in accordance with Poisson's
u«(, (0) is a bounded harmonic function in that disk in terms of its values on the
1(' = 1. Consequently, the 1('::; 1 including the point
formula, These values define a continuous function on function u «(, (0) is continuous on the closed disk
(0' where it is equal to a. Now, consider an arbitrary curve ,\ lying in the disk
1(1
( - (0 through values belonging to the disk
1(1 ::; 1,
Furthermore, on the circle
1(1
=
1,
(4)
Proof. As a preliminary, we point out that, by virtue of the smoothness of the curve
C, for any point a € C and any number (> 0, there exists an arc on C
containing the point a as an interior point such that the angle of inclination of the chord connecting any two points of that arc to the real axis differs from the angle of inclination of the tangent at the point a by an amount less than (. This follows from the fact that the angle of inclination of the chord is equal to the angle of inclination of the tangent at some interior po int of the arc intercepted by that chord.
<1
with endpoint
(0' and let l denote the corresponding curve in the domain B. Since u «(, (0) - .... a as
and arg X'(z) is a continuous func
arg 00' (Q=f)(Q - arg~- ; .
1 (this value being the same for approach from the two sides). We take this
limit as the value of u «(, (0) at the point (0' Obviously, the oscillation of the
11.
\(1::; 1
if the curve ,\ has a
definite tangent at the point (0' the curve l will have a definite tangent at the
Now let us assume that the condition of the theorem is satisfied and let us consider the function 1) Lindelof [1916].
- - - - - - - - - - - .------------_._----~~..::.:::::.::::-::::..-=-=-'---_.~::.:::::::;::-~::==.::::;;~-;;:::=-..=:~--=---:-..::...~..-:::::::::...~=.::.-::-..:::::::::::..-==::.:::.-::::::: ..::.:=:-.----:;-~--=-.==.-:::_-:;;;:;:;==;:,----'----:.~-:;=::,::-..:::::::~-------,-----:.
424
§ I.
X. FUNCTIONS ANALYTIC INSIDE A RECTIFIABLE CONTOUR
u (C, 't) = arg
1Il
(Cei~) "'stl't
(C)
1Il
,.
+ l(-l;;~ '1'1
,
> 0, a harmonic function iri the closed disk 1(1 ~ 1.
We assert that there exists a finite constant K such that lu «(, ,)1 < K for all (
which is, for arbitrary fixed ,
in the closed disk 1(1 ~ 1 and all , in the interval 0 <, < 1T. H this were not the
case, there would be a s~quence
'n)1
max 11:1S;1 lu «(,
--->
00
Irnl,
as n -->
00.
for n
1, 2, ... , such that m n =
=
Obviously, 'n
we can, by shifting to a subsequence of the arcs the sequence
l(nl
of points on
point (0 on the circle
1(' =
I( 1=
--->
'n
0 as n
-->
00.
Furthermore,
if necessary, ·arrange to have
1 at which lu «(n'
'n)1 = mn
'(I = 1
at the point (0' This con
1(1 =
denote the angle between the tangent to C at the point a,=
weal
I( \=
1 at the point a. Then, for given f
5> 0 such that, for
Irl
1 and let ¢ and the tangent
> 0, there exist "I > 0 and
< 5, we have
't) -lfll
<;
neighborhood of the point a that lies in the disk
¢' we have for every , > 0 and
I~
1(1
< 1. H we set U«(,,)
't)P(r,
t-6)dt+2~ ~
t/J +
I arg 00' (~) -
rp I
< e.
'This shows that arg w I «() is continuous in the closed disk
lal =
this function at the points a such that
1(' ~
I if we define
1 as being equal to the angle ¢ that
B now
'The continuity of arg X '(z) in
11 w '(t:). Since ¢
=
(j(s) -
t/J - 1T12,
follows from the fact that X '(z)
=
formula (4) also follows from what we have
shown. This completes the proof of the theorem.
Hp for all p > o.
1 -2 ~
2"
~
log 100' (re lt ) Idt =
V(ei/, 't)P(r, t-B)dt,
2"i
log I 00' (re n k) I
.!. V n-+co n ~ lim
k=l
.
-./ n
ll, I!
J!
n
~ hm log n-+co
2,,1 k
1 00 '
n
(re
n
) I
2'"
100' (re - n k )
I=
k=l
I
2"
log -2 ~ ~
't 1dt. 100' (rei)
I (4)
l
where (= rei e and I is the portion of the circle "I '
O. Therefore in I:J.,
= ;~ log
V(ei/, 't)P(r, t-6)dt
0/-"
(t/J -
, -->
=
1< 1 in accordance with Poisson's
2"
arc
the point a mentioned above, we have the limit relation u «(, ,) --> arg w ' «() as
have, for 0
formula
=2~ "'t" ~ V (e lt ,
1(\ < 1. Consequently, there exists an "II > 0 such that the inequality '(-al
Proof. From the inequality between the arithmetic and geometric means, we
Let us show that this inequality also holds in that portion I:J. of some
2~ ~
+ KtJ'l' I ~ per, t - 6)dt.
disk
w '( () €
on an arc of length "I containing points on each side of a.
V(r., 't)=
;
Theorem 5. Under the conditions of Theorem 4, we have log w '«() € HI and
Iu (c,
u «(, r) -
~
we have been considering.
denote an arbitrary point on the circle
to the circle
t-6)dt
425
But this last integral converges uniformly to zero as (--> a through values in the
1. 'But because of the smoothness of C, the sequence
point zo = w«(o) and the tangent to the circle tradiction proves the assertion made. = ei..p
~ per,
converge to a
of the values u «(n, 'n) converges to the angle formed by the tangent to C at the
Now let a
.
CORRESPONDENCE OF BOUNDARIES
I(I = 1
complementary to the
Consequently, in accordance with 'Theorem 1, the first interyal is bounded in the interval 0 < r < 1. But since the integral
"I)' There fore,
"'t" IV(r., 't)1~4: ~ per, t-B)dt+ 0/-"
2it
~ \ arg
o
00'
(re i9 ) I d6
-- ,"--
-
._--- ,--
",""-
-------- - -
____
,,_ ."
". __".,.-_".,""'L
.••
is, by Theorem 4, bounded in the interval 0
< r < I,
it follows that log w' (,)€ HI'
=
arg w'(re ie ), which, in accordance with Theorem
4,is continuous in the closed disk "I
s 1.
This completes the proof of the
=
~ Jew-etrf'l= I ~
w (,) denote a function that is regular in the
IB(s)-B(s')I~kls-s'la,
Then, w'(,) is continuous on the disk \,\
0<(1<1,
S
k=const,
(5)
1 and the following Lipschitz con
L
,..
"':' _;'-:
_.:;~:-.~:;=='=.~~-===.:~~~~
427
arg w' (e I6 ') , ~ k" I B- B' la,
w'
w ll
:;
W'II
in the disk
Iwl < M,
we have
I~lw-w'1 ~~=kMlw-w'l. ~ Mn-l
k=1
disk ,,\ < 1 and maps that disk univalently onto a domain B bounded by a smooth
C. Suppose that the angle of inclination ()(s) of the tangent to C as a function of the arclength s on C satisfies the Lipschitz condition
._
CORRESPONDENCE OF BOUNDARIES·
Furthermore, for arbitrary wand
closed Jordan curve
__
so that, again by Theorem 5 of § 5, Chapter IX, inequality (7) follows.
theorem. Theorem 6 (Kellogg). Let z
-,._
I arg w' (e i6 ) -
The second assertion of the theorem follows from Theorem 7 of § 5, Chapter IX as / applied to the function u(p, ()
"
_'~~:::::::_~
§ 1.
X. FUNCTIONS ANALYTIC INSIDE A RECTIFIABLE CONTOUR
426
_
k=1
= log w '(e i e) and w' = log w'(e ie') and use (7), we obtain
H we apply this to w
(6). This completes the proof of the theorem.
Theorem 7. 1) Let z = w(') denote a function that is regular in
"I
<1
that
maps that disk univalently onto a domain B bounded by a smooth Jordan curve C such that the angle of inclination ()(,) of its tangent to the real axis, considered as a function of arclength on C, is an absolutely continuous function and
ditions
I w' (e i6 ) - - w' (e I6 ') I.s; k 1 IBIlog w' (el 6)
B' 1\
(6)
log w' (e i6 ') I ~ k2 1 B- B'
-
la
(7)
()'(s) €
LP,
where p
> 1.
Then w"(,) € H p '
Proof. From the condition of the theorem, we find, by using Holder's inequality $
where k 1 and k 2 are constants, hold on the circle
\'1
=
I(j (s) -
1.
B(s') I = ~ B' (s) ds':;;; $'
S
S
0 B' (s)P dstP 0 dS)I/q ~M Is $'
s' liN,
s'
I
Proof. By Theorem 4, we have
arg w' (Q=B (Q - arg ~- ; , and, by Theorem 5, we have w '(,) € Hp for all p and formula (3'), we have, for ()'
> o.
(8)
where lip + llq
=
1. From this we conclude, by Theorem 6 (With a= 1Iq), that
w'(,) is a continuous function on the closed disk dO (s) = dO (s) - dO ds
< (),
8'
\ I
I,
w' (e ia ) 12 dB ~ dB ~ k 1 I B
B' jI/2,
arg w' (e i6 ') I ~ IB(s) B(s') ~ k 2 1s
1+\ B
- s' /a
~ (argw' (e I6 )+B + ;) E!.p.
B' I
+I B-
But, by Theorem 3 of §4, Chapter IX, and Theorem 2 of § 1, Chapter IX, we have
B' 1< k 2 1 B- B' la/2.
almost everywhere on the circle d
It follows on the basis of Theorem 5 of
§ 5,
8
I = ~ I w' (e i6 ) IdB ~ ks·1 B6'
and, consequently,
B' I,
k s = max I w' (Q Itl=1
I.
1'1 =
,e ,,'e
Ie
1000' (e I6 ) 00' (ei8)
,
,'e ») € LP. It now follows from the representation of the
so that ~ (e' w (e' )/ w (e' function
I
log 00' (e i6 ) dO
Chapter IX, that log w '(,) and
hence w'(,) are continuous functions in the closed disk \,' S 1. But then, s'
.1 w' (e i6 ) I,
we conclude that d() (s)1 d() € LP; that is,
8'
so that
Is -
Since
8
6
Is- s'l = ~ I w' (e i6 ) IdB ~
I arg w' (e i6 ) -
1'1 S 1.
On the basis of this fact
'W '(,)/ w '(,) by means of Poisson's integral that
1) Smirnov [1932].
428
X. FUNCTIONS ANALYTIC INSIDE A RECTIFIABLE CONTOUR
ffi.
(COO' (I;)) 00' (C)
'§2.
PRlVALOV'S UNIQUENESS THEORE."'M
429
Define fn (z) at the point z 0 € E as the maximum value of Ifn (z) I in the sector s. Obviously, this sequence converges to zero on E. In accordance with
E hp' '
%0
Then, by Theorem 4 of
§ 2,
Chapter IX, (w" «(V w I «() € Hp' Consequently,
w "«() € H p • This completes the proof of the theorem.
§ 2.
Egorov's theorem, the set E contains a closed subset P of positive measure on
co~verges uniformly to zero. The complement of P
Ifn (z)\
which the sequence
wi th respect to the disk
Iz I =
1 is an open set and hence consists of finitely or
countably many disjoint arcs. We replace those arcs that are less than a semi
Privalov's uniqueness theorem
Let us agree on some terminology. Suppose that B is a domain bounded by a Jordan curve C and that z 0 is a point on C at which there is a unique tangent to C and in a neighborhood of which the curve C lies on both sides of the normal. We shall refer to a continuous curve l contained in B and ending at point z 0 as a nontangential path if that part of l in a neighborhood of z 0 lies inside some angle of magnitude less than TT with vertex at the point z 0 and with bisector coin
circle with the sides of the right angles intercepting them from inside the disk
Iz I < 1
Iz I =
in such a way.that the sides of the angles intersect the circle
1 at
an angle TTl 4. On the other hand, we replace an arc greater than a semicircle with the two radii that pass through its endpoints. As a result, we obtain in place of the circle
Iz 1= 1
a rectifiable closed Jordan curve C 1 bounding the domain B 1
and contained in the disk
f (z)
Iz I < 1.
Uere, the set P lies entirely on the boundary
C l' Tbe function
ary of the domain B, we shall usually mean only points at which there are tan
points of the set P. Let us now map the domain B 1 onto the disk
gents to C possessing the property just described. Furthermore, if a function
w «() denote the inverse of the function executing this mapping. Correspond irig to the set P is a measurable set PI of positive measure on the circle 1(1 :51,
f(z) that is regular in B approaches a value a as z approaches a point z 0 on C along every nontangential path contain~d in C, we shall say simply that f(z) assumes (or has) the value a along nontangential paths. We turn now to the investigation of functions that are regular in a domain'
z
is continuous in
lJ
ciding with the inner normal to C. When we speak of a set of points on the bound
if we assign to it the value 0 at
I( I < 1.
Let
=
The function f 1«()
=
f(w «()) is regular in the open disk
1(' < 1,
continuous on
tbe closed disk \(1 S 1, and equal to zero at points of the set P l' Then, in accordance witb
§ 3 of
Chapter IX,
f 1 «() ==
0; that is,
f (z) == 0
in Iz 1
< 1.
bounded by a rectifiable Jordan curve. We begin with the exceedingly important
Thus, the theorem is proved for the case of a circle.
uniqueness theorem of Privalov.l)
Let B denote an arbitrary domain defined by a rectifiable closed Jordan curve
Theorem 1. Let f(z) denote a function that is regular in a domain B bounded
C. And let us map it onto the disk
'(I
by a rectifiable closed Jordan curoe C. Suppose that f(z) assumes the value 0
of positive measure of
along tangential paths on some set E of positive measure on C. Then f(z) == 0
transformed into a function
in B. Proof. Let us consider first the case in which B is the disk
Iz 1< 1.
About
every point z 0 € E, let us construct a right angle with vertex at z 0' with rays directed into
\z I < 1,
and with bisector directed along the radius to
us cut from this angle sectors
s,
for n
=
z=
O. Let
1, 2, ... , of radii lin. Consider a
sequence of functions fn(z), for n = 1, 2, ... , that are measurable
2)
on E.
=
'(I < 1.
We denote by E I the measurable set
1 into which E is mapped. The function
f 1 «()
that is regular in the disk
1(1
<1
f«()
is now
and that
assumes the value zero along non tangential paths on the set E' after deletion of a set of measure zero at points in which this mapping does not conserve angles. In accordance with what has been proved,
f 1«() ==
0 in
I" < "I.
Consequently,
f(z) == 0 in B. This completes the proof of the theorem. The following theorem 1) is of great significance in applications of this unique uniqueness theorem. Theorem 2. There does not exist a function that is regular in a domain B
1) Privalov [1919].
bounded by a rectifiable Jordan curoeand that assumes the value
2) The measurability of fn(z) follows from the fact that we can regard fn(z) as the limit of a sequence of functions fn,,(z) constructed in the same way but i~suing not from secturs but from the parts of them lying in <, as , -----> 1. The functions fn,,(z) are, for' sufficiently close to unity, defined and continuous on E.
0Cl
along non
tangential paths on a set of points E of positive measure on the boundary C.
IzI
1) Privalov [1919].
-
.-
--"~.
-
_'_
_"'..... "_·SL
._.
T'ti'""J'"'!"'!!!'" _
mf'-""~~~~"~.:~'-:-~~*""7'Y'C"':?":C~::?~~:''''';;'?':':''''"I:::'ii'''"'''',?;--::~7:'':;~-:::;:--.:~';~
m,. '!"ww¥'mPL"!ffi'i1
_!!9'!l'1!"E"'!!!'....-
§3. THE LIMITING VALUES OF CAUCHY'S INT:et;RAL
X. FUNCTIONS ANALYTIC INSIDE A RECTIFIABLE CONTOUR
430
Proof. As above, it will be sufficient to prove the theorem for the case of the Iz I < 1. Let us suppose that a function.
disk
f (z)
having the properties
such points, we may speak of nonrangential paths to C.
m~ritioJJ,~d /
We shall say that a function f(z ') defined on C is summable on C if the
",
in the theorem does exist. We follow the same reasoning as above except that now'
integral
Ie f(z ')dz',
meaning the Lebesgue integral S
we define the functions fn(z) on E according to the formula fn(z) = min (n)lf(z)l,
~f
Sz
and we apply Egorov's theorem to the functions ¢n tz) = 1/(1 + fn(z )). As a result, we can show that the function
f (z)
proof) and approaches "" on a set
ary 8
1,
is regular in the domain 8
1
But the function f(z) has only a finite number of zeros in 8
1,
exists. Now consider the Cauchy integral
This is
true because otherwise, these zeros would have a cluster point which must lie on the set P such that f(z)
---> ""
1
\
as z approaches them from inside 8
l'
p(z)lf(z) is regular and bounded in 8
1
values equal to zero on the set P. But (d. again proof of Theorem 1) this func tion would then have to be the function identically equal to the zero in 8
(1 )
'
l'
1
but in
This
obviously exists and is a regular function. On the other hand, if z lies on C, the integral (1) can be understood only in a conventional sense. Specifically, let z~
=
z '(s 0) denote an interior point of the curve C and let
C obtained after deletion from C of the smaller arc from the point z '(s 0
t \
As a direct consequence of Theorems 1 and 2, we have
B.
as
i -->
Let C denote a Jordan curve, closed or otherwise, in the z-plane. Since every point z' on it is completely determined by the arclength s measured along
dz'
1)
establishes a relationship between the limiting
values of the integral (1) as z approaches points on C and the existence of the singular integral (2). . Theorem 1. If Cauchy's integral (1) has almost everywhere on C definite
C from an initial point z ~ to z " we can rake s as parameter. Then, the para metric representation of C is z' = z '(s) for 0 ~ s
Cauchy's singular integral (2) exists almost everywhere on
Obviously, the function
of the integral (1)
z '(s) satisfies the inequality
I z' (81) -
z' (8~) 1'::;;;81 -
8~, 0.::;;; 8~'::;;; 81 '::;;; S,
exist almost everywhere on C. Conversely, if Cauchy's singular integral (2) exists almost everywhere on
C, then definite limiting values of Cauchy's integral C exist almost everywhere on
and hence is an absolutely continuous function of s. Therefore, the derivative dz '(s )Ids exists almost everywhere on C. Furthermore, in accordance with the
C. Here, in both cases the formula
§ 1,
we have jdz '(s)ldsl = 1 almost everywhere on C. But the curve
C has a unique tangent at every point z ~ €
lim z-+z~
C at which dz '(s )Ids exists and is
nonzero and C lies on both sides of the normal in every neighborhood of z ~. At
C, then
C and limiting values along nontangential paths lying on the other side of C also
(1) along all nontangential paths on each side of
Lemma of
(2)
what we shall mean by the integral (1) when z = z~.
limiting values along all nontangential paths lying on one side of
S S.
i) to
0, if this limit exists, is called Cauchy's singular integral, and this is
The following theorem the limiting values of Cauchy's integral
-
•
Theorem 3. If two functions that are regular in a domain 8 bounded by a have equal values along nontangential arcs, they are identically equal in
fez')
zr_z~
rectifiable closed Jordan curve on some set of positive measure on the boundary
Ci denote- the part of
the point z '(s 0 + i). Then, the limit of the integral
contradiction proves the theorem.
§ 3. On
dz'
where f(z ') is a summable function on C. For values of z not on C, this iritegral
and it has limiting
the case that we are considering it does not have any zeros at all in 8
f(z')
d Z'-Z
on Iz I =1 are points of
We denote by p (z) a polynomial having the same zeros in BIas f(z) has. Then, the function
d~s(S) d8,
(z' (8))
(d. preceding
P of positive measure contained in the bound
Iz I = 1 and the only boundary points of the domain 8
431
1) Privalov [1919].
d\:(z'~ dz'= d\ Z~(z').dZ'-+1Cif(z~) -Zo
(3)
-",_==,,_===,======""=====~=--,======,~~~!~~~~~~~~l~~"'~~_:~~~~_~_~_':~"';:;::;"=~;;~~':~~~':;:~~~~~=~~~":C',"~~",~~'"i"::"_::::~~~~::c~:~:::::~:~,,~~.'c:~;;;·~~-~'~--,"~:"~;;:'~7';;;;~~_""~':;;~~~":o~:::~~~~~_~:;':::~"'';;:~~~'~
432
~3. THE LIMITING VALUES OF CAUCHY'S INT~GRAL
X. FUNCTIONS ANALYTIC INSIDE A RECTIFIABLE CONTOt!R
= CI (z'),- I (z~) dz' _ CI (z~ - I.(z~) dz',
holds almost everywhere on C. In this formula, the limit in the left-hanli member
.,
the positive sign is taken when the non tangential path lies to the left of the
~ approaches
~(z') , dz', z -Zo
.
e
e=lz-z~1
Zo
(5)
is finite, and the derivative dz '(s)/ ds exists and is equal to unity in absolute
Proof. It will be sU_fficient to prove that the difference ~
z -
I
Now, suppose that the point z ~ = z '(s 0) on C is such that the function f(z')
taken when the path lies to the right of that tangent.
I(z') dz'- \
.,
C
approaches zero for all poirits z ~ € C as z approaches z~.
tangent to C directed in the direction of the integration; the negative sign is
z'-z
z - z
t
is understood to mean a limit along nontangential paths. In the right-hanJ\mem{Jer,
433
= Iz -
value. Let Xo denote the angle of inclination of the tangent to C at the point
z~ I
(+)
± 1Tif (z~) for all z ~ € C as z approaches z ~ along all nontangential
z 0 to the real axis. By letting z approach z ~ along a non tangential path I and setring z
z~+ ifel(1o+'!'),
we conclude that, for sufficiently small
10,
we have
Il/JI < l/Jo or !1T-l/J! < l/Jo, where l/Jo < p/2. We may assume that this last condi tion is satisfied for all points z on the curve I.
paths. Here, the sign is plus when the nontangential path lies to the left of the tangent to C at the point z~, and it is minos when this path lies to the right of
=
> O. Let h = h (.,,) > 0 denote a number
Now, suppose that we are given ." such that, for \s - sol
< h,
that tangent. H we can prove this assertion about the difference· (4), then, almost
Z'(S)--'-z~
everywhere on C and in particular at points at which f(z') is finite, the existence
s-so
of one of the integrals (4) will imply the existence of the corresponding limit for Let
the other integral.
10
denote an arbitrary number in the interval 0
This assertion is easily proved if fez') == 1 on C. Let z ~ = z '(s 0) denote an arbitrary
C\
in log (z ' -z) for one circuit around the curve C. The imaginary part of this from the point z to the point z'. But (log (z / - z))c does not- change if we re
f), Z /(s 0 + (0)) on C with a semicircle y with the same endpoints located on the other side of C from z. On the other hand, the increment place the arc (z '(s
0)
± 1Ti
(z') -
~
rp (a) da (1 +m) a-elei,!,
Therefore, we can write (5) as follows:
O. Obviously, the value of the subtrahend in (4) is equal to this last ,
can be made arbitrarily close to
10,
(f (z· (s» - I (z~» dZ~~S) ds -------:-.-----.:.:...:/;.,-.,....,-, C (S-8 (e'Io+a)_eie (10+'1')
\
I (z~) dz'= ) z'-z
CI t;
F (z,
increment de creased by the increment in (log (z ' - z ~))'Y' which, for sufficiently small
z -z
,
dz =
We have
0 -
in log (z ' - z~) for one circuit around this new path by an amount that approaches 10 - ....
1
< 10 < h.
If we now set s-sO=a, oe- i1o =m=m(a), (f(z/(s)-f(z~))(dz'(s)/ds)x e-(~o = ¢(a), where Iml <-." tor lal < hand ¢(O) = 0, we have
in log (z ' - z) for one circuit around the new path will differ from the increment zero as
(z~)
1 (z') - I
point on C at which Idz '(s)/dsl = 1. The first integral in (4) gives the increment increment is the increment in the angle of inclination of the vector extending
13 1<"l.
=ei1o+8,
(the sign depending on the position
z~) =
\
~
i
((l
(a) da C + m)ea,!,- eirpeie1eJI) (1 + m) a + .)
I
of the point z with respect to the direction of integration on C). This proves
'f (a) da 11
I
_\ -
~;.I
,
••
(6)
-I
We require also that the point z ~ be such that lilllo-.... o¢(a)/a
=
0, where
that in the present case, the differe nce (4) approache s ± 1Ti with the required choice of sign as
10 -->
O.
W(a) =
To prove this same assertion in the general case, it will now be sufficient
a
a
D
0
~ I tp (a) Ida= ~ If(z' (so
+ a» -
f(z'(so» Ida.
to show that the quantity
F(z z')=~ I(z') dz'- ~ I(Z/), dZ'-f(Z')[~ ~_ ~ ~]_ ,
0
z' - z
z' -
I
Zo
0
z' - z
z' - z' .
l:
0 I
Let us show that almost all points on C satisfy this requirement. Consider the set
!r 1, r2 ,
••• \
of all points in the plane with rational coordinates. Note that,
--~-~----
434
--~---------~--,-"---------_._--
----~._-
----"-~_._-
..._----"...,_._-:-.,----,,..
--
._-.
-~
-FE'':"-=''~~-o:
X. FUNCTIONS ANALYTIC INSIDE A RECTIFIABLE CONTOUR
for each
rk,
we have by Lebesgue's theorem,
s
~
d d S
§4. CAUCHY'S FORMULA
or, replacing the modulus in the denominator with the real part,
s+a
If(z' (s» - Tk Ids= lim _1 \' If (z' (s» - Tk Ids= If(z'(s» ~
a-+O a
R
~k \
&
\1 --=:
\
c?
....
s
t I
0< s
< S at
< s < S.
j
Therefore, if E k is the set of poin ts in the ibterval
which the above assertion does not hold or at which f(z '(s» = "",
l'fa\a) I da=
j+ 2.) .h
~max -
Let us now set
do
.h
I3 [1-h +-+-h 1 1 +1 ] ~12 max I
UEk •
12 R\1~(cos,\Jo_"I)
k=l
Then mes E rk
=
O. If So does not belong to E, then, for arbitrary
(> 0,
we can
Tk
IF (z, zo)'I ~ I\ d
1< T'
+~
~
If(z' (so +0» -f(z'(so» Ida ~+
~
I
I
0
R{1j)= ( 2
If(z'(so +0» '- Tk Ido
eieitlJ'f (a) da ((l +m) 0 -8ie1tlJ) (l +m) a
I+R ("I),
h
where
For a sufficiently small, ~
(l-"I)l a l
As a consequence of these inequalities, we have
such that
If(z' (So» -
12 ) +-1-"I
1
Finally, we. note that the last integral approaches zero as
a
' +cr-1 X\ If(z'(so»-Tk Ids~2+2 =6. t
But this means that for So outside E,
limiting values of !F(z, z~)1 as that F (z, z~)
a
(--->
0 because we
can take the limit under the integral sign. Therefore, we conclude that none of the (-->
0 exceed the number R(TJ)' But since these
limiting values are independent of TJ and since R( TJ)
W (a) ~ 0 as 0-0.
a
00
choose
e Jal
a
Consequently,
E=
-a-I- d~.
~
ill
then mes E k = O.
Ilf' (0) I
d
Integration by parts yie Ids for almost all s in 0
435
--->
0 as
(->
0, that is, as z
--->
--->
0 as TJ
--->
0, this means
z ~ along the curve l. This proves
.the assertion and with it the theorem.
Under the assumption that we have made, let us find a bound for each of the
§4. Cauchy's formula
integrals in (6). We have e
Rl=j
~
-e
If'(a)da (1 +m) a-tieitlJ
j
e
\' ~~
-e
l
accordance with Cauchy's formula in the case of a simply connected domain
or, replacing the modulus in the denominator with the imaginary part,
•
Rl~ n~".'~_"1 '+\'lep(a)lda~ : o -~e cos
"I I a 1< h
a
and over the other two arcs, which together we denote by C(h' We have R -, e1ttt
c....
eh
bounded by an arbitrary rectifiable closed Jordan curve •
The or em 1. Let f(z) denote a function that is regular in a bounded domain
2maxl
0 -
Let us now break the first integral in (6) into three integrals over the arc C h
r
The results expounded in the preceding section enabled Privalov 1) to give a complete answer to the question of the representability of analytic functions in
B, the boundary of which is a closed rectifiable Jordan curve C. Suppose that . f(z) has de finite limiting values almost everywhere on C along nontangential paths.
£
Suppose that these limiting values define a symmetric function f(z') on
Th~!!.Jor Cauchy's formula 1) Privalov [1919].
436
§4. CAUCHY'S FORMULA
X. FUNCTIONS ANALYTIC INSIDE A RECTIFIABLE CONTOUR
f(z)
= _1_ \ I(z') dz' 2111 ~ z'-z
paths lying outside C or, by virtue of the uniqueness theorem, equivalent to the {I)
. c
requirement that equation (3) hold outside C. The expansion of the left-hand side of (3) about z
=
00
in powers of 1/ z shows that condition (3) is equivalent to the
system of conditions (2). This completes the proof of the theorem.
to hold in B, it is necessary and sufficient that
rf(z') z'ndz ' -0 J -,
n-O,l, ... ,
Theorem 2. Let f(z') denot e a summable function defined on a closed recti (2)
\ ~(z') dz' _0. ~ z-z
fiable Jordan curve C. Then, for there to exist a function fez) that is regular in the finite domain B bounded by the curve C, that has limiting values almost
or, what amounts to the same thing, that outside jj
everywhere on C equal to fez ') along nontangential paths, and that can be repre (3)
sented in B in accordance with Cauchy's formula (1), it is necessary and suf ficient that the system of conditions (2) be satisfied.
Proof. Consider the function
cjI(z)=f (z)- _1_ \ I(z') dz'. 2111 ~ z'-z
(4)
Proof. H the function f(z) mentioned in the theorem exists, then, by 1heorem 1, the system oc'conditions (2) must be satisfied. Conversely, if the conditions (2) are satisfied, then outside
H the function fez) can be represented in the domain B in accordance with
B
~I- \ I(z') d ' 2111 ~ z' _zz
we have (3). Consequently, Cauchy's integral
_I \ ~ (z') dz'
Cauchy's formula (1), that is, if rjJ(z) == 0 in B, then Cauchy's integral
2111 cY z -z
(5)
has almost everywhere on C limiting values equal to zero along all paths lying
B.
has limiting values equal to f(z') along all nontangential paths in B almost
outside
everywhere on C. Conversely, if Cauchy's integral (5) has almost everywhere 'on
lar integral exists almost everywhere on C and that
Therefore, from the theorem of §3, we conclude that Cauchy's singu
C limiting values equal to f(z') along all nontangential paths in B, then
rjJ (z)
--+
_I \ 2111
0 almost everywhere on C along all non tangential paths in B. Con
sequently, by the uniqueness theorem of
§ 2,
rjJ(z) == 0 in B; that is, the function
I
(z')
d z'-z~
along all non tangential paths contained in
B. But, by formula (3) of § 3
with the positive sign taken, this last will hold if and only if Cauchy's singular
2111 ~ Z'-Z
in B, we conclude from the theorem of § 3 that the function f(z) has almost everywhere on C limiting values among aU nontangential paths lying outside B equal to
integral
_1 \ ~(z') , dz' 2111 ~ z -zo
exists almost everywhere on C and is equal to f(z~)j2. This, in turn, is by formula (3) of §3 with the negative sign taken, equivalent to the requirement that Cauchy's integral (5), which represents a function that is regular outside C, has almost everywhere on C limiting values equal to zero along aU nontangential
(6)
f (z) = -.!.., \ I (z') dz'
Thus, the function fez) has a representation in B by Cauchy's formula (1) if and only if Cauchy's integral has almost everywhere on C limiting values equal to
dz' _ I (z~) 2 •
H we now set
fez) can be represented in B in accordance with Cauchy's formula (1).
f (z')
437
_I \ 2111
I(z') dz' +/(z~)
d z' -
z~
.
2'
or, in accordance with (6), equal to fez ~). Consequently, the function f (z) answers the requirements of the theorem. This completes the proof of the theoreDL
438
§5. CLASSES OF FUNCTIONS. CAUCHY'S FORJlULA
X. FUNCTIONS ANALYTIC INSIDE A RECTIFIABLE CONTOU..t
I"
\
§ 5.
Classes of functions. Cauchy's formula
conver ge s uniformly in the interior of the disk
\ \
Suppose, just as above, that B is the finite domain in the z~plane bodnded by a rectifiable closed Jordan curve C. Let domain B conformally onto the disk
1'1
,=
X (z) denote a function that
< 1. Let z
~aps
1'1 = r
us set
If(q=f((O(~))~(O'(Q, epn(Q=f((On(~»Y (O~(~).
the
Since, for evew n, the function f(wn(')) is bounded in 2",
> 0, the class of all functions f(z) -t 0 that
~ rf(z) fPds cr
if
f(w(,nVw'(,)
(1 )
f (z)
the interior of the disk
')1
is summable
1'1 < 1
~ If(z) IP ds ~ M, CII
to the function
I ¢n (,)1
converges uniformly in
¢(,). Consequently, for 0 < r < 1,
~
which proves that the function f(z) belongs to the class
Ep'
Theorem 1. If f(z) EEl' then Cauchy's theorem holds:
lim ~ If(z)/Pds= ~lf(z')IPds.
(2)
C
r
~f(Z')dz'=O, This theorem follows from Theorem 5 of latter theorem to the function f(w
that are regular in B and have the properry that, for each of them, there exists in B a sequence of rectifiable closed Jordan curves Cn, for n
=
1, 2, ... , which con
verge to C and for which the integrals
tionto have almost everywhere on C definite limiting values along non tangential
2'l';l
1)
Obviously, every function
that is regular in B, there exists a sequence of curves C n with denote a function that maps the disk
1'1
< 1 conformally onto a finite domain bounded by the curve Cn in such a: way that W n (0) = w (0) and arg w; (0) = arg w' (0). Then, by the theorem on the con vergence of univalent functions, the sequence of the functions wn('),n= 1, 2, ... , 1) Keldy¥ and Lavrent 'ev [1937].
(,nw' (,).
e
f(z) =_1 \ j(z') dz'
E (, satisfies this condition. Let us prove the converse. Suppose that, for Wn ( , )
Chapter IX if we apply the
paths and for Cauchy's formula
are bounded by a finite quantity independent of n.
the property described. Let z =
§ 4,
Theorem 2.1) Let f(z) denote a function that is regular in B. For this func
~ If(z) IP ds cn
f (z)
(3)
C
Ep for p > 0, can also be defined in this way, but it can also be de
fined independently of conformal mapping, namely, as the class of functions f(z)
a function
=
We use the notation adopted above in the following two theorems:
r ..... l C
f (z)
P d6
~ I If (re i9 ) jP d6 ~ M or ~ If(z) jP ds ~M,
on C; and
The class
I
0
o
f(z) has definite limiting values
f(z ') almost everywhere on C, over all nontangential paths; If(z
lP d6 ~ ~ IIfll (e i9)
2"
belongs to the class Ep if and only
Ep , then
)
where M is independent of r. But the sequence
E Hp • Thus, by §40f Chapter IX and §1 of the present
chapter, if the function f(z) belongs to
i9
we have
(V. I. Sm irnov). By c hanging to the variable 'in this
integral, we easily see that the function
< 1, it follows that
2",
~ I Ifll (re o
are regular in B and have the property that the integral
is bounded for 0
1'1
¢n (,) E HP' Therefore, for 0 < r < 1,
under the mapping z = w(,).
Let u~ denote by E p' where p
< 1 to the function w(,). Let
denote its inverse
= W (,)
fWlction. Let Cr denote circular images in the domain B, that is, curv¢s in B corresponding to circle
439
(4)
z'-z
to hold everywhere in B, it is necessary and sufficient that f(z) be a member of
E l' Proof. Let f(z) denote a member of E l' Then, in accordance with what was said above, the function f(z) has almost everywhere on C definite limit values .. f(z') along nontangential paths and these values are summable on C. Consider
the integral 1) Smirnov [1932 a].
0='"
440
cr_~_,u,._
_._ ..•
=~_
_~'U
*.-SIi!!';~-"=FP!!'~"
__.._'__._~
~. __.
__,
_.__._'"...
_..._. _
.._
••.-•.._ =~.
.
X. FUNCTIONS ANALYTIC INSIDE A RECTIFIABLE CONTOUR §6. ON THE EXTRE\lA OF MEAN MODULI
441
2"
1=_1_ \ f (00 (e I9» 6)' (e i9 ) de i9 2nl ; 6) (ei~ 00 (C) --.
(5)
t\ f(z')z,ndz'=O,
We set 00
(C') 1
00
(C)
= w' (C) (~, _
C)
+ R (1;;, C).
sequently, in
B
H \'
1 \ f (z') dz' 2ni ~ X (z') - X (z)
2"
l=f(IIl(C»+2~1 ~f(w(ei9»w'(ei9)R(ei9,
C)de i9 .
(6)
o
r;; in
But, for fixed is regular in
the disk
'(I < 1
f(w «(»)w '«()R «(,
Ir;; I < 1,
the function
and continuous on
r;;)
1(1 ~ 1.
' ,... ,
n=O I
we conclude that the integral in the right-hand member of (7) is equal to zero. Con
When we substitute this into (5), we obtain, remembering that f(w(r;;))w' (r;;) € •
_ ..~.
R «(, r;;) as a function of (
H we now shift to the disk '"
1 \ f (00 (e i9
2ni
is a member of HI and consequently, by Cauchy's
f (z) X' (z) •
< 1, we obtain
2"
Therefore, the function
=
J
»
til'
(e iD) delS
pl8_t
,
=f(IIl(C»ro (C),
o
theorem, the integral on the right side of (6) is equal to zero. Thus, we have
that is, Cauchy's formula holds for the function f(w(r;;))w'(r;;), which is regular in
2"
1r;;1 < 1.
1 \ f (w (e i9» w' (e i9 ) dei9
2nl; w(ei9)_oo(Q =f(w(C».
Consequently, f(w(r;;))w'(r;;) €
HI; that is, f(z) €
£1' This completes
the proof of the theorem.
H we now shift to the z-plane, we obtain Cauchy's formula (4).1) §6. On the extrema of mean moduli Now suppose, conversely, that the function· f(z) has almost everywhere on C definite limiting values f(z') along non tangential paths, that these values are summable on C, and that Cauchy's formula (4) holds in B.
l' = _1 \ . 2nl
E
'I
Consider the integral
We keep the notation used in
and that w '(0) > O. Let us show that, in this case, the function
f (z') dz'
'-''I.
V
f_.\.
X' (z) ) lip
•
Fp(z)= ( X' (0)
We set
... Then, for z €
'..
1
....
=....
.!. . + Q(z', z).
and continuous on
(7)
B, the function Q(z " z) as a function of z' is regular in B
E.
> 0,
(1)
I=~lf(z)JPds (2) c out of all func tions F (z) belonging to the class £ p that satisfy the condition F (0) = 1. We shall also show that this minimum is equal to 21TW'(0). . Shifting to the disk
But for fixed z €
,p
and only this function minimizes the integral
B,
r=i,~~ + 2~i ~ fez') Q(z', z)dz',
§ 5 regarding the domain B except that we now
make the additional assumptions that the point z = 0 be longs to B, that w(O) = 0,
1r;;1 < 1,
we see that the problem of minimizing the irite
gral (2) reduces to minimizing the integral
Consequently, in accordance with Walsh's theorem,2) it can
be approximated uniformly in 11 by means of a polynomial to an arbitrary degree of accuracy. Since by Th eOrem 1 of § 4, 1) The proof given has the advantage thar ir does nor depend on §§ 3 and 4 and can be used to prove Cauchy's formula for domains with recrifiable boundaries orher than Jordan curves. On the orher hand, if we do use Theorem 1 of §4, and keep in mind the fact rhat, by Theorem 1, fc
.,tI.
2"
2"
I=~ IF(w(eI9»w'(ei9)\IPIPds=~ If(e i9)!PdB o 0
(3)
with respect to the functions f(r;;) = F (w (r;;))w' (r;;) 1/ p belonging to the class Hp that satisfy the condition f(o) = w '(0)11
p.
Now, the only possible extremal func
tion for the integral (3) is a function f(r;;) without zeros in '"
<1
because,
otherwise, by dividing f(r;;) by a Blaschke function b (r;;) we would get a function
~
,..".._--.':""_-;----,.....,-------:--
----=-,~~--_":"----
442
...
-"".---=--":"=-.-~_----.,':""",-----;;,
..-.--.:-:.--:----::;;;';'_.-:---=-"....-=•.•""-=---=. ,-;.•
-_.=",,-~---:_-,--~=,,,.::-"'""
.,. --.--.....------ ":",-.=.".._---:-._---=<;....
"'~""""'"--,=.---.,=-::""'------,.-.,""""""
...
~..,,----.._"
..
--,=-"?"'"'.,.~'".-
.•,""=--_~"'_.,-~'-----'-
"'=------
-----------
§6. ON THE EXTREMA OF MEAN MODUL'f
X. FUNCTIONS ANALYTIC INSIDE A RECTIFIABLE CONTOU\t
[1 U;) = b(0)[ (Olb«() € Hp for which [1(0) =
'(0)1/ p and for which
UJ
(3) is equal to the same integral for the function
< 1.
by !b(oW, where, obviously, Ib(O)1
Hp such that [(0)
= UJ
t~e\ integral
[(0 except that it is m~ltip1ielt
If we now consider the function'
'(0)1/ p without zeros in
'(I < 1
(0 €
and if we set
ro
f(C)P/2= ~ Cn~n,
co=
n=O
quantities depending only on 0 and M. Therefore, if we apply to the coefficients in these polynomials the condensation principle, we find a polynomial qn(z) that yields the smallest value for the integral (2). This proves the existence of extremal polynomials in
E (n).
k
2~ ~ If(eiD)IPdB=lirn2~ \
2n
r) i,; ~ ~ = e 0 d(
o
r_l
~
ro
If(re iD )2/P lide=lim
~ ICnlir-in~\coli.
r_1 n=O
Here, equality can hold only when [(0 == c
5/
p = UJ
iD
+c
(4)
•
which will playa significant role in all that follows. We have 2n
'(0)1/ P; that is, F(z) is given
Id(QI=e
'P) dO
0
(5) ,
where (= re i ¢. But, from Holder's inequality, we have, for 0
E p , let us now considet the set E
polynomials qn(z) of degree not exceeding n such that qn(O)
=
1 and let us pose
the problem of minimizing the integral (2).
~
I 00' (e iD) I), P (r, B- cp) dB
o
~
equal to M for one of the polynomials qn (z) E E
< A < 1,
2",
Let us show that extremal polynomials exist. Suppose that the integral (2) is
2n
U o
2"
1 00 '
(e iD) I per,
e-cp)dB/. Uper, B- cp) dey->., 0
that is, 2",
one of these polynomials and b «() is Blaschke's function for qn (UJ «()), then
(z) = (qn(z)/b(X(z)))P € we have
Pn,p (z)'
logj w' liD) I e e -C dO ---;0-
~ . ~ log I w' (e iD ) I Per. D-
by formula (1). This was what we wished to show. Along with all the functions F (z) €
We denote these extremal polynomials by
We now define the function
V 00' (0),
we have ~
443
2~ ~
E l' Therefore, in accordance with Cauchy's formula,
2",
I 00'
(e
iD
)
!AP(r,
o
e-cp)dB~(2~ ~
1 00'
(e iD) I P(r, B-cp)det.
0
If we now expand both sides in powers of A, subtract 1 from both sides, divide
f(Z)=_1 \ fez') dz' 27t1
t z' -z
by A, and then let A approach 0, we obtain the inequality
.
~
k
2~}
Iog!w'(eiD)!P(r,
B-cp)de~lOg[2~} !w'(eiD)IP(r, B-cp)dB].
Now, let us denote by 0 the distance between the point z and the curve C. For
Iz - z'l > 012,
On the basis of this inequali ty, we obtain from (5)
we have
2",
If(z)/
~:8 ~
Id(QI~2~) I w'(e iD )IP(r,
If(z')/ds so that, for 0
and, consequently,
Iqn(Z)IP~:8 ~ Iqn(Z')IPds~::. C
But the coefficients in the polynomial qn (z) do not exceed certain finite
B-cp)dB,
< r < 1, 2lt
2",
~ I d (reiD) I dB ~~ I 00' (e iD) I dB.
o
0
This shows that d(O E HI. Furthermore, it follows from (5) that
~
444
X. FUNCTIONS ANALYTIC INSIDE A RECTIFIABLE CONTOUR\
Id(eie)1
=
IUJ'(eie)1 almost everywhere on 1~1
=
§6. ON THE EXTREMA OF MEAN MODULI •
l-
when the function log IUJ I(~)I, which is harmonic in
I~I < 1, can
be represented in is bounded for 0
< r < 1. Therefore, in accordance with the familiar test for taking
the limit under the Lebesgue integral sign, we also have
2"
11l((c)1=2~ ~
.
~ (log I w' (reID) \ )~ dO o
< 1 holdsobl y
that disk by Poisson's integral: log
+
2"
It is obvious from (4) that the identity d(~):; UJ '(~) in I~I
log Iw' (e iD ) I per, 0 - ep)dO.
(6)
1
o
lim 2
r_l
'It
2"
~
+
1
§ 2,
we have log UJ
'(~)
IogUJ'(~) itself can be represented in I~I
from the proof of Theorem 5 of
§2
< 1 by Poisson's integral. It follows
that condition (6) is also satisfied in the more
general case in which arg UJ ' (~) is bounded in
I~ I < 1.
This will be the case, in
particular, for starlike domains and also for domains bounded by piecewise smooth curves that make nonzero interior angles with each other.
1)
(6 I/f)
'It ~
o
H we double this equation and subtract it from (6''), we get
HI and consequently, the function €
2"
log I w' (reiD) I dO = 2 \ l6g I w' (e lo) I dO.
Condition (6) is satisfied, for example, when C is a smooth curve because then, by Theorem 5 of
445
1
2"
2"
lim 2 \ I log 11Il' (reiD) II dO =-l r_1 'It.' 2'1t
o
~ I log 11Il' (e ID) II dO.
Keeping this in mind and reasoning as in the proof of equation (6) of Chapter IX, we obtain the limit relation
However, there
§ 4 of
2"
lim ~
exist s an example of a rectifiable closed Jordan curve C for which condition (6)
r-Io
1log I w' (reiD) 1- log I w' (e iO) II dO =
D.
is not satisfied. 2) We shall refer to Condition (6) in connection with the curve C Finally, from this we obtain equation (6) just as in the proof of Theorem 3 of §4, Chapter IX.
as condition (5), following Smirnov. 3 ) We note that condition (5) is equivalent to the condition
We now have the
2,.
loglw'(D)I=2~ ~ log\w'(eiD)\dO,
(6') •
Theorem. I} In the case of an arbitrary rectifiable clased Jordan curve C that encircles the point z
that is, to the condition d (0) = UJ '(0). This is true because (6') is obtained from
=
0, we have
lim
11_00
(6) with ~ = O.
To prove that (6') implies (6), we note that (6') can be written in the form
~ IPm p (z') /P ds =
C
2"
E(n)
becomes broader with increasing
n, the integrals in (7) do not increase. Consequendy the limit in (7) exists.
2"
~i~1 2~ ~ log I w' (reiD) I dO = 2~ ~ log I w' (e iO) IdO.
(6 ")
Let bn,p(z) denote the Blaschke function for Pn,p(UJ(~)) and suppose that
PII. P (w (C» = bn • p (C) hll , P (C). On the other hand, the inequality lcig x
<
x P /p
for p
(7)
where d (0) is given by formula (4). Proof. Since the family of polynomials
of the limiting re lationship
27t. d (D),
> 0 implies that the integral
Then, the equation bn.p(O)hn,p(O)
!
=
IPn,p(Z)jPds=J /Pn,p(w(eio»jP,[w'(eio)/dfJ =
---I} Keldy¥ and Lavrent 'ev [1937] exhibited a number of other cases in which (6) is satisfied. 2) Keldy~ and Lavrent ' ev [1937]. 3) Smirnov [ 1928].
1 implies that Ihn,p(O)!?: 1. We have
21'C
1) Keldy~ and Lavrent 'ev [1937].
§6.
X. FUNCTIONS ANAL YTIC INSIDE A RECTIFIABLE CONTOUR'
446
21<
~ I hn• P (e iO)
=
o
= ~ Ih~:;
iO (e )
o
d l / 2 (e iO)
I~ dt ~ 21t I
21<
~
hn•
P (0)
IP d
o
(0) ~ 21td (0)
g~6)ld(eiO)ldB+ ~
IHeiO)jPld(eiO)ldB= \ ;
= ),ld(0)ldO+1d(0)! )pg(B) dO - lim 71-+ ex>
~ 21Cd (0)
~ IPn. P(z) /P ds ~ 21td (0). E (n)
for which the ~alue of the
Id (,) I can be
represented in \ 'I
<1
Consequently,
t/J (0) > 0
log'f' (0) dO
logd(O)=
2~ ~
~J
p
(9)
But it follows from the theory of trigonometric series that corresponding to a bounded measurable function t/J(e ie ) is a sequence of polynomials an (e
I d (e iO) I d (0)
in e ie, uniformly bounded in
(Fejer's sums 1)
,
everywhere in (0, 277) to
t/J (e i 8)
with an (0)
=
~ log g (B) dB = O.
21<
lim ~
o On the basis of this, the interval (0, 277) contains a measurable set P on which
II g«()) are bounded, whereas on its complement CP, 1 211;
\logg(B)!dO
Cp
~
Ian (e iO) IP Id (e/o) IdB =
71-+000
21<
~
Cp
(0, 277), that converges almost
t/J (0).
Obviously, for these poly
21<
~
I<j> (e iO) IP Id (e iO ) IdO.
0
o
Ian (e iO) IPId (e iO ) IdB ~ 21Cd (0) (1 + 28)
or
Let us denote by ¢ «()) the function that is equal to
II g«()) on P and 1 on
~
CPo Define
C 21< I \ '0 2"p J log'f' (0) C dO e 0 e'• - C
=e
I \ - 2"p J log g (0)
p
<+ 'B
CdB e·B_C
Ian (X (z'» IP ds ~ 21Cd (0)(1 + 28)
(10)
> 1 +1 2£ .
(11)
and furthermore,
a" (0) We have
Since the function an (X (z)) is continuous on
{g~
)
For sufficiently large n, we therefore have
g(B)dO<8,
where (is a given positive number.
!.p('''lI' -
ie
nomials,
21<
<. +
(0) dO
1
we conclude that the function g «()) is summable in (0, 277) and that
=
(8)
<j>P(O)~I+£'
o
'f ..
+ 8).
and
logl d(e iO ) I dB.
Consequently, by setting
g(B)
...!.... \ log g ~J =e CP
- 2-.!.- \ log g (B) dO
=e
21<
2~ ~
d (0) = 21td (0) (1
21<
\ 2...!.... ~J <j>(O)=e 0
in accordance with
Poisson's formula, we have, in particular,
the functions g «()) and
21t8
Furthermore,
integral (2) will be arbitrarily close to 277d(0). Since, by (5), log
+
C
It remains now to construct a polynomial q(z) E
(r)
Id(eiB)jdB
Cp
and, consequently,
,II
447
Now,
IP ·1 d (e iO) IdO
21<
ON THE EXTREMA OF MEAN MODUCI
almost everywhere on P, almost everywhere on
CPo
---S 'f' II 'f ie ~oo ine I) peCIIca y, I fie ) "-' -"-n=ocne ,then
Un
B (e
it can, in accordance with
i8
)=
~n-l « k)/) ik e k=O n n e •
£.
=~==
__
":::::=::0:;'::-==:::-==:~~-:::,:':::~=::==-=:-'=::"-:==-~~:='=:::='_S===-~===~-::::""'~~~
~~~~~'::=::~;=~::::::::::::=:::='~:=~;;''==,?==::~~==::E=:::-=:::-=:::'~::~_:''~:-::
~"-:":".'~_~:- _::.-:':::':_:::~:;"'~,~
;~E~;;;:-
Walsh 's
1)
=
f&
the polynomial P(z)
/1
1, we have
I
~ P (z)
\P ds ,::;; 21td (0) (1
=
;;".:::~:~~:::~~~:.:::-~:~::::':~'';;.E::-~~"~~;;::::';':~,;;:;;~~,:
.':"'C'-':::'~::+-~~;~:~~':::;::-:::~·::-;:-:::
449
Equation (2) then follows on the basis of the theorem in §6. This completes the
theorem, be uniformly approximatedjn-B by a polynomial p(z) that also
satisfies inequalities (10) and (11). Then,
-:::::: '" w:::~;~.
§7. APPROXIMATION IN MEAN
X. FUNCTIONS ANALYTIC INSIDE A RECTIFIABLE CONTOUR
448
P(O)
';'~:::::~~=~::'=~~~-:::"::~:::-":'~;::~2~~-;;::::::'~~.,0".:i::E-:~:::~~:::':::~::0:
proof of the theorem.
p(z)/p(O),
Since
+ 2e)~ =
21td (0) (1
+ e'),
V."/!l",, = V
P~ (z) =
C
d (0)
x: (z)
t
where (' and ( are arbitrarily small. This completes the formula of (7), and hence almost everywhere on C, condition (5) is a necessary and sufficient condition for
the theorem.
§ 7.
F;(z) to be identically equal to F /z), where F 2(z) i~ defined by equation (1) of Approximation in mean and the theory of orthogonal pol~omials
§6. Therefore, the preceding theorem leads to
As an application of the results of §6, let us now look at questions of
Theorem 2. For the limit relation
approximation in the class E 2' 2)
lim
converges in mean on C to the function (1)
lim ~ I P~ (z) - Pn.~ (z) I~ ds = O.
(2)
p.
polynomials in z is complete in a domain B jf, for any function F(z) E E 2 and
n-+oo C
) IPHz) - Pn.~ (z) I~ ds = o~
< 1,
any (> 0, there exists a polynomial P(z) such that
~ I P (z) - P (z) l~ ds
we have
I
i9 Pn.dw (e » -
V/(~~') I~ I
C
I9 d (e ) I de
~
o
B, it is necessary and sufficient that the curoe C satisfy condition (5).
IPn.~(w(eiO»V d(e i8)_ V
d(O)I~de
Proof of the sufficiency. Suppose that condition (5) is satisfied. Then, in accordance with Theorem 2, we have equation (3). Now," if F(z) E E 2' then
2",
=
~ IP'I, ~ (w (e/ 9» I~ I d (e iO ) Ide + 21td (0)
f(~) = F (w (~)) Jw '(i) E H 2 and, consequently, in accordance with Theorem 2 of
o
§4
2",
-2m (V d(O)
~ Pn.~(w(eIO»
of Chapter IX,
yd (e 1ti) de)
21<
lim ~ Ij(eIO)_j(rei9)J~de=0
o and, since Pn,
2(W (~)) Jd
r-+ 1 0
(~) E H 2' this is equal to
or
~ IPn, ~ (z) I' ds - 21td (0). c
lim ~ IP (z) - j(rX (z» V
r-+ 1 C ~,:
1) Cf. Walsh
< e.
Theorem 3. For the family of all polynomials in z to be complete in a domain
2",
=
444). We turn now to approximation in mean. We shall say that the family of all
that is,
2",
(3)
to hold, it is necessary and sufficient that the curoe C satisfy condition (5) (cf.
P~(z)= -V d~(~;»' P~(O)=l,
Proof. Shifting to the disk I~\
~ IP, (z) - Pn. , (z) I~ ds = 0
lI-+00C·
Theorem 1. The sequen:ce of the extremal polynomials Pn. /z) defined in §6
h9601 p. 47 of 1965 ed.
2) Smirnov (1928]. We note that the theorems that follow have been generalized by Keldy~ and Lavrent'ev to the case of the class E p '
x: (z) I~ ds = O.
This means that, for given (> 0, there exists an r in 0
~ I P(z) -j(rx (z» V
C
x' (z) l~ds < E.
such that
'-
..-
0Il_""_"""'."",',",~
..•. .--,_J<......_ _""""'....._"""""""".... _.,...;.~"~,:-....;;"~~;,.~;;;;;~~;;;...,.;;.;::.,.."";::;;.,.;;'::."i\,,;,,.<;.
,,-,,,_'O;R'J'I_.
~_",,
,-.~,.,.;:;..*~~,.:,_=«':"i
'"'~3''',,":,·~~,."~
....,'<,
"·"~"'""·'''·'7''··'''·''''''''·''~_)''-'''''''''·'''·"_''·'1''
X. FUNCTIONS ANAL YTIC INSIDE A RECTIFIABLE CONTOUR
450
But the function f(rx(z)) is continuous on
B.
If(rx (z)) Let us form the polynomials Pn(z)
V X' (0) -
=
Q(z) I
B,
lim P n (z)
< e.
in B.l) In particular, limn-->ooPn(O)
~ IF (z) - Pn(z) I~ds ~ ~ (I F(z) - f(rx (z)) V X' (z) I c .. c
Pn. 2 (z)
Therefore, if we set Pn(z)
V x' (0))
""';_'n·.~·~.".."""; ...;;;".';"'';';';;;-;''';';';'''.~,;,;;:;'",,,~:.;
451
On the other hand, we obtain from (4), by use of Cauchy's formula, the result that n-+oo
Q(z)Pn,2(z). We have
+1 f (rX (z)) (V x' (z) -
.".'."".....'.""-"',,~,,_,"""~,,"'."".,'
§7. APPROXIMATION IN MEAN
Consequently, in accordance with
Walsh's theorem,Orhere exists a polynomial Q(z) such that, in
,. e."'. "<~-"'. '."';""1-'7-,.,.•.,,,,,,,,,,,,,,,,,",,.,,,,,,, ,-_<.''''-'~~''''''_)~''''<''''''<'''~",,,'''''''"
=
= V x' (z)
Jx '(0).
= Pn(z)/Pn(O), Pn(O) = 1, we have, in accordance
with (5), 1
+ IPn. ~ (z) (j (rX (z)) V x' (0) .
Q(z)) I)~ ds.
lim
~ I Pn (z) 12 ds =
21t0l' (0).
n-+oo C
I'
Consequently, by applying Minkowski's inequality, we obtain
t9
( C~IF(Z)-Pn(Z)I~dS ,
nI
In accordance with §6, this yields d(O)
V x' (z)
l~ ds yt9
+ max If(rx (z)) IV x' (0) (~C I F 2(z) -
Pn. 2 (z)
~
F (z) - f(rx (z))
'(0); that is, condition (5) is satis fied. This completes the proof of the theorem. UJ
The results that we have obtained are closely connected with the theory of
C
z~B
=
I~ ds //9
orthogonal polynomials for a domain B. By a system of orthogonal polynomials, we mean here a sequence of poly
+ e (~ Pn. 2(Z) \2 ds //9. 1
nomials Kn(z), for n
C
Here, the right-hand member is, for sufficiently large n, less that
2V;.
Since (
is an arbitrary positive number, this proves the sufficiency of the condition of the
=
0, 1,"', such that, for each n, Kn(z) is of degree nand
('
-
JKm(z)Kn(z)ds= C
{I..
if
0..
if
m=n, m #- n.
(6)
theorem. Proof of the necessity. Suppose that the system of polynomials is complete , for the domain B. Then, for the function
J;?T;) E E 2,
there exists a sequence
~ I V x' (z) - P n (z) j 2 ds=0.
(4)
n-+oo C
H we now apply Minkowski's inequality twice, we have 9
C
2
9
C
C
~ IF (z) 12 ds =
C
which, in accordance with (4), y ie Ids
lim ~IPII(z)12ds=21t
11-+00 C
•
(5)
11
1]
2
(8)
I Ck 1 ,
k=!
where the c k are the Fourier coefficients for F(z). Proof. Let p(z) denote an arbitrary nth-degree polynomial and let
1) Cf. Walsh [1960], p. 47.
(7)
B, it is necessary and sufficient that an arbitrary function F(z) in E 2 satisfy the closedness condition
C
C
n=O, 1, ... ,
C
Theorem 4. For the family of all polynomials in z to be complete in a domain
,
2
C
cII=~F(z)Kn(z)ds,
are called the Fourier coefficients of the functions F(z).
t ~ nIP n (z) - V x' (z) 1dst2 + (~1 x' (z) Ids t (~I x' (z) 1ds t ~ (~I VX' (z) - P n (z) I~ ds //2 + (~I P n (Z) 12ds //2,
(~1 P n (z) I~ ds
tion of the polynomials Kn(z). Now, if F(z) EEl' then the numbers
of polynomials Pn (z) such that
lim
Obviously, an arbitrary polynomial in z can be represented as a linear combina
1) See proof of Theorem 6.
UE
452
§7. APPROXIMATION IN MEAN
X. FUNCTIONS ANALYTIC INSIDE A RECTIFIABLE CONTOUR
453
that converges uniformly inside B.
represent it in the form
Proof. Since the minimum of the integral (9) out of all polynomials in z yields
II
P (Z) = ~
k=O
·ckKk (Z).
(8 ')
.
a polynomial defined in accordance with formula (8') with c ~
1, ...
For fez) € E 2'
,n,
lim SI F (z') - Pn (z') 19 ds II_ex>
~ \ F(z) - p(z) lilds= SI F('t) 19ds+ ~ Ip(z) 19 ds c
C
C
9ds+
1
II
n
~ I Ck 19
C
k=O
ICkI9+
k=O
= 0,
(12)
F(z)-Pn(z)=_l \ F(z')-Pn(z')dz' 21':1 ~
~
I Ck-
z'-Z
.
H E is a closed set in the domain B and if 0 is the distance from E to C, then,
n
II
-2m(~ CkCk)=~IF(z)19ds- ~
C
k=O
C
c k> for k = 0,
if condition (8) is satisfied. Now, by Cauchy's formula, we have in B
C
- 2m (~F(Z)p (z)ds )=~ IF(z)
=
it follows that, for these polynomials Pn(z),
CkI 9.
for z € E,
k=O
IF (z) -
It follows that the minimum for the integral
~ I F(z) -p (Z) 19 dC
Pn (z) I ~
2:e ~ IF (z') -
Pn (z') 1 ds
~
(9)
ice -V~~"-IF-(-z'-)-Pn-(-z'-)/-9-d-s-.l-e-ng-th--'-C.
C
yields a polynomial p(z) determined in accordance with formula (8') with c ~ = c k. Here, this minimum is equal to
~ IF (z)
II
9 1 ds
-
From this .we conclude in accordance with (12) that pn(Z)
--+
F(z) uniformly on E.
Since Pn(z) is a segment of the Fourier series (11), this proves the theorem.
kJ;;{J I Ck 19.
(10)
In particular, if we apply Theorem 6 to the function
f (z) =
.JX '(z) and keep
in mind the fact that then For the system of all polynomials in z to be complete, it is obviously necessary and sufficient that the difference (10) approach 0 as n function
--+
00
for an arbitrary
F (z) € E 2' that is, it is necessary and sufficient that equation
211:
cn
= SV X' (z') Kn (z') ds = ~ K n (Ill (e IS )) -v-;J(ei6 ) dO C
0
(8) hold.
= 21tKn (0) y-;;;-' (0),
This completes the proof of the theorem. we obtain
By combining Theorem 4 with Theorem 3, we immediately obtain
Theorem 5. For an arbitrary function fez) € E 2 to satisfy the closedness formula, it is necessary and sufficient that C satisfy condition (8).
ex>
(~ Kk (0) Kk X' (z) = 21t k=~
We now have
Theorem 6. When condition (8) is satisfied, an arbitrary function F(z) € E 2 00
k=O
ckKk (z),
(Z»)2 ,
~ I Kk (0) I~
k=O
can be expanded in B in a Fourier series
F (z) = ~
Theorem 7. For curves C satisfying condition (8), we have in B the formula
Ck
= SF (z') K k (z') ds, C
(11)
where the convergence of the series in the numerator is uniform inside B.
(13)
§ 1.
GLUING THEOREMS
455
B (2) in such a way that the points x and g(x) in the interval l are mapped into
the same point w of that interval (see Figure 18). Let us now map the univalent 2
CHAPTER XI
""
domain made up of the images G(l) and G(2) of the domains B(l) and B(2) under these mappings and of the interval l uni
I
SOME SUPPLEMENTARY INFORMATION
-1
ICC
•
'I
I
~1
valentlyontothelune B(l)UB(2)Ulin the w'-plane in such a way that the points
§ 1.
± 1 are mapped into themselves. Under this mapping, the domains G (l) and G(2)
Gluing theorems
Gluing theorems establish the existence of analytic functions that obey
are mapped into domains adjacent to some
certain relationships on the boundarie s of certain domains. The nature of the se
smooth curve
relationships is clear from the statements of the theorems we now present.
Al
that is analytic at all
interior points (see Figure 19). H a func
Figure 17
mapping o( the interval l: - 1':; x ~ 1 into itsel(, mapping each endpoint onto
tion w' = !(w) provides this mapping, then the function w' = !(gk(z)) = gk +2(z)' k = 1, 2, will provide mappings of the
itsel(. Suppose also that, as a (unction o( a complex argument, g (z) is regular
domains B(1) and B(2) onto these same adjacent domains. In accordance with
and univalent in a su((iciently narrow circular lune with vertices at the points
the symmetry principle, the functions w' = gk + 2(Z), for
Theorem L Suppose that a (unction x '= g(x) sets up a continuous one-to-one
k = 1, 2, also map the
± 1 and containing the interval 1. Under these conditions, there exist two (unc tions w = (l(z) and w = !2(z) that map the semidisks B 1 : Izi < 1, ~(z) > 0, and B 2:
Izi < 1,
(rom the disk
< 0, Iwl < 1
:25(z)
respectively onto two disjoint domains G 1 and G 2 obtained by making it a smooth cut A that is analytic at all interior
~1~ ~J
points.1) Furthermore, the mappings are such that, on l, i\:;~
/1 (X) =/2 (g(x», that is, such that cmy two points x, g(x) € l that lie on the boundaries o( the domains
-1
Figure 18
Figure 19
Bland B 2 are mapped into the same point A.
Proof. It follows from the conditions of the theorem that the function z g(z) is regular and univalent in a sufficiently narrow segment
the interval l and the circular arc contained in ~(z) ment, ~(G(z)) ~
o.
8(1)
1=
bounded by
> 0 and that, on this seg
We can take the segment B(1) in such a way that the values
"double segments" B(3) and B(4) bounded by the interval 1 and the circular arcs inclined to l at the angle rr/2 n -
1
in an analogous manner. Let G(3) and
G(4) denote the images under these mappings and then let us map the domain
UG(4) UAl
U8(4) U1 in
of the interior angles at the vertices are equal to IT/2 n , where n is an integer.
GO)
Let B(2) denote the segment symmetric to B(l) about the x-axis (see Figure 17).
such a way that the points ± 1 are mapped into themselves. Under this mapping,
In BO) we consider the function w w
=
=
g 1 (z) = g(z), and in B (2) the function
g 2(z) '" z. For arbitrary x E l, these functions map the domains B( 1) and 1) Thar is, such mar any inrerior arc of ir is an analyric curve.
454
univalently onto the lune B(3)
the domains GO) and
the w "-plane in
G(4) are mapped into domains that are taogent along some
smooth curve A2 that is analytic at all interior points. By applying the symmetry principle n - 1 times as indicated, we obtain the mappings mentioned in the
------"----·"-"-_"
456
~ _ _,
..;...,,>.....___....,...:·...
.u,·..................-_.......n - ..... ~ ...-=-~
~=O;;
...
...==;:.;""';.;;;.-;..:'""i';'.Zii,~
. '".;......:·.:.,·;·:~~::.~<.,,:'
,,:.':.""~,;~',;.-~:.~n:,~:;-,~
;.;.;:;;;;..:,.;::..:. 'p'_.:" .;;":;
::..-i.,,;:;.r;i·';'O:,,~·;;.'..\,
..,'~~.":.,,:
':.-'-~::':~
§ 1.
XI. SOME SUPPLEMENTARY INFORMATION
theorem. This completes the proof of the theorem.
""~':"';::':'.;. :.;~':;':
...:::';:''''''-
·..;;_·."';,;,,~'"".':'O~~~,,:.:·:l%"'~;_....
:-=-;;:.:;', ~;:.:'.~~~~i".i'."""~-= .. ;~,:;;:,::~_1,j.:;':~,,;;~,P:7~,,,,;=,,·,;· .. :·:.~'~i·~~'·"'~
GL DING THEOREMS
457
1 z
F(z)=--Z.
Lavrent'ev's theory of quasiconformal mappings, Volkovyski'12 ) succeeded in obtaining several other gluing theorems that play an important role in the theory of Riemann surfaces. In these theorems, the requirements imposed on functions of
We pose the extremal problem: Out of all functions F(z) € minimizing the integral
the type g(x) are weaker.
IF
Another procedure for obtaining gluing theorems is based on the use of the boundary properties of analytic functions that were obtained in Chapters IX and X. Specifically, let us prove Theorem 2. 3) Let a and b denote two distinct points on the circle
1
!IJl, we have IF
=
m. Let us suppose the
opposite. Then, there exists a sequence of functions Fn(z) = z-r + z + fn(z),
for 71 = 1, 2, ••• , such that I F n --> m as 71 - 4 <Xl. Let us define
In (z) =
00
~ C~)Zk. k=1
Then, 00
~ kl Ck(n)l~ r ~k = I F , n,.oJ-I I c.(n)l~ ~ 1"1m ~ _ , 2, ... n r-lk=1
I
w=F(z)=-+alz+ .. ., z
( 4)
Let m denote the greatest lower bound of the integrals (4) in the family ~. Let
defined on Yl.that possesses the following properties: 1) g(z) is regular and has nonzero derivative at all interior points of the arc Yr; 2) the function z '= g(z) maps the arc Yr bijectivelyonto the complementary arc Y2 on the circle Izi = 1 and maps the endpoints into themselves. Under these conditions"there exists a function
!IJl, find the one
=/z/
us show that, for some function F(z) €
Iz I =
and let Y1 denote one of the arcs connecting them. Let g(z) denote a function
Izi ::; 1
;:-.:..::\'";:.:.:.::.;-."....-~-
An example of such a function is the function
Theorem 1 is a modification of a gluing theorem of Lavrent 'ev. 1) By using
that is regular in the closed disk that satisfies the equation 4 )
~::;;:, :~";,'::;
(1)
Izi
Therefore, the functions fn(z) are Uniformly bounded inside the disk
except at the points 0, a, and band
the condensation principle is applicable to them. Let
lfnk (z)l denote a sub
sequence of functions that converge s uniformly inside the disk function fo(z). Obviously, fo(z) € H 2 •
F(z)=F(g(z))
(2)
< 1 and
Iz I < 1
to the
Furthermore, in accordance with Minkowski's inequality, we have, for
0< r < 1,
at all interior points of the arc Yr'
2~
Proof. Consider the family !IJl of functions F(z) of the form
( ~ 1/0(e
1
o
F(z)=-Z+z+/(z) where f(z) E H 2 and f(o) = 0 (with regard to the class H 2' cf.
§4
)-
(3)
I) Lavrent 'ev [1935l 2) VolkovyskiT [I946l 3) Schaeffer and Spencer [1947l 4) In a monograph by the same authors [1950] the existence of a function F(z) that is univalent in Iz I < 1 is proved.
0
(> 0,
ia
0
+ (~o 1/ (reiD) For given
2~
In (e ) I~ dots ~ ( ~ 1/0 (e iD) _
k
of Chapter
IX) that satisfy almost everywhere on Yr the relationship
m(F(g (z))) = m(F (z»).
ia
10 (reiD)
I~ dO y;s
In (reiD) I~ dO) /s + (~ II" (re ia ) -in (eia) I~ dO) /a" J
k
1
(5)
0
we choose r in the interval 0
< r < 1 in such a way that the
first and third integrals of the right side of inequality (5) are less than (. Then, we can choose N > 0 in such a way that the second integral is, for than (. Consequently, for
71
> N,
71
> N,
less
the integral on the left side of inequality (5) is
less than 9(. This means that the sequence of values of this integral converges
458
§ 1.
XI. SOME SUPPLEMENTARY INFORMATION
to zero as n
-->
00.
Then, as we know, the sequence of values of
almost everywhere on
Iz I =
.8
in (e'
459
I
) converges
Obviously, this function belongs to the family ~ for arbitrary real (. We have
I to (o(e i 8). But equation (3) holds almost every
IF.=IF.+2s ~~ atCf~(Z)ID'(z»ds+Si
Fn (z). Consequently, this will also be true for
where on Yl for the functions
,.
GLUING THEOREMS
Izl
B
IID'(z) lids.
Jzl
Fo(z)=z-l+z+{O(z). This means that the function Fo(z)E~. Furthermore,
By virtue of the extremality of F 0 and the arbitrariness of (, this leads to the
since
condition
B
Izl
for 0
< r < 1,
If~ (Z) I' ds =
lim
~ ~ If~ (zH' ds ~ lim IFn = m, n-oo
~ S at (j~ (z) ID' (z» ds = O.
II-ool z /
o + ivo and lIl(z) = u + iv, we have, by Green's formula and the Cauchy-Riemann conditions, for 0 < r < 1,
But, if we set (o(z)
it follows that IF 0 ~ m, which means that IF 0 = m. This proves
that our extremal problem does indeed have a solution. Let
F o(z) denote any of
i
the extremal functions. Let X (e 8) denote an arbitrary real function of () that is twice differentiable on the arc Yl and is constant in a sufficiently small i8 neighborhood of each of the points a and b. Let us define X(e ) on the arc Y2
-
~
= U
~ vo~~dX+'Vo~;dY
vodu=-
IZI=r
)zl=r
ihJo + ou OU o) d \ -_ J\ J\ (ihJ ox ox ox Ox S= J
by the equation
= X (z).
(5 ')
Izi < 1,
and has on
lzl
= 1 a real part equal to
ds.
S
lim
vo(z)du(z)=O
r-l/zl_r
X(z). This function is
constructed in accordance with the Schwarz formula, on the basis of which, when
to/'
at V 0 (Z) ~ (z»
Therefore, condition (8) can be written in the form
Let us now denote by ¢ (z) a function that vanishes at z = 0, is regular in 'the disk
\" ~
Izl
Izl
x (g(z»
(8)
/zl
and also in the form
2"
we set
lim \ Vo (re ) O~ u (re ) dB r-l ~ l9
cp (z)
=
00
~ ak zk,
(6)
k=O
is bounded in
Iz I < 1
\ ~
and the integral
~ I Vo (re i9) Ii dB o
X (e l6) e- ki6 dB" k = 1 2, •••. is bounded in 0
H we integrate the right-hand member by parts twice, we see that
lakl
~
MIP,
< r < 1.
the integral sign as r
for k = 1, 2, .... Here M is finite and independent of k. This shows that the
-->
Therefore, in equation (9), we can take the limit under
1. When we do this, we obtain 2"
~
function (6) is continuous on the closed disk jzl ~ 1, and by (5') satisfies on
Yl the relation
Vo (e
i9
)
:6 X (e
i9 ) d{} =
0,
that is,
at (cp (g(z») = at (cp (z». Let us set III (z) = ¢ (z) - a 0
(9 )
2"
2"
ak = J.21t
= O.
From Poisson's formula, we see (integrating by .parts) that the function
.au(re i 8)j ae
we have
i9
and let us define
P * (z) = Po (z)
+ sID (z).
(7)
2"
~ Vo (e i9 ) dX (e i9 ) = O. o
By using equations (2) and (7), we see that, if a
= eia.
and b
= e i /3,
then
460
XI. SOME SUPPLEMENTARY INFORMATION
§2. MAPPING OF SIMPLY CONNECTED RIEMANN Sl1RFACES
~
univalent in a sufficiently small neighborhood Uz
~ (Vo (e i9) - Vo (g(e I9» dx (e i9) = O.
(10)
11
H we now set B
V(e~= ~ (Vo (e i8) - Vo (g(e I9»de
(10 ')
11
and integrate by parts, we obtain
of the point z 0 and it maps
Uzo into some neighborhood Uwo of the p~int W o = g(z 0) € Y2' Here, the subset U; 0 of the neighborhood Uzo that is contained in the domain Izi > I is mapped into the subset U~o of the neighborhood Uwo contained in the disk Iwl < 1. Thus the function Fo(g (z » is regular in U; • Consequently, the neighborhood Uz is o 0 partitioned by the cirel e lz I = 1 into two subsets, in one of which the function F o(z) is regular, and in the other the function F o(g (z» is regular. These two functions are equal almost everywhere on the common arc y of the circle. Further
~ V(e i8) dOl d i9 _ X(e ) de _ o.
more, the values of these functions on the boundaries of the corresponding domains
l
~
0
461
(11)
11
or on the boundaries of suitable subdomains are such that their absolute values are square-integrable. But then, Cauchy's formula and theorem can be applied.
This equation is established for an arbitrary function X (0), that satisfies the conditions mentioned above. Now let Xl(e
ie ) denote
an arbitrary function defined on Yl that has a con
When we use these, we can show by the usual line of reasoning that each of the functions
F o(z) and F o(g (z» is an analytic continuation of the other across the
arc y. In particular, these functions are regular a6d equal to each other on the
tinuous first derivative with respect to () and vanishes in sufficiently small
arc Y itself. Since Zo is arbitrary, we conclude that the function Fo(z) is regular
neighborhoods of the points a and b. Then the function
at all interior points of the arc Yl and that equation (2) holds for it everywhere
9
X(ei~= SXJ (e i9) de
on
Yl'
This completes the proof of the theorem.
II
satisfies the conditions mentioned above. Therefore, equation (11) yields ~
~
18
i9
V (e ) :0 Xl (e ) dO
§ 2.
Conformal mapping of simply connected Riemann surfaces
Let us now consider the question of the conformal mapping of simply con
. O.
nected Riemaru;t surfaces onto a one-·sheeted disk. In the constructive definition
11
We now have the conditions of a well-known lemma of du Bois-Reymond in the
of Riemann surfaces, we take· for the simplest plane figures not circles, as is
calculus of variations. Th erefore , we conc lude that V (e i e) = cons t on Yl' It then follows from (10) that
sometimes done, but triangles, including curvilinear triangles.
Vo (g(z» =Vo(z),
Specifically, suppose that there are over the w-plane and parallel to it a finite or countable set of triangles .6 1, .6 2 ,
••••
By "triangle" we mean here
either a finite domain bounded by a closed Jordan curve consisting of three ana
and hence
lytic arcs that meet each other at nonzero angles or a domain obtairied from such
.3 (Po (g (z») =.3 (Po (z»
a domain by inversion about one of the "vertices". I) We shall refer to the three almost everywhere on
Yl'
This, together with the relationship
boundary arcs as sides and we shall refer to their endpoints as the vertices of the
at (Po (g(z») = ffi (Po (z» ,
triangle. These triangles are connected, Le. glued, along the sides in such a way
implies the validity of the relation
that, for each k the triangle .6 k is glued along some or all of its sides to another
Po (g(z» = Po (z)
(12)
almost everywhere on Yl' Now, let Zo denote an interior point of the arc
Yl'
The function w = g(z) is
triangle in the set which has no other points in common with .6 k • The triangles together may overlap but their points lying over the same point in the w-plane I) The difference between the two cases mentioned in the definition of triangles disappears if we shift to the Riemann sphere.
~;::~~===.:::.-=-~=_:::~=,-:,,,===:;;:;~:;;:;~~--=-=-=--:,,=:~-==.:~~===,~==---::...-:;;;~~~~~~~~~.
462
__ -=~'~===""::-=:::::::::~~~=~~~~_="=--='=-~==""""~~~~~~~:==:=~~~~~=::~=--'-"="""=--=~~c"=~~~~~~~~",:::~~-:,~:~",:,~-"_~."~~,,,,,.:::::".":~-=-:~:;:-~~;,~;~;;:;;;::~~~~~,==:~~-;:;:~..;:,:;:-~-=:::::""--,,,,~=~~~,;,~=:;;:~~c::~~==::=::,,_~
XI. SOME SUPPLEMENTARY INFORMATION
§2. MAPPING OF SIMPLY CONNECTED RIEMANN SURFACES I
6"
463
are treated as different points if they are not identified under the gluing described
that do not appear in B n and one side of
above. We shall denote the points of the triangles by the same letters as the
side not belonging to B n of the two trian~les appearing in B. Furthermore, let us assume that, in the case of a finite number of domains B n , this last coincides with B. In the case of an infinite number of domains Bn , these constitute a
points in the w-plane over which they lie. Consider a Riemann surface F consisting of (1) the interior points of all the
or by adjoining to B n the common
triangles 6 k , (2) the interior points of all those sides of each triangle 6 k that
sequence exhausting the domain B. Obviously, such a construction can always be
are also sides of other triangles adjacent to 6 k , (3) the vertices of the triangles 6 k in which a finite numb;r of triangles 6 k adjacent to each other are connected
done beginning with B 1 = 6{. The triangles 6 k corresponding to the triangles
6"
in circular order. We note that the sum of the angles of the triangles at each vertex
appearing in Bn (for n = 1, 2, ... ) constitute a simply connected Riemann surface F n contained in F n + l' and an arbitrary point of the ·surface F lies in
is a multiple of 217. H this sum exceeds 217, the vertex is called a branch point
F n beginning with some n. Obviously, the surface F 1
of the Riemann surface. A characteristic property of connectedness for surfaces is assumed satisfied for Riemann surfaces, namely, if any two points lying on F lie in the triangles 6 k and 6 k "
6 k 2'
••• ,
F contains a sequence of triangles 6 k l' 6 kn , each adjacent to the next, such that 6 k 1 = 6 k , and 6 k n = 6 k
rl
<
00,
=
6
1
Let us suppose that we can assign to the surface F a one-sheeted simply
=
f l (z)
such that
Iz I < r l' f l (o)
for some n, the surface F n is in one-to-one correspondence with a disk where rn
<
00,
0 and f ~ (0)
connected domain B consisting of triangles 6" of the same kind. Suppose also that there exists between the triangles 6 k and 6" a one-to-one correspondence
for F n + 1 (with the disk in question possibly replaced with the domain
that always assigns to two adjacent triangles on F two adjacent triangles in B.
or the entire z-plane). Here, we need to consider two cases:
Theorem 1. An arbitrary simply connected Riemann surface F is in one-to one correspondence with a disk
Iz I < r,
where 0
< r :5
00,
=
=
=
1. Let us show that this will also be true
Iz \ <
00
1. Suppose that the surface F n + 1 is obtained by adjoining to F n the tri
In this case, the Riemann surface is said to be simply connected. With regard stitutes the subject of the present section:
Iz I < r n'
the correspondence being defined by a meromorphic function w
fn (z) such that fn (0)
to simply connected Riemann surfaces, we have Poincanf's theorem, which con
where
is a point
6 1 and ['(0)_= 1. Let us now suppose that,
(fixed) in the interior of the triangle '.
is the image of some disk
under a suitable univalent function w
angle 6 k along one of its sides l. In such a case, F nand 6 k can be regarded as the single-valued images of the semidisks B 1: Iz I < 1, ~(z) > 0 and B 2 : Iz I < 1,
~(z)
<0
under suitable meromorphic functions w
or the entire z-plane,
=
cPl(z) (see Figures 20 and 21)
fl
1.
loY
where the correspondence is defined on the disk or plane by some meromorphic function. 1)
,."
Proof. Suppose that the triangles 6 k in the composition of the surface Fare in the relationship described above with the triangles
6"
constituting a one
sheeted simply connected domain B. Let us construct in B a finite or countable = 1, 2, ... , such that B n C B n + l' for n ;:: 1, is obtained from Bn either by adjoining to Bn one of the triangles 6"
Figure 20
set of simply connected domains B n, for n
1) If r
< 00,
F is called a surface of the hyperbolic type: if r = 00, it is called a surface of the parabolic type; in the case of the entire plane, it is called a surface of the e llipt i c type. (In the last case, the meromorphic function in question is obviously a a rational ftaction.) A simply connected Riemann surface F is always of a definite type. This is a simple consequence of Liouville's theorem.
Figure 21
Figure 22
cP/z) in such a way that both mappings map the arc l into a common boundary diameter. Obviously, the function g(x) = cP/cPl1 (x )), satisfies the con
and w
=
ditions of Theorem 1 of
z ' = t{; 1 (z) and z
1=
§ 1.
On the basis of this theorem, there exist functions
t{;2(z) that map Bland B 2 univalel1tly onto two complemen
tary domains G l and G2 of the disk the points x and g(x) (for - 1
:5 x :5
Iz'l < 1
(see Figure
22) in such a way that
1) on the boundaries of Bland B 2 are
ma pped into the same point of the disk Iz' I < 1. But then the functions
464
_ _ ''''','''
'
'
'
'
'
'
.
_
'
'
'
'
'
'
'
'
'
~
.
'
_
'
'
'
_
'
__"_"'''__
.
"
~
_
'
_
.
_.
"
,.
"
'
••••• ,• •
~
_
.
_
_
'• •,
.
•• __ , _ .
~
_
.
g.
~
• • • • _.
"
u_
' . _ " ' _ " _ _ ",, _ _ ,,., _ _
XI. SOME SUPPLEMENTARY INFORMATION
§2. MAPPING OF SIMPLY CONNECTED RIEMANN SURFACES
465
w = cPI(l/lll(z)) and w = cP2(l/l2 1(Z)) map the domains G I and G 2 onto F n and 6 k
convergence of the corresponding normalized functions w = cPv(z), where cPv(O) =
respectively in such a way that the common portions of the boundaries of these
W
o and
cP~(O) = 1 for v = 1, 2, •.• , which was proved above for the surfaces F n
1, 2, ... , we conclude that the surface F n + 1 is the image
functions coincide. In accordance with Riemann's theorem on analytic continua
themselves, where n
tion, we conclude that both these functions define a meromorphic function w =
of the disk Iz I < r n + 1> where r n + 1
F(z) on the disk Iz I < 1 that maps that disk bijectively onto the surface Fn + I' By a suitable normalization, we easily obtain the fun ction w = fn + I (z ), fn + 1(0) =
boundary arcs because otherwise we would run into a contradiction with Theorem
w o' f~+I(O) = 1, which yields such a mapping of the disk Izi < rn+I' where
2 of
rn +I
<
00
.:::;
00,
under the meromorphic function w =
fn + I(Z), fn + 1(0) = w o' f: + 1(0) = 1. Here, r n + I can be
§ 2,
00
only if F n + I has no
Chapter X, but then F n + I coincides with F.
1£ the number of surfaces F n is finite, we conclude that the surface F is in
on Fn + I'
one-to-one correspondence with a disk Iz
2. Let us now consider the case when the surface F n + I is obtained by
al < ( constituting a
neighborhood of the point
take the limit as n
~(z)
<0
-->
00,
~=ljIn(Z)=hl~1(In (Z)),
adjoined can obviously be regarded as the single-valued images of the semidisks
< 1,
or with the entire
To do this, consider the functions
from it and the disk K, with a sufficiently small arc I' c I with endpoint at a Izi
00,
sequence of functions fn(z) and we need to irivestigate what happens when we
a
on F n + I but cut along I. The surface F n with points belonging to K de leted
2:
where r':::;
On the other hand, if the number of sudaces is infinite, we shall have a
F n + I' we remove from the surface F n a suffiCiently small one-sheeted or multiple-sheeted disk K: Iw -
1< r,
z-plane, the correspondence being defined by a suitable meromorphic function,
adjoining to F n the common side I, not be longing to F n, of rwo triangles 6 k and 61< ' contained in Fn' 1£ a and b are the endpoints of the arc land a lies in
8 1: \zl < 1, ~(z) > 0 and 8
=
The function
(see Figure 23) under meromorphic
,=
q.n (0) = 0,
l/l '(0) =
n= 1, 2~ ...
l/ln (z) is regular in the disk Iz I < r n and the values that it
assumes in that disk all lie in the disk have
1jI~(O)= 1,
1'1 < rn + I'
By the Schwarz lemma, we
1 .:::; rn + I/r n . This means that r n does not decrease with increasing
n. Therefore, the sequence
lrnl
approaches a finite or infinite limit
r
as n
-->
00,
We now set
"ll
CPn•• (z)=!.;;'Cf.(z))=z+ ... ,
v
We see that, for every fixed v (v= 1,2, '''), the functions cPn,v(z), n=1, 2,"', are regular and univalent in Izj
tions has a subsequence cP n (V)
}z)
Consequently, the sequence of these func
that converges in
Iz I < r
V
to a univalent
k '
Figure 23
function cPJ-z). Here, we may assume that the sequence of the numbers functions. Here, both functions map the common boundary diameter onto the portions of the boundaries lying over the circle Iw -
al
= (. Therefore, reasoning
as incase 1, we can show that the sudace FnUL' is the image of a finite disk
contained in the sequence of the numbers n~l)
cPJ.V) (c,bv,l(z)) = cP (V) I< ,v
indefinitely so that as a result we exhaust the entire cut I. 1£ we now use the
• I
This shows that the function w
,=
nC;:)
is
1). Since.
,
1
=f )cP-;; 1(,)), iriverse to ,= cPv(f-;;l(w)) is
independent of v· Here, if G v is the image of the disk ping
=
(z) in the disk Izi < r l , so that cPV(cPV,I(z)) = cPI(z),
it follows that cP)f;l(w)) = cPI(f11(w)) in 6
under a suitable meromorphic function. When we note that the surface FnUL' is obtained from F n by shortening of the cut I, we continue this shortening process
n I<
(with v
cPv(z)' then the function
I zl < r V under
the map
..
466
-
--
u
_
... _ ~
_-
""~-
..
--",·_-o... ,.-_.~·- ......'
•
~~
§ 3·
XI. SOME SUPPLEMENTARY INFORMATION
w == (v(cP:;; l(m ==
-
._.:.
-":"::':'"0.",••;7::;". - .... ~,~_.-
EXTREMUM FOR BOUNDED FUNCTIONS
I
1/
== 1, 2, ... , coincides with the entire open disk
Suppose that r ==
00.
Since the image G v of the disk
Iz 1< r v
I(I < r.
'(I < r v /4 for all 1/= 1,2"" (by Koebe's theorem) the domain G contains in this case an arbitrary finite disk and
consequently must coincide with the entire finite plane. Consequently, by the Sch~arz lemma, we have in
IcPv(z)1 Iz I < r v
Izi
•
the disk
Iz I < r v
1/
denote a number
set
1/,
<Xl
under the mapping (== cPv (z)' contains the disk
means that the same will be true of the domain G for all
G is obviously contained in the disk 1(1 < r.
1(1
< r,
== 1, 2, ...• But since
the domain G coincides with the
disk
=
t/J «().
This completes the proof of the theorem.
Poincare's theorem assigns to each simply connected Riemann surface F a
It is easy to prove the converse, namely, that to every function (z) that is
meromorphic in the disk jz I < r, where r::;
21C ~
This
that disk bijectively onto F.
k=2
\
1/
1(I < p.
IJ.k·)z",
we have 1
the domain G v, being the image of
function that is meromorphic in some disk (or on the entire plane) that maps
ep.(z)=z+ ~ ~
PS-PI
the right-hand member of this inequality is greater
than p, it follows that, for such values of
tion w
Let us now take arbitrary p, PI' P2' 0 < P < PI < P2 < r. Let
< r v < r. If we
1/
pf
~
Thus, in the case of an infinite number of triangles 6. k , the surface F is the single-valued image of the disk 1(\ < r, where r::; 00, under the meromorphic func
I
•
Since for sufficiently large
under the map
ping (== cPv(z) is contained in the disk
Now, suppose that r is finite. Then, we have
-
G of the sequence of
= 1, 2, .. '. It remains to show that the limiting domain
the domains Gv'
467
that is,
t/J(t;)
maps the one-sheeted domain Gv onto the surface F v It follows that Gv C Gv+l' 1/
';;":~-i..~';""-'o~~",--"._-
+ ~2 ~ I (.) pg
I ep. (Pile i9) I.g du CJ
S
-
k
IJ.k
00,
can be assigned a simply connected
Riemann surface onto which the function (z) maps the disk
i
00
~. PI!:::S; r~ pg.
Izi < r
bijectively.
Specifically, we partition the disk jz I < r into small triangles, in each of which
III 2k
the function (z) is univalent, and consider the surface
F composed of
the images of these triangles.
Consequently, 00
~I
Thus, simply connected Riemann surfaces can be regarded as the geometric
rB rS ,.,;'( ) III p2k .,;:: ---=----._ ....
rS
II '-"'=
k=2
equivalents for functions that are meromorphic in a disk (or on the entire -plane).
S
PII'
This conclusion is completely analogous to the conclusion that follows from
•
univalent functions when we use the existence theorems established in Chapters II and V.
Therefore,
---.--
I '-"'= ~ Vrs-r. IIJ. (.) k r.
I
k=2
00
Pi _
§ 3. An extremum for bounded functions in multiply connected domains In § 2 of Chapter II and § 1 of Chapter VI, we considered questions dealing
PI
Using this inequality, we have on the circle co
k=2 3, ...
1 k-l"
Izi = PI' ~_~
~ Vr
~
l
-
r
k=2'
r:
with an extremum of the quantity ('(0) in certain classes of analytic functions
_p_~_
defined in a given domain B. This led to important existence theorems dealing with mappings of the domain B onto a disk.
k-l
P.
=Pt-
V r2 r.
r:
pi
--, PI-PI
We now answer one of those questions.
'*~
_r~~~
468
~
~~_W _ _ W"'1!i1iiiii1Uiiii&i1i~ _ _'m_ili~.1~lJi&~:~~-m>t'JiI.~~!'""j;'~>?""lJ'offi"~'!'>&. __ ~ 'IJZt:L~';;~';WS~~:~'t;.*,~i!n?.m:e,"§J!i.ib.o;:;,'i:lHr"",-,~;§~,""",r;>,~"'~~Rir&'--;";f~~~~\1i""ja
§ 3·
XI. SOME SUPPLEMENTARY INFORMATION
m
in a given n-connected domain B with non degenerate boundary continua, that are given finite point a € B. Then the quantity
1f'(a)1
=
°
§ 5 o[ Chapter
I
VI).
When conditions (2) are satisfied, we have by virtue of the extremality of the function (o(z)
That the extremal problem posed in the theorem has a solution is obvious.
> 0,
the point z
Since I~=lw/(z) = 1 so that the function (z~, "', z~) can be expressed as a linear combination of the functions F~(z ~ , ... , z ~), k = 1, ... , n - 1, the nth of conditions (2) is a consequence of the remaining conditions and we can drop it.
K I , " " Kn : K = U nk -_ I Kk •
Let [o(z) denote one of the extremal functions. Since I[~ (a)1
=
a log II; (a) I-log I/~ (a) I =
is a simple zero of the function [o(z).
m
~
(g(a, Zk) -
m.
at' " ' , zm in the domain B. (Of course, it is not assumed that these are all the
= k=1 ~ (g(Zk'
zeros of the function [o(z) in B.) Consider the Green's function g(z, () for the domain B and the harmonic
I=
log 1/0 (z) I
+ u (Z),
(1)
m
where the Zk belong to B, k
=
1, ...
1, " ' , n - 1, is minimized when zi.
where
the matrix (Zk U
11~:1II,
(z) = ~ (g(z, Zk) - g(z, Zk», k=1
where, in turn, z~ € B for k
=
in B. Also, for [I(z) to be single-valued in B, it is
and satisfy conditions (2) for 1 =
Zk for k
=
1, " ' , m. But then, the rank of
k=l, "', 2m,
1=0,1, ... , n-l,
(4)
= Xk, k = 1, ... ,2m, does not exceed n - 1. This is true because if the rank of this matrix were equal to n (which can be the case only when 2m > n - 0,
then the functions
g/
=
F /(z ~ , ••. , z~), for 1 = 0, •.. , n - 1, would map a
neighborhood of the point
m
(z) = ~ (h (z, zk) -
h (z, Zk»
k=1
(Xl' •••
,x2m) in 2m-dimensional Euclidean space
onto an open set in n-dimensional space containing the point (0, 0, ... , 0). The intersection of this set with the go-axis (g\
be single-valued.
=
g2
= ... = gn-\ = 0) is an open
segment containing the point (0, 0, ... , 0). C.onsequently, the function
Since, in accordance with (10 'H) of §6 of Chapter VI, 21tlool (Zk),
(h (z, Zk»Kl
=-
where w/(z) is the harmonic measure of the curve
21tlooz (zie), 1= 1, ... , n,
K/ with respect to the domain
B, it follows that single-valuedness of v(z) in B leads to the n conditions on 1) AbHors [1947].
=
,m,
with xi.
1, ... ,m.
necessary that the conjugate function
=-
(3)
= xi. + ix~ +k, zk = Xk + iX m +k)
m
(h (z, Zk»Kz
~ 0.
k=1
log 111 (z)
'lI
g(Zk, a»
Po (z;, ... , z~)= ~ g(zi" a),
-gn, defined by
<1
a) -
This leads to the conclusion that the function
function h(z, () associated with it. In addition to [o(z), consider the varied func
Obviously, \[\(z)!
g(a, Zk)
k_\
Suppose that the function [o(z) has, in addition to a, zeros at the poin ts
tion [I (z) €
(2)
k=\
Proof. Without loss of generality, we may assume that B is a finite domain bounded by n closed analytic Jordan curves
Pl(Z~, ... , z~)= ~ (ooz(zie)-OO/(Zk»=O, 1=1, ... , n.
at a
is maximized by a (unction
ma pping the domain B onto the complete n-sheeted disk (see
469
the z k:
Theorem 1.1) Let ~ denote the family o[ all [unctions [(z) that are regular bounded by unity in modulus in B and that satisfy the condition [(a)
EXTREMUM FOR BOUNDED FUNCTIONS
F o(z ~ , ... , z ~) would' not be minimized when z i. = z k' k = 1, ..• , m. From what we have shown, we conclude that there exist real numbers 11.0 ' At' ... ,An-I' not all equal to 0, such that the 2m equations hold: -
+ A iJP iJxk + . . • + ,iJP"'_l iJxk =
A0 iJx iJPo
k
1
1
11.11.-1
0, k = 1, ... , 2m.
(5)
470
§ 3.
XI. SOME SUPPLEMENTARY lNFORMATION
•
471
the boundary. Here, the zeros on K are counted with half multiplicity. I)
Keeping (2) and (3) in mind, we conclude from (5) that, for the values Ak mentioned above, the partial derivatives with respect to x and y (where z = x + iy) of the function
This result leads to the first important conclusion regarding the extremal func tion fo(z). The zeros of the function fo(z) in the domain B are also zeros of the
-
vanish at z
EXTR EMUM FOR BOUNDED FUNCTIONS
= Zk,
for k
A~(Z, a)
+
AIOO!
= t, ... ,m.
(z)
+... +
AIt_IOOIt_1
function w' (z). The number of these zeros does not exceed n - 1. If there are exactly n - 1 of them, then Ao';' 0. 2 )
(z)
This function is obviously not a constant
since· the Ak are not all zero.
Now suppose that z l ' function
f o(z)
q, = m(p (z,
~», 0011 (z)
m
-
AoP (z, a)
(6)
Zk)
k=1
= m(Wk (z»
is harmonic in Band U(z) S 0 in B. Since this last inequality is satisfied in the part of the domain B contained in a small neighborhood of an arbitrary point
It-I
(z) =
S n - 1, constitute all the zeros of the
U(z)=Iogl/o(z)l+g(z, a)+ ~ g(z,
(see §6, Chapter VI) and let us define W
m
inside B. Then, the function
Let us define analytic functions p(z, () and Wk(Z), k = 1"", n, by
g(z,
... , Z m'
+ ~ Alwl (z).
of the boundary K, if we shift to tQedisk by means of a univalent mapping and use the boundary properties of nonnegative harmonic functions in a disk (see
1=1
From the latter equation, we conclude that the derivative w'(z) is not identically
§2
of Chapter IX), we conclude that U(z) has definite limiting values almost every where on K along nontangential paths and these values define a summable func
zero but that it has zeros at all the points z l " " , zm' Since the real part of the function w(z) is constant on each boundary curve
tion on K. Now, let
Kl, for l = 1, ••. ,n, in the case in which w '(z) has no zeros on K, we obtain
f 2(z)
denote the varied function obtairied from the relationship
log Ilg (z) I=log 1/0 (z) I
the result
+ u (z),
(7)
where u(z) is a harmonic function in B. To make f 2(z) single-valued in B, we
It
(arg w' (z»K = ~ (arg W· (z»KI
require that the conjugate v(z) to u(z) be single-valued in B. For If2(z) \ not to
1=1
exceed 1 in B, it will be sufficient to require that u(z) be bounded in Band
It
=
! [(arg ~:)
KI -
(arg
1=1
~;)KI J=
27t (n -
2),
that U(z) + u(z) be nonpositive almost everywhere on K. This is true because, in the first place, by virtue of (6), and (7), m
1/2 (z) l=eU(Z)+U(Z) . e -g(z, a)- k-1 l: g(z, zk)
where ds is the element of arc length on the boundary K. We note that the func
-
tion w' (z) is regular at a when Ao = 0 and we note that w'(z) has a simple pole
B when Ao';' o. Therefore, we conclude from Cauchy's theorem on the number of zeros and poles that the number of zeros of the function w '(z) in B z = a in
distinct from z if Ao';' O.
=a
does not exceed n - 2 if Ao
=0
and is exactly equal to n - 1
This result will also hold when w' (z) has zeros on K, as follows
from the same theorem of Cauchy but this time with consideration of the zeros on
10
I/o(z) I
(8)
B. Furthermore, under the assumptions made, u(z) also has definite limiting
1) For a direct derivarion of rhis theorem for rhe function Wi (z), consider neighbor hoods not only of its zeros lying in B bur also of irs zeros lying on K. Then Cauchy's theorem can be applied ro the logarithmic derivarive of W'(z), If we then let the neighborhoods of the zeros shrink to a point, the difference between rhe number of zeros and the number of poles of the funcrion w '(z) in B will be equal to (argw'(z))K' This last expression means the sum of rhe increments in argw '(z) over the set of arcs con stituting K and not encircling the zeros of w' (z). 2) (A more complete study of the zeros of the function f O(z) has been made by Garabedian [I949})
472
§3-
XI. SOME SUPPLEMENTARY INFORMATION
EXTREMUM FOR BOUNDED FUNCTIONS·
values almost everywhere on K. From Green's formula
O(Z)+U(Z)=2~~ (U(Z')+U(z')dg~:z)ds,
n
(where
'= f
by taldng the limit as
f -->
B
Here,
} (U(Z') +u (z')
dg~;
z) ds
f
0
u(z) ~ 0 in B. Therefore, in accordance with
'=
I
f - ....
d~W k\ u(z)-----an:-ds=O,
1 0 (z)
log"lf~ (a) I= log
u(a)=_l \ u(z') 2~
u (z),
;
dg(z', a) dn
1-
and if we denote by Kf,I,It'
f,
where
where k
l
where the Ak' for k
K I , I, I
I- l,
(10)
.....,. •
B that is close to K l and defined
then the single-valuedness of v(z) in
II
~
(001 (z)
,UK 11=1
I,
J
I, II
ddU ds
n
(OOt(z)+e)
K" t, t
t, I
- a) :: ds
~
=-
+ 26
~ ~: ds =
~.t,t
K
there
E
outside
,dg(z,
a)+n~l, drok) .i.J An lin
AO _.~
1, ••• , n - 1, are constants and
'=
on
E
,
k=1
from the domain
B to the disk
1'1 < 1,
as was done in
f
is a constant of either
:~ ds
§3
of Chapter VI, then
use Poisson's formula to construct a harmonic function on that disk from the new boundary values, and finally return to the domain
ds~O
a curve in B close to K k and defined
f,
\"
1(1,
> 0,
sign but sufficiently small absolute value. (To construct u(z), we need only shift
B can be expressed in the .form of n - 1 equations (for sufficiently small f):
I,
.. -
B with it.) For this function
u(z), conditions (11) take the forms
_ A
o
~ dg drot ds + n'\11 A ~ dliJk • dlJ>t d = dn dn .i.J k dn dn s
is a sufficiently small positive number,
f
with the same
dv(z)=-
on
U(Z)={,,(
l.to (a) I+ u (a).
Furthermore, if we denote by Kf,l,l a curve in
= J\" t. t
(11)
Turning now to the function (6), let us suppose that, for some q
f0
Then, by virtue of the extremaliry of the function fo(z), we have u(a) ~ O. By using Green's formula (as derived above), we see that this leads to the inequality
(v (z»K
[ =1, ... , n - 1.
exists a subset E of K of positive measute such that U(z) ~ - q for z E E. Then,
and let z approac h a, we obtain
'= f
0) equations
for u(z) we may take the function defined by the boundary values
1.= log Iz-a 1+
log Is (z) z-a
by the equation WI(Z)
ds=O,
K '0. t. I
is a fixed small positive number. This is equivalent to satisfaction of
(9)
(8), If 2(z)1 S 1 in B. Thus, f2(z) € !JJl. If we rewr ite (7) ill the form
'=
du dn
O. (As always, the normal n is directed inside the
domain B.) It follows. that U(z)
by the equation WI(Z)
\
J
[=1, ... , n-1.
is a sufficien dy small positive
f
the n - 1 limiting (as
2~
+ 2a
k=1
number), we obtain Green's formula for the domain
U(Z)+U(Z)=
drot (z)
u(z)----cTrZ ds
UK., t, k
Ka where K f is the set of curves g(z', z)
~
473
0 }
,
(12)
k=1
[=1, ... , n-1, and, by virtue of the choice of the Ak' for k
=
0, 1, •.. , n - 1, they must be
satisfied. But conditions (12) constitute a system of n - 1 homogeneous linear equations with n unknowns Aft> k
'=
0, 1, ... , n - 1. Therefore, this system has
nonzero solutions. Let us now denote by AIt, k
'=
0, ... , n - 1, any nonzero sys
tem of solutions and let us construct from them the function u(z) in (10). We then obtain an inequality in which the left-hand member includes the factor maybe of either sign. This leads to the condition
f,
which
,-, ==
-----,~-
474
=
"'"
~:
--,-._~~~~?~~~~~.~~-b~~~~~
§3. EXTREMUM FOR BOUNDED FUNCTIO~
XI. SOME SUPPLEMENTARY INFORMATION
Now, the conditions under which
II-I
-.A } (dg(z, a»)S ds+ ~ A' } drok. dg(z, a) ds=O. ~ k_ k=1
dn
o
dn
...'t""~~~~-~~~~:W~'~~F--"'1:;M~~~~~~~~~~tw:",~~~~~
(13)
dn
f o(z)
is single-valued in B take the form
I
m
~ (~A0 dg(z,a)+A dn I
dUl1
dn
+ . . . +A
II-I
where the VI are integers. Obviously, VI over 1 = 1, .•.
dUl Il- 1 )SdS=0. dn .
,n,
2:
1, for 1 = 1, ... , n. H we sum (14)
II
m+l= ~ 'Il~n. ~=l
II-I
dg +
~
dUlk
~ Ak dn =0.
-"0 dn
But m ~ n - 1. Consequently, m = n - 1 and VI = 1 for 1= 1, ... , n.
k=1
Summarizing what we have said, we conclude that the function
Since, in addition, everywhere on K we have
+~ ~
f o(d
has
exactly n zeros in the domain B and that it is equal to unity in modulus on K.
II-I
dg -A O ds-
(14)
we obtain
It follows that, almost everywhere on E.
,
wl(Zk)='Il, [=1, .... n,
k=1
H we multiply equation (13) by -A o and equations (12) by Al (for each 1 =
~
!
21t(arg!0(z))K/=wl(a)+
1, '" ,n - 1) and then add all the resulting equations, we obtain
475
Then, by evaluating (arg (f o(z) - wo» K, we prove that the function f o(d - w 0
dUlk AkdS=O.
has, for each w 0 in the ,disk Iwol
k=1
<1
exactly n zeros (counting each one
according to its multiplicity) in B, and this means that the function w = f o(z)
we conclude that the function
maps the domain B onto the n-sheeted disk Iw I < 1. This completes the proof of 11.
W' (z) ="'- AOP; (z, a)
+ ~ AkWf, (z),
the theorem.
k=1
By means of a third, more complicated variation of the function
f o(z),
one can
which is regular on K, vanishes on E. But then w '(z) ;;; 0 in B and hence the
show 1) that the extremal function is unique up to a constant factor e i a.. However,
function
we shall not stop here to do this. Let us denote by F(z, a) that one of the extremal functions of Theorem 1 for
/I
- Aog(Z, a)
+~
AkCl)k
(z)
which the derivative is positive at the point z = a, thus emphasizing its de
k_1
pendence on a. This functions also appears in the construction of extremal func
is constant in B. However, this cannot be the case because of the choice of the Ak for k
= 0,
1, ...
,n. This contradiction shows that V(z) = 0 almost every
where on K. But then, if we apply Privalov's uniqueness theorem (d.
§2
of
(bapter X) to a suitable subset of the domain B. we conclude that V(z);;; 0 in
B; that is, in B we have
tions of certain other extremal problems associated with the same class of bounded functions. 2 ) We present one theorem of this kind:
Theorem 2. Suppose that a function f(z) is regular in a finitely connected domain B with nondegenerate boundaries and that If(z)1
1!'(Z)I~(I-I!(Z)lg)F'(z,z),
m
logl!o(z)I=-g(z. a)- ~ g(z, Zk)
!
(f'(z, Z)=aF~,
(z)=e
F(z, a)+a 11
::VIN
_"
lel=I, 11l1< 1.
m of ( )
JO Z
=e-p(z.
a)-
~ /=t(Z,
zk)
in B. Then, in B,
z) Ic=J,
(15)
with equality holding at the point z = a E B only for functions of the form
k=1
and
1) Cf. AhHors [1947J. 2) Alenicyn [19sOal (See also Alenicyn [196IJ.)
(16)
476
§4. THE THREE-DISK THEOREM
'-. XI. SOME SUPPLEMENTARY IN FORMAnON
at some point Zo in the open annulus q
Furthermore, in B,
If' (z) I ~ F' (z, z), with equality holding at z
=
a € B only for f(z)
f(z) for which fez 0) > O. Let us denote the extremal functions by f (z, p), thus
= (. F(z,
a), where
kl
f(a)f(z~) satisfies the conditions of Theorem
=
1.
pointing out their dependence on p. Then, we can easily see that between suitable extremal functions we have the relationship fez, pq)
=
1, and conversely,
a. Obviously, equality holds only for the function (16). The second part of
the theorem follows immediately from the first. This completes the proof of the theorem.
~;
~
Of:
Lemma. If the function fez) is regular in the annulus q ~ [zl ~ 1 except possibly at a single simple pole on the segment - 1 < z < - q and if !f(z)1 ~ 1 on
the circles Iz I = 1 and Iz I = q, then If(z) I ~ 1 on the open segment q < z < 1 with equality holding only in the case f(z):= const.
Izol < r 2
1
log~
log
(1)
10 ra
raM~ gn;
rl' r2'
M1, and M2, and where A is real. Since this last
function is regular in the annulus only for the exceptional values of Izol at which A is an integer, it is only in these cases that inequality (1) is sharp. The
problem arises of finding a sharp inequality in the remaining cases. We shall now investigate this question.
I)
We note first of all that, by means of simple transformations, we can reduce the problem posed to the case when rl
=
q,
r2 =
1, M1
=
p, M2
=
1. Hence, we
can consider functions f(z) that are regular in the closed annulus q ~ Iz I ~ 1 and satisfy the inequality If(z)1 ~ 100 the circle
This function also satisfies the conditions of the lemma and it assumes real
< Iz I < 1
< 1. Let us now consider the case in < z < - q. Since /F(-l)! ~ 1 and IF(-q)1 ~ the interval -1 < z < - q every real value of
=
jzl =
1 and the inequality If(z)1 ~ p
q. The question then deals with the maximum value of
If{zo)1
which F(z) has a simple pole on - 1 1, it follows that F(z) assumes on
absolute value greater than 1. On the other hand, it easily follows from Cauchy's
F(z) assumes every value w such that Iw I > 1 at exactly one point of the annulus q < Iz! < 1. Consequently, IF(z)! must not exceed 1 anywhere on the interval q < z < 1 and, in particular, F(zo) ~ 1. H equality holds here, then F'(zo) = 0 and a neighbor hood of the point z 0 would be mapped by w = F(z) into a multiple-sheeted neighborhood of the point w = 1 or of the point w = - 1. But then, F(z) would theorem on the number of zeros and poles that the function w
assume certain values w such that
Izi < 1.
Robinson [1943].
IwI > 1 at
=
certain points of the annulus q
<
Thus, we have shown that F(zo) ~ 1 with equality holding only when
F(z):= conSt. But F(zo) = f(zo) and, consequently, f(zo) ~ 1. Now, if f(zo) = 1, = 1 and then F(z):= 1. On q < z < 1 and - 1 < z < - q, this yields ~ (f(z» := 1, Consequently, the image of the interval - 1 < z < - q will be the
we have F(z 0) 1)
+f-(z)~~
and F(z) Ie. const, then obviously F(z 0)
Izol,
Izi
We may assume that
values for real values of z. If F(z) is regular in the open annulus q
with equality holding for some Zo in the open annulus r 1 < only in the case of a function fez) = czA, where c and A are constants
on the circle
< z0 < L
!.. rl
Izi ~ r2'
depending on
Suppose that q
F (z) ="2 (f(z)
If(z) I ~ M 1 for r 1 ~
< z < - q.
fez 0) > O. Let us define
of Chapter VIII, log!..
1. We
then have fez, p):= 1.
pole on the interval - 1
H a function fez) is regular in the closed annulus rl ~ !zl ~ r 2, if If(z)! ~ M1 on the circle Izi = r 1, and if If(z)1 ~ M2 on Izi = r2' then, as was shown in
~
Proof. We may confine ourselv~s to the case when f(z) actually does have a
§ 4. The three-disk theore m
§4
zf(z, p). This last means
shall now consider this case. We note that the case p = 1 is trivial because we
(g(z) + a)j (1 + iig(z», where a is a constant such that ja I < 1, is of the type in question. By Theorem 1, we have Ig '(a)1 ~ F '(a, a), which is inequality (15) with =
=
that the case of arbitrary p can be reduced to the case in which q
if g(z) satisfies the conditions of Theorem 1, then the function fez) =
z
We can also assume that Zo
is positive and that the maximum sought may be found only among the functions
(17)
Proof. H a is any point in the domain B, then the ·function g(z)
(f(z) - f(a»/(1 -
< Izol < 1.
477
---.......
478
.......
XI. SOME SUPPLEMENTARY INFORMATION
entire line ~ (f(z))
=
1, and the image of the interval q
that line. Also, the line ~ (w)
=
§4. THE THREE-DISK THEOREM
< z < 1 is a segment of > 1. On the other hand,
1 lies entirely inside Iwl
ir follows just as above that the function f(z) assumes every value w exceeding
< Izi < 1.
f o(z)
< Iz 1< 1
in the annulus q
479
is a point on the interval - 1
< z < - q.
Now, if f(z) is an arbitrary function satisfying the conditions of the theorem, the function f(z)1 f o(z) satisfies all the conditions and consequently,
tradiction proves that fez 0) must be less than 1. This completes the proof of the
If(zo)/fo(zo)! ~ 1; that is If(zo)1 ~ Ifo(zo)!. Here, equality holds only for a func tion of the form fez) == do(z), where 1(/ = 1. This proves the first part of the
lemma.
theorem.
1 in modulus at exactly one point of the open annulus q
This con
To prove the second part, we note that the extremal function
Theorem. Out of all functions fez) that are regular in the closed annulus
Izi ~ 1 such that If(z)1 ~ 1 on the circle jzl = 1 and If(z)1 ~ p on the circle Izi = q, where q
q~
q < Zo < 1 is attained only by functions of the form fez) = (fo(z), where 1(1 = 1 and w = fo(z) is a function that maps the open annulus r < Izi < 1 univalently onto the open disk Iwl < 1 with a cut along some arc Iwl
=
p, largwl
< o.
The explicit expression for fo(z) is given by the formula
fo(z)=z
e('1j, q)_ 6 (P;, q) ,
(2)
f o(z)
can be
characterized by the following considerations: 1) f o(z) is regular in the closed
Izi ~ 1, 2) Ifo(z)! = 1 on the circle Izi = 1, 3) Ifo(z)1 = p on the q, 4) f o(z) is positive for positive values of z, and 5) f o(z) has a unique slmple zero lying on the interval - 1 < z < - q. To see this, suppose that fl(z) is another such function. Then, the functions fl(z)/f o(z) and fo(z)lfl(z) both satisfy the conditions of the lemma. Consequently, on the interval q < z < 1 we have Ifl(z)1 ~ Ifo(z)1 and Ifo(z)1 ~ If1(z)/, so that Ifo(z)1 = If1(z)1 and, again in accordance with the lemma, this can be true only when f 1(z) == f o(z).
annulus q ~ circle
Iz I =
With this in mind, let us now look at the function (2) with €)(z, q) defined by (3). Since the infinite product (2) converges for all z 1= 0, it represents a regular
where
function in 0
a (z, q) = n (1 + q211-1 z) (1 +q2n-1 Z-l)
< Izi <
and the points z
00
zeros of it. Consequently, the function (2) is a regular
00
(3)
11=1
except at the simple poles - q 2nlp, for n
=
± 1, ± 2, are simple function in 0 < Iz I <
q2n-l, for n
= -
0,
=
0,
00
± 1, ± 2, ...• Since all these poles
q < Iz I < 1, the function f o(z) is regular in it. Further more, the function fo(z) has simple poles z = - pq2n-2, for n = 0, ± 1, ± 2, ... , lie outside the annulus
(the function El(z, q) differs only by a constant factor from the familiar theta function &(z)= I:=_coqn2z2n). Proof. That the function
f o(z)
mentioned in the theorem exists is shown as
follows: we map the disk Iw I < 1 with cut along the arc Iw I = p, larg wi the annulus q '<
Izi < 1
one of which, namely,
in such a way that the point w
=
< 0 onto
1 is mapped into z
By the symmetry principle, the entire w-plane with cuts along the arcs Iwl
wi < 0 and Iwl = lip, larg wi < 0 is then mapped univalently q , < Iz I < II q with linear directions conserved at a point. larg
=
=
1.
p,
onto the annulus
§ 5 of
< 0 < TT,
TT.
-
p lies in the annulus q
co
II (1 + 110 (z) I= I z I·
II +
q211. ;)
=
f o(z)
(I + q2n-2 ~)
n=! 00
n=!
q211-2pZ) ( 1
+ q2n . p~ )
_, IT (
1 +q211
There
the quantity q' assumes the value q
mentioned in the theorem. The inverse function w
< Izi < 1. Finally, on the
1,
Chapter V, the quantity q is a con
tinuous function of 0 that varies from 0 to p as 0 varies from 0 to fore, for some 0 in the interval 0
Iz I =
(1
I
In accordance with Theorem 2 of
circle
z=
1I=!
f) (l +q211-2~Z)
(1 +q211-2pZ) (1 +q211
Ii)
=1,
then serves as the
required function. It follows from the condition of symmetry that the only zero of that is,
If 0(z)1
=
1 and, analogously,
If o(z)\
=
p on
Iz I =
q. This shows that the
_
IiUlil 1Wi
.....
aii\lmilllO'il;Iiiii
F
'.1
W!
-.-_-
......
m'm
I!i
fill..
• ..
m::,;tril'[!iWfl1U'~=~~~M@~~~~~~!%a~~'J! ;!~~~~
,
XI. SOME SUPPLEMENTARY INFORMATION
480 function
f o{z)
§s.
satisfies all the conditions 1)-5) enumerated above. Therefore, it
coincides with the function
f o{z)
mentioned in the theorem. This completes the
the n the A~, for m
TRANSFORMATION OF ANALYTIC =
1, 2, ... , are independent of th e values at, a 2 ,
A~= Transformation of analytic functions by means of polynomials
In this section we shall establish a property of transformations of analytic
-np3 n {
(
Ap Jor m=pn for other m.
-+0:0
••• ,
On
o
(p= 1, 2, ...)
(4)
Furthermore, if we set
functions by means of polynomials, which we shall then apply to the derivation of
-
D= lim
certain theorems dealing with the transfinite diameter of closed sets of points in the plane and with polynomials.
481
and are expressed in terms of the Am according to the formulas
proof of the theorem.
§ 5.
FUNCT~ONS
1I!.D/I""AI" II
IAml, D* =
m-+oo
1I!.2.ITA""*T
-
lim
II
m-+oo
IA~ I,
(5)
1)
The property referred .co is described in the following theorem:
we have
Theorem 1. Let f{z) denote a (unction that is regular for large z and has the expansion 00
ak
~
1.A
fez) =
(1)
Z'i'
k=!
D*=~.
(6)
Proof. It will be sufficient to prove the theorem in the cwo special cases when p{z) = aoz and when p{z) = Z n +
at zn-t
+ ... + a,., because we can then
obtain the proofof the theorem for the general case by considering first f{z) and p{z) = aoz and then the functions f1{z) = f{aoz) and p{z) = zn +
in a neighborhood o( z = 00. Let p{z) denote the nth degree polynomial p{z) = aoz n + at zn-t + .•. + On , where a o f. 0 and n ~ 1. Suppose that the func tion f. (z) = f(P{z)) has the expansion
(a/ao)zn-t + ... + a,,/ao. In the case p{z) = aoz, we have a; = a
c/ ak, for
k
=
1, 2, •... Consequently,
by making a substitution in A~ and by then taking the common factors out of the 00
~
ak'"
f* (z)= ~ Zfi' k=!
(2)
rows and columns of the determinant, we arrive at (4) and then at (6). Let us now consider the case when p{z) = zn +
I( we set
at zn- t
+ ... + an. Suppose
that the function f{z) is regular in Izi > R. Let r> R be such that minlzl=rlp{z)1
al 2
A m =ja
a2 ... am as ... am+l
f(P(z))=-~ \ 21t1 d ~ where the integral is over the circle C:
am am+l ..• a2m_l a*1 a: ... a~
A~=la2*
>R. Then, by Cauchy's formula we have in Izi >r
(3)
a: ... a~+!
a~ a~+!
1_\
dl' ...,
(7)
I(I = R in the counterclockwise direc 0 as a function of z in a neighbor
tion. Let us expand the fraction 1/{p{z) hood of z =
00
in a Laurent series 00
_I_ _
... a:m-
-
f(C)
p{z)-c -
~ ~
k=!
1
bk(C)
-zr,
(8)
which converges uniformly on C with respect to (if Iz I > r. From this result we 1)
Goluzin £19461>].
obtain the equations
=.~~==~===~~~=-~,~=~=--=-------====-===~~~~~===-----~===-=~===--=-========-~~~~~~=~~=~~-===~~~~-==--=O=~~.
----'========~.""~==_====_~___=e=___=
~,
XI. SOME SUPPLEMENTARY INFORMATION
482
bl (C)=b 2 (0= .. . =bn_t (Q= 0,
bn (C) = I,
+ .,b. «()~O.
+ gbn (C) = 0, b 1(C) + al b2n_2 (Q +...+ an_Ibn (C) = 0; bm (C) + a,l b-m_l (C) +...+ amb n (C) =Cbm_n (C), bfl +2 (C)
+
b..... (C) !1.1bfl+l (C)
§s. (9)
}
!1.
(10)
2fl _
m_
The fact that (11) holds also for n
<m
~
••
I
483
b. (C.), ..• , b.+n_l (C.), C.b. (C.), ..• , C.b.+n_1(C v) C~b. (C v), ... ,C~bm+V-2n_l (C.). After p such operatiOns, we obtain a determinant in which the vth row has the form
b. (C v)' .•. , bv+n-l (C.), Cvb. (C.), ..• , C.bv+n -1 (C.),
m>n.
(11)
2n - 1 follows from (9) and (10). When
we substitute the expansion (8) into (7) and keep (2) in mind, we obtain by equat ing coefficients of like power s
C~b (C.) ... , C~b~ (C.), ... , cJ:b.+q_1(C.). Let us substitute into (13) the determinant thus transformed and let us put the integral signs back into the rows. We now change the symbols for the variables of integration
a% = 2~i ~ f(C) bk (C) dC,
TRANSFORMATION OF ANALYTIC FUNCTIONS
k=l, 2, .,.
(12)
'v;
specifically, we denote the variable of integration in all elements
of the vth column by
'v'
Then, putting the iotegral signs back in front again, we obtain
c
Let us now return to the determinant A~. Let us replace the coefficients a~
10
it with their integral representations (12) denoting the variable of integration in the vth row (for v = 1, 2, ... , m) by 'v' Then, by the properties of determi
A~ = (2~ir ~ .. j
b1 (C1)
f(Ct) ... f(C m )
nants,
I~~ (C~) . .
bn (C n )
Cn+1b1 (C n +t )
bn+l (C n ) Cn +1b2 (Cn +l )
, b m (Cl) ... b m+n (Cn> Cn+lbm (Cn+l)
A~= (2~ir ~ ~ ... ~ f(C 1) c c
... C2n bn (C2n )
t(C m)
. . . C9nbn+l (C 2n ) C:n + 1b2 (C2n+1) .. , Cfnbq+l (Cm)
C
b1 (Ct)
X (~(~g)
bg (C t ) bs (Cg)
.....
bm ~m) bm+t (Cm )
bm (C1) b m+1 (C2)
C:n-j-1b l (C2n+1) ... cfn bq (C m ) dCI ...
dCl '"
dCm'
(13)
bgm _1 (C m)
dCm.
(14)
.. C2nbm+n (Cgn) C:n+ Ibm (C2n+1) . .. Cfnbm+q_1(C m) Let us now perform the same operations in the determinant in this equation, but this time on the rows instead of the columns. After p steps, we arrive at the
< n, we have o ~ q < n, and let us
For m
A: = 0 from (9). For m ~ n, let us set m = np'+ q, where
determinant (see. p. 484) composed of p 2 n x n matrices, 2p rectangular matrices, and one
transform the determinant in the right-hand side of
q x q matrix. It is convenient to regard the last 2p + 1 of these matrices as
formula (13). Specifically, let us add to each column from the last to the the n preceding columns multiplied respectively by Cln,
an-I' ., • , a l
use (11). We then obtain a determinant of which the vth row (for v
=
en
+ 1)st
and let us
1, 2, •.. , m)
has the form
truncations of n x n matrices, of which we shall have occasion to speak below. On the basis of (9), in every square matrix of the type indicated, all the elements to the left of the secondary diagonal are equal to zero. Furthermore, if in each matrix we add the first k rows (k
b. (C.), .. 0' b.+n _ 1 (C.),
C.b. (C.),
C.b.+1(C.), ... , C.b m+'_n_1(C.).
Then (for m ~ 2n), let us add to each column from the last to the (2n + l)st
=
1, 2, .,.
1
n - 1) multiplied respectively by
a k , ••• , a I to the (k + l)st row and keep (9) in mind, we arrive at a matrix in which only
the elements of the secondary diagonal are nonzero. Since these operations are performed simultaneously on the complete rows of the determinant (15), it is trans
the n preceding columns, multiplied by Cln, Cln-l' "', a l respectively. Then,
formed into a determinant in which only the elements containing bn remain nonzero. But
again using (11), we obtain a determinant whose vth row has the form
bn = 1. Consequently, the determinant (15) and hence A~ do not depend on a p
'" , an'
_~~._~_"':.,. ~._ ." .
·...-
""'......_"'.. ~.,... _"'~~
;;;;;;""'55i5;...... _ _._ _ ~..
--"'---.";-""lli1t
""'__"".
-·_.w,,",,-
";~":T;'.,n-::c~".-<>,~;""",,;;;.,...., -_
_.",,~--,
--""",,'tt{f'~"';"';:;::::' "'-'''-_.'''--,,'". "--,~",,,~,,.
_d::'",',.,... •__. -"....."'~,....,. -
-'"",""...1'"._,"'''''_.,....-
_.,~_""=,,~
".
~ ~-'k:.• "''''"~=;..~.:o.:- .... _,
.,... --',...;;,.~~~''''"''· .. ,
---........,
XI. SOME SUPPLEMENTARY INFORMATION
484
§5. TRANSFORMATION OF AN ALYTIC FUNCTIONS
,-..
On the basis of what has been shown, to calculate A~, we can now set
....
11"\ '-'
at' " ' ,
~
.......
,....,
""
'+
J
>t:>
~
,...., '+
.......~
... ...+
'-'
>t:>
cJ: . ,,,
....... EO '-'
""
>t:>
~
.......
1
~ >t:>'"
+ Q,~ .......
................
485
G.n
=
O. Then, aZ
=
a v fa k
=
n v (where v
1, 2, ... ) and aZ
=
=
0 for
all of k. Consequently, A~ is of the form
J
'-'
...+""I
J
"1
J...
>t:>
'+
>t:>""
~""
""
I:!
~
>t:>
a.t: .......
~
I:!
~
>t:>
"J:
~
+
+
at
ap+l
a~
~
~
at : ~ ················~~·····::··············· ~~
~
.. ..···r..··..··
.......
'+
J...... J
+ ~
~
q .-"-..
I
,....,
~
n .-"-..
.
:
....... ~
~;~~
..
't[ a
.
p +l
:
'-'
...""
'..."
>t:>
>t:>
A* aa ~ 111
+
+
~
..... --
ctl'
~~"".'.'.~".'.'.":'."'
.. '.'
""."
(16)
.., ............................ , ..................................
..............................................
..
~ ~ I:!
>t:>
Ji
....... ...
+ I:!
~
,....,
...
S '-'
>t:>
>t:>'"
J
S
........ '"
~
J
'-'
'+I:! I:!
.......
'+I:!
J
'-'
J
I:!
.......
... ~
.......
~
>t:>
~
>t:>
J
.......I:!
....... I:!
... ...+
'-'
+
~
I:!
'+
I:!
I:!
>t:>
I:!
.......
~
J
'-'
+ .. I:! .......
ap+t:
I I:!
I:!
'i'3
--.
,....,...
I:!
ap+t
~
'-'
>t:>
...
>t:>'"
J... J'+.. ... >t:>
..1 .......
m=np+q,
.J
where the unfilled-in spaces are occupied mostly by zeros. We now put the
>t:>
columns numbered n, 2n,"', 2np into the 1st, 2nd, ..• , pth positions (without
'-' I:!
..
+
changing the order of the remaining columns) and we put the rows numbered n + 1,
~
'lfi ....................................... ': ...........
--;:
~ I:!
>t:>
~
..
.......
I:! .......
'-'
:lI:!
>t:>
.......
..:;
~
'-'
>t:>
>t:>
'"
~
'iI:!
>t:>'"
,..... ..:;
'-' I:!
>t:>
.......
J
'-' I:!
>t:>
J
.. .. ..:;
,...., ....... '-' >t:>
--;: ..... '-'
1
>t:>
I:!
.......
,...., ~ J '" j
..
,....,...
~
~
>t:>
....... '-'
..:;
.
.................. -
'" ~
'+I:!
>t:>
a p+l
:\
~
~ I:!
...
.......
~
I:!
>t:>
~
.. ..
.......
~ .Q
a, ... >J,
3 ...
... + '-'
--.
.......I:!
I
'-'
I:!
'+I:!
>t:>
~
>t:>
~
,....,
,...., ~
>t:>'"
'tJ
..:;
'-'
.
""
>t:>
':J'
+ 1, ... , n(p - 1) + 1 into 21ld, 3td, ... , pth positions. Then we arrive at a determinant
that is equal to the determinant Ap multiplied by a determinant in which, for every nonzero
q, the qth row from the bottom consists only of zeros, so that the determinant is equal to zero. On the other hand, if q = 0, this last determinant will have the same form as (16) with p replaced by p - 1. Repeating the same operation, we show after p steps that the determi nant (16) is equal to ± A~. This completes the proof of (4). Therefore,
-
D* = lim
m-+ 00
"!.li~ II A~
I
- n2J1..2r
I = plim ..... oo
II
--- --
I A:p I = plim .....oo
n 2J1..2ri'A1n
II
I Ap In =
'!rn II D,
which is equivalent to (6). This completes the proof of the theorem. We turn now to some applications of this theorem.
Theorem 2. For functions that are regular in an infinite domain B with
-----..-......
§6. ON p. VALENT FUNCTIONS
XI. SOME SUPPLEMENTARY INFORMATION
486
487
boundary K consisting of a finite number of closed Jordan curves and having the
arbitr~rily dose to d(K). This proves the sharpness of the inequality mentioned
expansion f(z) = I:=1akl zk, in a neighborhood of z =
in the theorem. This completes the proof of the theorem.
00,
the inequality D :5 d(K)
Theorem 3. For any complex numbers a and b, an arbitrary positive integer
given by Theorem 3 of §2, Chapter VII, is sharp. Here, Dis defined as in Theorem 1 and d(K) is the transfinite diameter of the set K.
n, and a con tinuum E such that d(E) < via
Proof. We shall go through the proof only for the case in which the curves
p n(z)
=
zn +
C 1 zn-1
-
b 1/4, an arbitrary polynomial
+ ... + C n assumes either the value a or the value b some
mentioned in the theorem are all analytic. Then, if we denote by g(z, (0) the
where in the complement of E with respect to the z-plane. The number
Green's function for the domain B, we conclude that this function is harmonic on
\II a -
K. Therefore, for sufficiently small 0> 0, the equation g(z, (0)
on E.
= -
0 define s a
set of closed analytic Jordan curves approximating the curves in K and defining domains
8.
B8 containing
If g8(Z, (0)
is the Green's function for the domain
B 8, if ;Vn.8(Z), for n = 0,1,'" are the Cheby~ev polynomials for the boundary K8 of the domain B 8 which have all their Zeros on K8, if ;;;n. 8 = max z E K 8 j;Vn. 8 (z) I, and if d(K 8) is the transfinite diameter of K 8, then one can easily see that g 8 (z, (0)
=
log (1 - II z)
= -
=
t!'
n2nt
= lim n.... 00 Vll1 n ' = 114
for
II z + 1hz 2 + •... H we now apply Theorem = (Pn(z)b )/(a - b), we see that D* =
1 to this function and to the polynomial p(z)
via - b II 4
for the function
.
(§ 3, Chapter VII). The convergence in this last formula is uniform in every closed subset of the domain B 8, and in particular on B. From this last it follows that ther~ exists a positive number N such that, for n > we have l;Vn. 8(Z)! > ;;;n. 8 on B. This means that, for n >N, the set E*= E(!t n .8(Z)! :5mn.8) lies entirely outside the domain B.
On (z) -
a
~
a%
~ zk • k=1
H all the zeros of the polynomials Pn(z) - ~ and Pn(z) - b lie in the continuum E, the function f'
=
VIla - b174:5
d(E). But this contradicts the conditions of the theorem. Consequently, the poly nomial Pn(z) assumes in B at least one of the values a, b. That the inequality for d (E) cannot be
Now consider the function 00
f(z)
=
!
1
(17)
-;;k'
k=1
By setting up the determinants A k for this function, we easily see that 1, 2, .... Therefore,
Iz I > 1,
the function f (z)
f* (Z)=logPn (z) -b =
mn,~
n-HJO
it is regular in
~
Itn~dz) I =eg(z,ooH~
d (KtJ = d (K) e-~, lim
=
~
Proof. It was shown in ~ 2 of Chapter VII that D
g(z, (0) + 0 and, consequently, n
A 2n + 1 = 1, for n
bl!4 cannot be replaced by a larger number without additional restrictions
D 2: 1 for it. On the other hand, since
it follows from Theorem 3 of
§ 2,
improved is shown by the example of the polynomial Pn (z) = zn + a' and the set E = E(zn E e), where e is the segment connecting the points o and b - a. In this case, we have, in accordance with Theorem 2 of § 1, Chapter VII, d(E) = Via - b II 4 and this polynomial p n (z) assumes the values a and b respectively at the point z = 0 and at the points V~' e 2Ttki 1 n, where k = 1, 2, .•. , and all these points lie on E. This completes the proof of the theorem.
Chapter VII, that D:5 L
Consequently, D = L From the function (17), we now form the function
f * (z)
=
f(;Vn. 8(z)j;;;n. 8) = I:=1 aZlz k • This function is regular outside the set E* and, in particular, in the domain B. By Theorem 1, D * = n"/;;;n. 8 D = '$,n. 8' But 8 limn.... oo~n, 8 = d(K8) = d(K)e- . Consequently, for sufficiently large n we have D* >d(K)e- 8 - {;. Since {; can be chosen arbitrarily small, we can make D*
§ 6. On A function w =
f (z),
p-valent functions 1)
which is regular or meromorphic in a domain B in the
complex plane is said to be p-valent (for p plex value is assumed by 1) Goluzin [1940].
f (z)
=
1, 2, ... ) in that domain if no com
at more than p points in B, that is, if the
-------------
-,_._-==-=--===::==----=-~=-=--=-=:=.:"===--------=-_._---'---..:...-==-----=-=---:::=~--=-=:-==-----=-...::-'_.-:==-=--------::-:=-::::=~:-_-.---:-~---==---:-_--=-----==-=-:-.:::=------::"_=-----=--==--=-----------------
------- ,
.......
4BB
96.
XI. SOME SUPPLEMENTARY INFORMATIbN
Riemann surface onto which the function w
=
f(z) maps B bijectively covers
Let us inde.x these points A l .k> A 2 • k , · · · , Ank,k' We denote their distances from Rnk • k • Let us suppose that they are indexed
the point w = 0 by R l • k • R 2 ,k>""
In the present section, we shall consider the following classes of p-valent
in such a way that R l.k
functions:
1= 1, 2,'"
Sp is the class of functions of the form
w=f(z)=zP(1 +alz+a. zi + ... ),
(1) ,
,nk)
n
~
k=1
w=F(q=~p (1 +? +?s + ... ),
"I
except for a pole at
[el.k R~. k +el.k R~. k+ ...
+ enk,k R~k.k] AlPk.
(5)
(2)
,= "".
But if we move along Gk (for k
= 1,
2, ..• , n) from w = "" to w
= 0, it
follows
from geometric coosiderationsthat th~ numbers of sheets St p lying above segments
:l; is the class of functions in :lp that do not assume the value zero in the domain
We assign to the point AZ,k (for
<1' deqeases. Then, the integral (4) is the limit of the Riemann sum
:lp is the class of functions of the form
1'1 > 1
> R 2 ,k > ... > Rnk • k •
the number eZ,k> which is defined to be + 1 if the coordinate <1'
increases as we move around L p in a neighborhood of A1,k and equal to - 1 if
IzI < 1.
that are p-valent and regular in
489
are finitely many points of the curve L p that lie over Gk (for k = 1, 2, '" , n).
each point in the w-plane with no more than p sheets.
that are regular and p-valent in
ON p-VALENT FUNCTIONS
> l.
of the ray Gk between A1,k and A z + l,k (for 1= 1, 2, ... ) are respectively equal to p - :l:=1
fs.k,
For these classes of p-valent functions and also for certain 9f their sub
and these numbers do not exceed p. Therefore, I
~ es.k~O, 1=1, 2, ... ,nk' k=1, 2, •.. , n.
classes, we now give a number of sharp inequalities regarding the initial coef
s=1
ficients and we shall establish the analogues of the area, covering, and distortion
Thus, if we set
theorems, which are well known for univalent functions. I
. 10 • Lemma. If F(,) €:lp (where p ~ 1), then, for A> 0 and p> 1, ~
..
a: )1
s=1
I d(J~ o.
iO A F(pe )
Proof. The function w = F(,) maps the domain
I"
a',k= ~ es. k'
(3) all the numbers aZ,k 'will be nonnegative.
>P
Furthermore,
for p> 1 onto a
Riemann surface St p bounded by an analytic curve L p and having the maximum number p of sheets in a neighborhood of w =
al. k=el, .II'
Let Rand <1' denote polar coordinates in the w-plane. Consider the integral
-
a'_J, k
= e"~ k'
1=2, 3, '" , nk'
Consequently, we can rewrite the sum (5) in the form n
SR AdlIl, L
az. k
00.
A> 0,
p
(4)
over L p in the direction that pu ts St p on the right. This integral is the limit of the ,Riemann sum constructed as follows: From the point w = 0, we draw a suf
.II-I
the magnit.udes (nonnegative) of the angles between pairwise-adjacent rays. There
i•
.II)
•••
.
lt follows that this sum is always nonnegative. Therefore, the integral (4) is also nonnegative.
fici endy large number n of rays G l' G 2' ••• , Gn that are not tangent to L p and do not pass through multiple points of L p. We denote by ~
).). + ank , k Rnk.). k] AcI>k' R 2. k) + a2. k (R ).). k - Rs. +
~ ...:::.J [al. k (Rt, k -
w
=
On the other hand, since the curve L p has the. parametric representation '8 (J S 217, we have
F(pe' ), where O:S
"___
""
~~~~:~~.ft:i'&!:.9:§f~~~~~""-.A..~t'?§"..E~~~.~c~~~J,i'.·'~'!:'~~~":':":~:
.. ~::':': .. :::::""'~i!:'!:.~~~iiE'~:.,::,::-,::~-.::::::::~,~'¥D
--........."
490
§6. ON p-VALENT FUNCTIONS
XI. SOME SUPPLEMENTARY INFORMATION
belonging to the class I~.
2"
5. RAdlP= ~ R>'~~ dO. p
491
Proof. from the lemma as applied to the function (2), we have, for A> 0 and
0
p> 1,
But if we set F(~) = Re itb , where ~= pe i8 , we have
o~
p
~
oR
op
(M="Rop .
211:
t\ I F (pelS) I~A dB ~ O.
(10)
Consequently, 2"
2"
0
~
SRAdlP=p ~ R),-l~~dO=t~ Lp
For \~I
\ RAdO.
> 1,
we set
[F(q]A = C"p (1
(6)
+Cl C-l+ ... ).
Then, (10) yields
00 the basis of what was said above, we then get (3). This completes the proof
co
V (n-ApL.,1l_1Dl I en I' .tt..
of the lemma.
2°. Beginning with this lemma, let us prove the following theorems: Theorem 1. lf the function (2) belongs to the class I
p
11=0
(where p :::: 1), then
1£ we let p approach 1, we obtain
co
~ n Iap +n I~ ~p+(p-l) lall~ +
11=1
... +1 ap _ll¥.
co
(7)
~ (n -
:p !
Ap) I CII I~ ~ Ap.
(11)
11=1
Proof. From (3) with A = 2, we have co
_
~O, co-I.
p~ IP-n) Ialll~
But the expansion [F (~)]), in
> 0, p> 1,
I~ I > 1
begins as follows:
[F(C)]A~C),p(1+A~+ ... +A~::::+
11=1
... ).
or co
~
(p - n) I anl~ p~ (p-n)
Consequently, c n
-1> O.
n=1
k, k + 1, ... , 2k - 1. Keeping this in mind,
=
Ap) A~
(n -
1, Theorem 1 coincides with the theorem on areas for univalent functions. Theorem 2. 1) If a function (2) belongs to the class I~ (where p ?: 1) and if
al
Aa n , where n
2k - 1:
By letting p approach 1, we get (7), which completes the proof of the theorem. For p
=
we obtain from (11) the following inequality for A < k and n = k, k + 1, ... ,
I
an 19 ~ Ap,
=
= a 2 = ••• = ak-l
that is, 9
= 0 (where k?: 1), then, for n = k, k + 1, ... , 2k - 1, we
I an 1
~ ,.----f!.. 1-
'
_\..
have
1£ we now set A = n/2p, we obtain
lanl ~ 2pn with equality holding only for a function
F(C)=C
P
(8)
(12)
Equality can hold in (12) only if c v = 0 for v -f- n, that is, only if
2p
(1 +C~r,
Ian 1~2~.
1111=1,
(9)
1) For p = 1, Theorem 2 was proved by the author [1938], where it was erroneously formulated for the class II instead of the class I{.
'1p
p
F (C) = C
(1 + C~
t.
I"II =
1.
The function (13) be longs to the class I~ because the function
(13)
._-----._._-~-~._-
492
XI. SOME SUPPLEMENTARY INFORMATION
§6. ON p-VALENT FUNCTIONS
2
~(l+C~t-=C+
By using inequalities (14) and (15) to find a bound for the right-hand side of
...
(7), we obtain immediately
Theorem 4. I[ the function (2) belongs to the class I~ is univalent in
"I
> 1. This completes the proof of the theorem. I a.g I :s;;;p (2p -
(14)
1),
,
wi th equality holding only for a function of the form
F (C) =
+tf
~P (1
P ,
([or p ~ 1), then
CX)
~ nllXp+nlll~Bp,
Theorem 3. If the [unction (2) belongs to the class I~ (where p ~ 1), then
.. ICtt I :s;;; 2p,
493
(17)
n=1
where B p is a finite quantity depending only on p. In particular, 1) p = 2,
171 I= 1.
(15)
co
~ n 1lXp+n Ii ~ 18;
Furthermore, for n = 1, 2, 3, ... ,
11=1
I I~ An,p,
(18)
(16)
(Xn
with equality holding only for a function of the form where A n • p is a [inite quantity depending only on nand p.
Proof. The first of inequalities (14) follows from Theorem 2 for k prove the remaining assertions of the theorem, we set
F(~)=CP(1 +t+~+
..·
=
r
~ (n-I)lcnlg~l.
F(q=~8(1 +t)8, 1711=1.
11=2
for all n = 2, 3, .... Since the a,., for n
easily expressed in terms of the c n, for n n
= 2,
=
Ic 11 :s 2.
=
2, 3, ..• , are
1, 2, •.• , this proves (16) for
3, "', because, in addition, the inequality yield the inequality
Jall:s 2p
as is shown by. the example of the function
p,
F(q=(~+C+ft,
and the equation
we have
which belongs to the class I aI' ••• , a
2P
-
l
~p+P (P.,-I) 4=p(2p -1),
for arbitrary c and for which the coefficients
p
can be made arbitrarily large with sufficiendy large c. Further
more, an arbitrary polynomial
Ctl ll
zP
+
a
l
zp-l
+ ... + ~ belongs to the class I
p•
H we note that the function (2) belongs to the class I~ if and only if the
function
with equality obviously holding only foe the function (15). This completes tQe proof of the theorem.
(21)
We note that constants analogous to. An,p and Bp do not always exist for the class I
In particular, since
I~ I=pCg +P (p -:; 1) c~, Ia.g I ~p I Cg I+P (P-:; 1) I
(20)
11=1
with equality holding only for a function o[ the [orm
CX)
c1 = a/ p
(19)
oX>
~ n!lXp+n Ii ~ 300;
1'1
Ie n I :s 1
171 I = 1;
2) for p = 3,
for > 1. Then, from the lemma, with A= 2/p, we obtairi, by letting p approach 1 from above,
It follows that
+tr,
F (~) = ~i (1
1. To
n
1
F(
=zP(l +at z + ... )
-- __.
··
••.
_r_"K'.~"'"""'""""'-"""'"_=,,_._·-
.._..
_.~_._.'_.~.
_
~._.:..;;~.:;;;:.:,:~~;;.
..~:.-.._.~.
':':'::~~''':;-:::,",~~_;:
•
--
494
~
• .::
;-'-'~'
.;:;;:;::--.,.
'.T
_
.~,~
'"'
. _ '__
:.'_~:'-:_.'
-
~..
-
',:-"~:",
-.
__ • __
~:.: "''""'.~ ',r
.:'.;""....
XI. SOME SUPPLEMENTARY INFORMATI0N
§6. ON p-VALENT FUNCTIONS
belongs to the class Sp and if. we note that, for a 1 = a 2 = ... = ak-l = 0 (k:;:. 1), we have a 1 = a 2 = ... = ak-l = 0 and a" = - an> where n = k, k + 1, ... , 2k - 1, then, in accordance with Theorem 2, we obtain the corresponding theorem for the
cf(z) _
c - f (z) -
z
P+ (
ap
+ 1-)c z i P+
495
...
also belongs to Sp. Therefore, by Theorem 5,
class Sp: Theorem 5. If the function (1) belongs to the class Sp ([or p a 1 = a2 = ... = ak-l = 0 (for k? 1), then, with n
?
lap++1~2, lapl~2,
1) and if
k, k + 1,"', 2k - 1, we have
=
,
lanl~2pn'
so that
(22)
1+1~4,
Equality holds in the above relationship only for a function of the form
zP
F(z)= (1
+
that is,
~' 1"l1=1,
(23)
1
Icl~4'
1JZn) n
belonging to the class Sp' In particular, for an arbitrary function (1) in the clas.s
30. From what we have already shown, we can obtain some results concerning coverings.
belonging to the class Sp (where p
? 1)
lR completely covers the disk
maps the disk
IwI ~
1/2 P
Izj < 1.
f(z) =
l1,
=
In the case of the func
in addition to the preceding, a p = 0, then, from (26), we have
lei?
n,
1/2; that is,
Now, let the function (1) be any function in the class Sp. Then, the function
+ 1;
though not always any larger disk with center at w
Iwl < 1/4.
tion (24) the surface !R does not cover the point w = - 1/4. This proves 2).
surface
Then,
= '" = a p- 1 = 0, then R completely covers the disk
(26)
lR completely covers the disk \wl < 1/2. In the case of the function (25), the R does not cover the point w = - 1/2. This proves 3).
Theorem 6. Let !R denote a Riemann surface onto which a function (1)
2) if a 1 .= a 2
!R completely covers the disk
This shows that
Sp, we have la 1 1 ~ 2p.
1)
.,'~ .•~._ .,..?-__, _,"_;:;;""' ~ . -;;"'.,.•;'" '_'"''"'''.'-''''''__'''
.............
f 1(z) =
\wl < 1/4
0 since the function
zP _mi=ZP+2z iP + ...
p+1
V I (zP+1) =
zP
+ p+ I z'P+1 + .,. a
(27)
also belongs to the class Sp. This is true because, if there were points z l' (24)
z2' ••. , zp+ 1 in the disk
Izi < 1
at which fl(z 1) = fl(z2) = .•. = fl(zp"t1), we
would have belongs to the class in question; 3) if al
=a2 =
••• = a p
= 0,
then !R completely covers the disk
though not always a larger disk with center at w
=
I(z\p+1)= I
Iwl < 1/2
p+1 (Z2 )=
... =
I
(zpp+i + 1),
0 since the function
so, that by virtue of the p-valence of the function f(z), among the numbers
f(Z)=l ZPZiP=ZP+Z3P+ '" belongs to the class in question.
(25)
z~ + 1, zi + 1, z~ + 1
••. ,
= zi + 1.
z~ ~~, there would be some that are equal. Suppose that
Then, z2
only terms with powers
= Tfz l '
zn(p
rf+l = 1. But since the series (27) contains
+ I)+P, for n = 1, 2, .,. , we would have
Proof. We begin with part 2) of the theorem. Suppose that the function
11 (z~)
I(Z)=ZP+apZi P+ ... belongs to the class Sp. Let c denote a value that f(z) does not assume in
Iz I < 1.
Then the function
and consequently the result Tf
= 1;
rf = 1.
= 11 (1j Z1) = "lP/1 (Z1)
From the equations
that is, z 1
=
if = 1
and TJP + 1
z 2' which proves the p-valence.
= 1,
we obtain
~--- ...... ---
496
§7. ON TIlE CARATHEODORY-FEJER PROBLEM
XI. SOME SUPPLEMENTARY INFORMATION
Now, if c is a value that f(z) does not assume in the disk
Izi < 1,
then
which belongs to the class
!;
497
for arbitrary·. p> 1.
f1(z) does not assume the valueP+..jc. Consequently, in accordance with 3) we have
P+~ ~ 1/2, that is,
Ie I ~ I/ p P + 1.
§7. Some remarks on the Caratheodory-Fejer problem
This completes the proof of the theorem.
and on an analogous problem 1)
4°. Let us find another bound for the absolute value of the derivative of func tions belonging to the classes
fu subsection 1° of the present section, we shall give a simple derivation of
!p and !;.
Theorem 7. If the function (2) belongs to the class in the disk
1(1
!p (where p
~ 1), then,
> 1, we have
the results of Caratheodory and Fejer 2 ) dealing with the problem of the extend ability of a polynomial in z (by adding terms of higher powers of z) to a power series representing a rational fraction with constant modulus on the circle
I p' (1:) I :s;;;;; c (lXI,
Clg, ... , IX p _ l )
Iz I ==
1
and on the minimal property of the maximum of the modulus of such a fraction in
ICI~IP I • IC
the· open disk
Iz I < 1
among all functions that are regular in that disk. This
derivation is based on simple results dealing with the solution of the problem of coefficients in the case of bounded functions. In subsection 2°, we consider the
where c(a 1 , a 2 , ••• , CIp-1) is a finite quantity depending only on al' a 2 , ••• , a p - 1' If the function (2) belongs the class (where p ~ 1), then, in the ais.k
extendability of a polynomial in a similar fashion, starting with results dealing
1(1) 1,
with the problem of the coefficients in the case of functions with bounded mean
!;
values of the modulus on concentric circles. Here, the minimization of the maxi
I CIP
I F' (C) I ~ c (P) I C1_ 1 '
mum value of the modulus is accordingly replaced with minimization of the maxi mum of the mean value of the modulus.. A particular case of the extendability of
where c(p) is a finite quantity depending only on p. Proof. From (2), we have, iii the disk
a polynomial that is considered in subsection 2° has more than once found appli
1(1 > 1,
cation (by Landau, Fejer, Szasz, and others) in inequalities relating to bounded 00
IP
(C) I ~ pIC Ip-I + (p - 1) !IXIIIC IP-II +
... +
1 1Xp-1 I
+
!
n 'I ~I~':!
11=1 00
=pIC\P-I+(P-l)l lXlIICI P-II + ... +llXp_d+] l~l·vnIIXJI+/l1 ~p ICI P- +(p -1) 11X111 CID-\I+
1
... + IlXp _ll 00
11'
!
n IlXp +II Ill,
11=1
We note that the order of the inequalities given by Theorem 7 is sharp since
F(C)=P
CI-l
PC
_ 1'
obtaining sharp inequalities in all cases, thus removing this gap at least theoreti
of the solution given by I. Schur to the problem of coefficients for bounded func
from which Theorem 7 follows by virtue of Theorems 1 and 3.
for the function
section 2°, our second method of extension enables us to indicate a path for
tion of the Caratheodory-Fejer problem, give a simple derivation of the final form 00
1l=1
pi F'(p) = pl-l
the particular extendability referred to does not exist. As will be shown in sub
cally. We note further that in subsection 1° we shall, by beginning with the solu
11=1
+ 1/ 1: I C12~1I
functions. However, the problem of sharp inequalities has remained unsolved if
tions in the case of interior points of the domain of the coefficients. We shall use the following notation: 3 ) We denote by B the class of func
+ C lZ + •.. that are regular in the open disk Izi < 1 and that satisfy in thaC disk the condition If(z)l:s: 1. We denote by HI the class of func tions f(z) == Co + C 1 z + ..• that are regular in the disk Iz I < 1 and that satisfy the condition tions f(z) ==
Co
1) Goluzin [1946a].
2) Caratheodory and Fejer [1911).
3)
---............
498
XI. SOME SUPPLEMENTARY INFORMATION
for 0
499
form (l)corresl'onds to a point (cO' cl"", Cn-I) on the boundary of
21<
2~ ~
§7. ON THE CARATHEODORY-FEJER PROBLEM
IJ(re/
B )
IdO ~1
B(n).
Following Schur, we define a sequence lfk(z)l inductively by
~ (z) --:- J(z) ~ (z) = ~ Ik-l (IZ) - Ik-l (0) o
< r < 1.
'k
Z 1
-
k-l (0) /,-k-l () Z
-I'
k
=1
J
2
"
(2)
•••
for n ~ 1, the points in which are the complex n Let us show that Ih(o)! < 1 for k = 0, 1, " ' , n - 1. Since the h(z) belong to B whenever they are defined, we have Ih(o)! ~ 1. Let f ,AD) denote the first tuples O. Beginning with the con Consider the space
tR n ,
cept of a neighborhood, we define cluster points, interior and boundary points of a
to (2),
set, open and closed sets, and convex sets in the usual way. The space ~n can,
+
f. (.{) = E, f k :...! (z)
in case of necessity, be regarded as a 2n-dimensional Euclidean space with
Ik-l (0) zlk (z) 1+Ik-l (O)z/k (z) ,
Cartesian coordinates corresponding to the real and imaginary parts of the nwmbers
Co' Cl'
••• ,
Cn- l'
We denote by B (n) the set of points (c 0'
C
l' ••• ,
C
n-I)
€
tR n
such that the
we can show by induction that
numbers co' cl' " ' , c n- I are the first n coefficients of some function in the class B. We denote by H I(n) the analogous set for H l' These are bounded sets in ~n because, for all k
=
0, 1, ... , we have
Ie k I ~ 1
for both the functions in
B and the functions in HI' by virtue of the integral representation of the C k. Since the principle of compactness holds for functions in Band HI and since the limit. of a sequence of functions be longi ng to either of these c lasses also belongs to the same class, it follows that B (n) and H ~n) are closed sets. Further more, since for functions fI(z) and f 2 (z) belonging to one of these classes the functions AfI(z) + (1 - A)f 2(z) for 0 < A < 1 also belong to the same class, it follows that B (n) and H ~n) are convex sets. It is also obvious that the origin (0, 0, ••• , 0) is an interior point of these sets since the polynomials Co
( )
J .-k Z
nator by 1 +
Z
ClO
+~n sz + ... + ;;ozn-l + 1Z +... + n_1Zn- 1 . Cl
Cl
Proof. 1) Let us suppose that a function fez) €
B other than a fraction of the
1) The basic idea of the proof is borrowed from Schur. See Schur [1917] and also Bieberbach [1927].
'a
( 1)
1, k
= 1,
2, ... , .
(3k. In particular, fo(z)
=
f(z) is also
+ ... + zn-II-1 and this, by the hypothesis of the theorem, is
impossible. Since
Ifk(o)! < 1
for k
=
0, 1,'" , n - 1, the functions (2) are defined
for k = 0, 1, ... , n - 1 (that is they have denominators not identically equal to
0). For these functions, the inverse formulas
Ik dO)+zlk(Z) k=n-I, ... ,1,Jo(z)=J(z) I Ik-l (0) Zlk (z) ,
+
(4)
are also meaningful. Now keeping the values h(o) for k = 0, 1, ... , n - 1, let
us calculate the functions
Cl n 1
I Ek I=
equal to 1 because otherwise we could arrange for
this by changing the arguments of {30"'"
C1 z + ... + Cn- 1zn-I belong, for small Cko both to B and to HI' B(n)
£k
,
a fraction that reduces to the form (1) when we multiply its numerator and denomi
Jk-l (z)
there correspond in B only fractions of the form
~k+~k_1Z+'''+~ozk
= Ek ~o + ~lZ +... + ~kzk
Here we can assume the
=
10 • Theorem 1 (Schur). To points (co' cl"", Cn-I) on the boundary of
(3)
f :(z),
for k
J:-I (Z)=Jn_I(O), J't-I(Z)
I All the
f :(z)
=
n - 1, ... , 0, from the formulas
Ik_I(O)+zl:(z) k=n-1, ... , 1. (5) 1+ I k - 1 (O)zl% (z) J
belong to B. From (4) and (5), we have
ik-I(z) - J:-I (z)
(1 -Ilk_I (0) II) z (fk(z) (1
n (z»
+ I k - l (0) zlk(z»(1 +I k_1 (O)zll (z»
Ilni\/pr~av
nf
Z (Ik (z) -
f%(z»
rpk
W~~hinO'tnn
(z),
I ihrarv
--~
500
§7. ON THE CARATHEODORY-FEJER PROBlEM
XI. SOME SUPPLEMENTARY INFORMATION
where the ¢k (z) are, for k = n - 1, ... , 1, regular in
I(z) - It (z)
1z I
to some positive q is a function f(z) € B, then there exists a q '< q to which
< 1. Therefore
= zn-l
corresponds the function (q I Iq)f(z) € B, and we see that (qc o' " ' , qc n- l )€ B(n) for all q < q o. Therefore, by virtue of the closedness of B(n \ we conclude that
In-l (0».
Since the point z = 0 is a zer 0 of the right-hand member of multiplicity no less than n, the first n coefficients in the expansions of
f (z)
and
f ~(z)
about z
(q OC 0' 0
••• ,
q OC n-l) € B(n). It follows from the definition of q 0 that
are identical; specifically, they are co' .•. , Cn-l' On the other hand, since fk(O),
(qoc o' " ' ' qocn-l) is a boundary point of B(n). By Theorem 1, corresponding to it in B is a fracrion of the form (I). The fraction R(z) = A¢ (z), where A = II q 0'
for k = 0, 1, ... , n - 1, are rational functions of co' ... ,cn-l' co' •.. , cn-l
is of the form (6). It is regular in Izi < 1 and has CO' cI"", c n -
and si nce CO' Cl ' for k
=
... ,
Cn-l in our notation satisfy the inequalities
Ih(o)1
=
< 1
ncoefficients in its expansion about z
0, 1, ... , n - 1, these same inequalities are satisfied for all systems of
c·n-l ) sufficiently close to the preceding ones. The functions f ;(z) calculated from f k(O) corresponding to c~, c;, ... , c ~-l belong (by virtue
of formulas (5)) to the class B. On the other hand, if we take for f o(z) the poly nomial c~ + ... + c~_IZn-1 and then calculate the h(z), for k = 1,2"", n - 1, from formulas (2), these fk(z) agaili satisfy.(4). Just as above, we can show from (4) and (5) that the (unction !(z) -
f ~(z)
f ~(z)
uniqueness of the fraction R(z) and proceed to the second part of the theorem. f~; ,
Let f(z) deqoteany of the functions mentioned in the theorem with finite Mf • 10 such a case, the difference f(z) - R(z) and hence the function (a o + ...+ ~_lzn-l) [f(z) - R(z)] have the expansions about z
point of B(n) iii contradiction to the assumption. Therefore, only fractions of the
00
(ao+ ... +an-1Zn-l)/(z)=).,(a.n_l+ ... +~zn-l)+ ~ a"zk. "=11
H we now integrate the squares of the moduli of both sides of "this equation with respect to () (where z = re iB ) over the circle Izi = r, where r < 1, we obtain 2"
form (1) can correspond to boundary points of B(n>. This completes the proof of
;7t ~ Icto +...+ an_1zn-1 I~ II (z) I~ dO
the theorem. Theorem 2 (Caralheodoryand Fejer). For any polynomial Co + c1z + ... + cn_1z n- 1 ~ 0, there exists a uniquel) fraction of the form
R (z) =)., CI n_ 1 +... +;ozn-l
+ ... +
CIn-lZIl-1 ,
= A~ (I an_ I~ +...+ 1ao IlIrll(n-O) + 1
"',
),,>0,
(6)
cn-l' this function and only it minimizes the quantity Mf = m8X!Z I
Proof. Let qo denote the least upper bound of the set of positive numbers q such that (qc o"'" qCn-l) G B(n). The number qo is finite and positive since B(n) is bounded and since the origin is an interior point of B (n). H corresponding
1) By uniqueness is meant that any two such fractions are identical as functions of z, that is, after all possible cancellations have been made.
~
ex:>
~
Ia" III r ll".
1I=n
that is regular in Iz 1 < 1 and that has as its first n coefficients in its expansion about z = 0 the numbers co' Cl"", cn-l' Out of all functions f(z) = Co + c1z + '" that are regular in Izl < 1 and have the same first.n coefficients co' c l'
0 that begin
=
with terms of degree not less than n. This yields
are
" .. , ,cn-l; • ' . (cO"", . ")' co' th at 'IS, t h'I S f unction correspon d s to t h e pomt Cn-l which therefore belongs to B (n). This shows that (c 0' ••• , c n-l) is an interior
as its first
O. Except for the uniqueness, this proves
now has a zero of multiplicity at least
n at z = O. Consequently, the first n coefficients of the function
CIO
=
l
the first part of the theorem. We leave aside for the moment the question of
numbers (c""'" o
t
501
H we now replace If(z)1 with maxl z 1= r approach 1, we obtain
(Mj-).,II)(I ao
If(z)! ::; mRXlz !
/11+ ... +
"If and let r
ex:>
1 an_lll1)~
~
I all
III.
k=1I
Consequently, "If ~ A =
MR (z) with equality holding only "'hen all the n, n + 1, ... , that is, only when f(z) = R(z). This proves the se cond part of the theorem. ak =
0, for k
=
The uniqueness of R(z) now follows easily. Suppose that, in addition to R(z), there exists another fraction R "(z) the first n terms in the expansion of which are the same as those in the expansion of R(z). Let us denote the number
--'.=~'
~~~~~..,...~~~~
-
XI. SOME SUPPLEMENTARY INFORMAT'ION
502
§7. ON THE CARATHEODORY-FEJER
above as applied once to R(z) and fez) = R·(z) and once to R·(z,) and fez)
2:
Adn _ 1 = Adll_ 1l =
completes the proof of the theorem.
A~O
In what follows, the fraction R(z) mentioned in Theorem 2 which is uniquely
determined by co' ••. , C n-l' will be denoted by R(z, co' .•• , Cn-l) and the
C n-l)
is equal
o o
Dn(A) =
0 0
Co
o
o o
-A 0 O-A eo... 0 0
ell Cl
•
coao,
+ ~o~n-~ +.~. ~n~la~, coal
CIaO'
}
+
(10)
replacing all terms with their conjugates, we obtain a system of 2n linear homo geneous equations in 2n unknowns
to the largest positive root of the equation of degree 2n
0... -A...
503
If we combine these equations with the equations obtained from (10) by
Cn -l)'
Theorem 3 (Caratheodory and Fe jer). The number A(c 0' •.. ,
-A
PRO~EM
=
A and A 2: A· so that A = A· and hence R *(z) == R(z). This
number A corresponding to it by A(C o"'"
=~"'"
""""_....
determining A, a o' ... , c1n-l:
corresponding to A by A·. Then, on the basis of the minimal property proved
R(z), we see that A·
-~~~
Cn _
Co
Cn _1l
1
Co
o o
-A
For this system
to have nonzero solutions, it is necessary that its determinant be equal to zero,
Cl
o o
a,.-1' ... , i'io• a o' •.• ,an-I'
and this leads to equation (7). which the corresponding A must satisfy. Since the required fractions exist (the fraction R(z, co' ..• , C n-l) can serve as an example), equation (7) must have positive roots. For each such root, the system of equations (7)
=0.
referred to has nonzero solutions in the unknowns
Cln-l' ... , ao' a o' .•. , ~-1"
However, a system of values of these unknowns is necessary in which ak and i'ik are, for 0, 1, ... ,n - 1, complex conjugates. Such a system exists. Specifi
Cn _l
If all the
Co' C l ' ••• , C n - l
en_I!'"
0
Co
are real, then A(c
0...
0' .,. ,
-.A
cally, if f3.-(n-l)' " ' , f3.-0' {:30' "', {:3n-l is an arbitrary nonzero system of solu
c n -l) is also the greatest
dn(A)=1
A
o -
0 '" A ...
Co
Cl •••
0
Co
Co
Cl
13-k. (8)
-A
ao
+... + + '" + an_ z n-
(9)
A> 0,
1 ,
=
0 are equal to
co' .•. ,c n - l ' These fractions include the fraction R(z, co' .. , , c n -
1 ).
If we
cross-multiply in the equation
- + +-aozn-l _ n a o + ... + n z
an 1
'"
Cl _l
1 -
= -
Yk =
f3k
+
(n - 1), "', - 0, 0, ... , n - 1 and Ok = i ({:3k - f3.-k), for k =
0. 0, ., . , (n - 1) is also a system. Here, at least one of these
=
Y-k and Ok
=
8-k>
this proves the assertion made. Thus, we have
shown that corresponding to every positive root of equation (7) is a fraction (9)
the first n coefficients of which in the expansion about z
A
also provide a system
solutions is other than the zero solution because, if this were not the case, the ditions Yk
aoz n I 1
130' 13-0' ...• 13-(n-1)
system (:3k would also be the zero solution. Since Yk and Ok satisfy the con
Proof. We seek all fractions of the form
A an_I
for k
- (n - 1), .. , ,-
1=0. Cn _l
(3n-I"'"
of solutions, as one can easily verify. Therefore, each of the systems
of the absolute values of the real roots of the equation 1) of degree n
-
tions, then the numbers
Co
+ + CI Z
with the property required. Suppose now that all distinct positive roots of equation (7) are constituted by Ao"'"
Av , where Ao > ... > Ay and suppose that corresponding to them are
the fractions Ro(z), ••. , Ry(z) of the form (9). Let us show that of these\only
Ro(z) is regular at •••
and equate coefficients of like powers of z we obtain a system of equations for
the roors of equations (7) and (8) are real, bur this factor is of no significance ro us.
We note first of all that the fractions Rk(z), for
k = 0, 1,"', v, have no zeros or poles on the circle Izi = 1 since \Rk(z)! = Ak, for k = 0, 1,···. v, on Izi = 1. Suppose now that Rko(z)'is regular-in the . closed disk
I) FuUhermore, by virtue of the familiar properties of characteristic equadons, all
Izi < 1.
Iz I 5: 1.
Let
R k 1(z) denote an arbitrary fraction among the fractions
R k (z). The difference if; (z) = R k 0 (z) - R k 1 (z) has an expansion about z = 0 that begins with terms of power no less than n. Therefore, the number r of roots of
_ _ _ _ ilfima
'lIIIl~
_!IIi
_:wwm
1!!_~~~_'la\Ilo~_~milll!i;;l1J1l1U'H_~f~~·~ikl:I~;·~II~,y~~~i'I"-'QIo~~";i:.y.~~.;I_~!iiZ_II!l_~
_ _~ ~ R I I : I < 1 l i ~
----"
504
S7.
XI. SOME SUPPLEMENTARY INFORMATION
¢ (z) (counting each root according to its multiplicity) in the open disk
Iz I < 1,
and the number a of poles in that disk does not exceed n - 1; specifically, it is equal to the number of all the poles of Rk 1(z). Furthermore, on
I~ (z) I~ II Rll o (z) I-I Rill (z) II = IAllo Consequently, ¢(z)
Izj =
t.
0 on
Izi =
1. Therefore, as
Iz I =
1,
Alii I -=F 0.
z moves
arg R k /z) acquires an increment of 217(t - 0)' where t is the number of zeros of
jz I < 1.
This number does not exceed n - L Therefore, the increment
in arg (¢(z)/R k1 (Z» as z moves around Izi = 1 is equal to 217(r- t) ~ 217. Con sequently, the circle Izi = 1 includes a point Zo at which ¢(z)/R k l(z) is posi
final form of the solution by Schur of the problem of coefficients for bounded func The~em 4 (Schur). For the point (cO"'" cn-I) to be an interior point of B(n), it is necessary and sufficient that the inequality
O
I Rll o (zo) I =
I rp
(Zo)
+ Rill (zo)
1=
1 Rkl (zo) I
I1 + R:~(~o) I > IRkl (zo) I= Akl'
°1
° Dk(l)= I ° ° ° ~o
tive. We have
AII =
505
tions in the case of interior points of the domain of the coefficients.
1 around the circle
1, the function arg ¢(z) acquires an increment 217(r- 0), and the function
Rk 1(z) in
ON THE CARATHEODORY-FEJER PROBLEM
~1
~o
~k-l
~k-'j
.
Co
"
...
Cl ••• CII_l
° .Co ... ... 1 Co 1>0, k=l, ... ,n, ° °° ...... °
... 1 ° ... 1 •.• ° . . . °. ° ... ° ° ... 1 CII_i
(11)
~o
be sa tisfied. Proof. We note that the determinant Dn (A) (d. (7» is an even function of A because, if we divide each of the first n rows and the first n columns by - 1, we obtain Dn(A). Therefore, the determinant D n(I) is of the form (11).
> Ak l' This shows that Ak 0 must be equal to AO; that is, the fraction (9), which is regular in \zl < 1 corresponds to the greatest root of equation (7).
(c 0' ••• , c n-I) lies inside B(n), then the point (qc 0' ••• , qc n-I) lies in B(n)
Therefore, R(z, co'"', c n-
for some q
that is, Ak 0
1)
= Ro(z) and A(co"'"
c n-l) = AO' which proves
> 1. Consequently, there exists a function f(z) € B whose expansion
begins qc 0 + .•. + qc n_ I z n-
the fir st part of the theorem. Now, in the case of real co' " ' , cn-I' if we set ak = Xk + iYk, the system (10) can be broken, by separating the real and imaginary parts, into distinct sys tems each consisting of n homogeneous linear equations in the unknowns Xk for one system and Yk for the other. Here, the determinant of the first system is the determinant (8) and that of the second system is d(-A).
We first prove the necessity of the conditions of the theorem. H the point
For the system (7) to
have nonzero solutions, it is obviously necessary and sufficient that one of these smaller systems have nonzero solutions, that is, that A be a root of the equation
d n (A) d n (-A) = O. Furthermore, just as in the proof of the first part of the theorem, we can show that A must be the greatest root of this equation, that is, the great
1
+ .•.. On the other hand, the function f 1(z) = I f(z)/q = Co + ... + c n- I zn- + ... satisfies in Izi < 1 the inequality Ifl(z)1 ~ 1/ q. It then follows from Theorem 2 that A(c 0' .•• , c n-I) ~ 1/q. But then, in accordance with Theorem 3, Dn (A) has no roots in 1 ~ A < + 00. Since Dn(+oo)=+oo,wehave Dn (I»O. Furthermore, since (c o,···,c n-l)isan interior point of B(n), it follows that the point (co',··, Ck-l) is obviously, for k < n, an interior point of B( k) in the space 'R k • Consequently, by analogous reasoning, Dk (l) > 0 for k = 1, 2, •.• , n - 1. This proves the necessity of con ditions ( 1 1 ) . ' ~ We now prove the sufficiency of the conditions of the theorem. H conditions,
est of the absolute values of the roots of equation (8). This completes the proof
(11) are satisfied, then the equations Dk (A) = 0, for k = 1, •..
of the theorem.
roots in the interval 1
Before turning to the following problem of the Caratheodory-Fejer type, we shall give an application of what we have just discussed to the derivation of the
~
A<+
00.
,n,
cannot have
To see this, let us assume the opposite. Let
D j.L(A) = 0 be the equation with smallest k = fl that has roots in the interval 1 ~ A < + 00. Let us denote by Dk,I(A) (where k, 1=1, 2 " " , n) the minor of the determinant D j.L(A) obtained from that determinant by removing the kth row and
_:,. _.
- ------------------_._--
506
~
-~---~
_
-_.
____"c____:_=~o----~==-·=---:",~--~------
..,..."____,,._,.._:_:,,....•
~,_____o_"____,~=_cc-___".-____,_=-~
XI. SOME SUPPLEMENTARY INFORMATION
§7. ON THE CARATHEODORY-FEJER PRO'tJLEM
the lth column. Let us denote by DkZ,kZ(>') the minor of DjJ-(>') obtained by remov ties of determinants, we have
cto
21J.
~ D k , k (A),
(12)
(A) D I • I (A)
D k,I
(A) D I , k (A) = DIJ. (A) D kl , III (A).
(13)
+ ... + ct,z' :;t: ° in
f(z) =
"i.:
=0
k
c k x
belonging to class H I' find those that maximize the quantity
/I-I
I~ In the present case, with real >.., we have: Dk,k(>") are real, Dk.,z(>") = Dk,z(>"), and DI,I(A)=->..Dj.L_I(>"). If >"*21 is a multiple root of DjJ-(A) = 0, it follows from (13) that all the Dk,k(A *)
are, for k = 1, 2, ... , 2p., different from 0 and
have the same sign. It then follows from (12) that Dk,k(A *) = 0, for k = 1,
2, .•. , 2p., since D ~ (A *)
=
I z I ~ 1.
Proof. Let us solve the following extremal problem: Out of all the functions
11=1 Dk, k
(1)
O~'1
where
=
507
(cto+ ... +CV')2(?0+ ... +~/I_,_tZn-H)(~n--,_t+ ... +~oz/I-v-J)
ing the kth and lth rows and the kth and lth columns. From the familiar proper
D~ (A)
_
..,...._::_c_,.~==_~ __===_______,____--~
TkCk
k=O
I
for given complex numbers Yk not all zero. By virtue of the familiar boundary properties 1) of functions belonging to the class H I' the function 00
O. 10 particular,
f(z)
=
~
Ckzk
E HI
k=1
D t . t (A*)=- A*DIJ._t (A*) = 0, which cannot be the case. This shows that the equation D jJ-(>') ple roots in 1 S A < +
00.
Furthermore, the interval 1 S >..
<+
00
=
0 has no multi
cannot have more
than one simple root because, otherwise, when we denote by A* and A** two
tangential paths iri the disk
IzI < 1
Ck
D jJ-_I (;~.) = 0 has a root between A* and >.. **. Since by assumption this is impossi
(Here
f(O
ble, it follows that the equation D jJ-(>") = 0 has no more than one root in the inter
k=O
has no zeros at all in I S A < +
00,
> O. Consequently D jJ-(>") = 0
which contradicts the assumption. This con
tradiction shows that Dn (>..) = 0 has no roots in the interval 1 S A < + fore
A(C o"'"
cn-I)
00.
Theorem 1, the point
(c 0' ••• , c n-I) is an interior point of
B(n).
This com
pletes the proof of the sufficiency.
k=O, 1, ...
(2)
_1 \ f (C) ( 21ti J V Tn-l
+
..•
+ T rn-t) dC.
(3)
0-'
But, in addition to formulas (2), we also have the formulas 1 21ti
Cf
.)
k
(C) C dC = 0,
k=O, 1, '"
I C1=1
is an example of a function in the class B that is different from a fraction of the form (1) and that has an expansion beginning z 0 + ..• + C n-I zn-I + .. '. By
dl.,
ICI=I
There
< 1 and, consequently, the fraction R(z, co'"', cn-I)
Ck+1
means the limiting values mentioned above.) On the basis of this,
vallS A < +
and D jJ-(I)
J
I C1=1
n-I
~ Tk Ck -....
00
\ fCc)
~ 21ti
DjJ-_I (A **). But D ~ (>.. *) and D ~ (>... *) obviously have the same sign. Therefore
However, D jJ-(+ 00) = +
and it can be expressed in terms of these
_ -
DjJ-_I(>"*) have the same sign, as is true, analogously, for DjJ-(>"**) and D jJ-_I (A *) and' DjJ-_I(>.. **) must have the same sign. This leads to the fact that
1 along all non
limiting values in accordance with Cauchy's formula. Consequently, we obtain
for the coefficients c k the integral representations
consecutive roots and use (12) and (13), we can show that D~ (>.. *) and
00.
Iz I =
has definite limiting values almost everywhere on the circle
This enables us to replace the parenthesized expression in (3) with an arbitra~y, function, regular in the closed disk first n coefficients equal to
"I
Yn-l' •••
S 1, whose expansion about
,=
0 has its
'Yo' For such a function, we may take,
2°. Theorem 5. Corresponding to the points
1) If we remain within the framework of the elements of the theory of functions, we can interpret the class HI. in what follows as functions satisfying the additional condition of being continuous in the closed disk Iz lSI.
§7. ON THE CARATHEODORY-FEJER PROBLEM
XI. SOME SUPPLEMENTARY INFORMATION
508
where a~ + ... + a~zv;i 0 in
0
in particular, a fraction R(z) of rhe form (6) of subsecrion 1 consrructed from
,
'~\
these data. Then, from (3) we obtain
.\
n-I
arg
'YTkCk=~ ~ f(C)R(C)dr 2'ltt --
_
Izl $1.
Therefore, on
1'1 ==
509
1,
f (C) R (C) _ f (C) (ii~ + .,. +ii~C')
cn 1 - arg Cn • 1 (a; +a;C')
+ '"
f(C)
9
It =1
k=O
and, consequently, n-I
I
2:
TkCk
I ~ 2~ ~ \/(e) R(Qdr: I It!=1
k=O
= 2~
~ I/(QdCI:,,~;;A=A(in-l' ... , To);
(4)
Itl=1
i8 which is regular in I~ < 1 and for which the integral (1/277) f~1< 1¢(re )ldO, is bounded for 0 < r < 1, has an imaginary part whose limiting values coincide
that is,
almost everywhere on n-I
I~
TkCk
k=O
I~ A(Tn-I, .•• , To).
(5)
1'1
== 1 with the values of an analytic function of the arc
length. If we represent J(¢(z)) in the disk
Izi < 1
in terms of its limiting values
in accordance with Poisson's formula (which is possible since ¢ (z) itself can
J (¢ (z))
be so represented in terms of its limiting values, we conclude that Here A(Yn-l"'"
0
Yo) is the quantity referred to in subsection 1
•
Let us investigate whether equality can hold in (5) and, if so, for what func tions. For equality to hold in (5), it is necessary that equality hold in both places in (4), that is, that the function f(z) satisfy the two conditions
analytic function of the arclength on
Izi
== 1 and is continuous on
2)
2~ ~
=
canst almost everywhere on
1'1 ==
S I.
Izi
Thus, we need to seek the extremal functions only among the functions f(z) that
1'1 ==
lor,
what amounts to the same thing, the condition that arg
1;
$ 1.
are regular on jz lSI. From what was said above, condition (1) is equivalent to the condition that the last argument in (6) have a constant value on
I) arg f (C~"R. (C)
Izi
But this in turn implies that ¢(z) and, consequently, f(z) are regular on
is an
acquire an increment 277(n - v-I) as 'moves around the circle
1'1 ==
CV)
1 in the
positive direction. This means that the function f(')/(a~ + .•. + a~ ,v)2 has
I/(QdC!= 1.
It/=!
exactly n - v-I zeros in the open disk
But the fraction
1'1 < I,
counting multiple zeros in
accordance with their multiplicity and counting half the multiplicity for zeros on
+ ... +Ci R(Z)=A a + ... + an_1z n o iin _ t
Il-. -r=;,-. -EA
I'I ==
n-I
oZn-l _ 1
Z-
TJk
Ik Z
_
lei-I,
1. Furthermore, in accordance with the symmetry principle as applied to
l'llkl~],
f(C)
k=1
cn-'-1
(a~
+ .., +a;C')2
if we cancel all common factors in the numerator and denominator, assumes the
it follows that this function is regular on the closed disk
form
pole at ,==
R(Z)=A
a:+ ao+
+Cioz ' +a~z"
,,~n-I,
00
preceding zeros about the circle
I
"I ~ 1
except for a
and that it has zeros symmetrically distributed with respect to the
(C) = C (a~
I"
= 1. Consequently,
f«J
+ ... +a;C~)~ (~o + ... + ~n_'_lcn-'-l). X (~n-.-l
must be of the form
+ ... + ~oCn-H),
(7)
~-----~~,
510
XI. SOME SUPPLEMENTARY IN FORMAnON
where c
=
§7. ON TilE CARATHEODORY-FEJER PROBLEM
const. If we choose C in such a way that condition 2) is also satisfied,
we obtain a function fez) €
H I for which equality can hold in (5). We can show
by direct verification in (4) that, for an arbitrary choice of (30"'"
(3n-v-I and
511
Proof. By beginning with Theorem 5, one can prove the existence of the ~1
,I
::l
required polynomial P(z) of the form (8) in just the same way as the existence of the fraction R(z) in Theorem 2 was proved on the basis of Theorem 1. Leaving
the corresponding choice of c, equality does indeed hold in (5). Thus, we have
aside for the moment the question of uniqueness, let us prove the minimal property
found the entire class of solutions of our extremal problem. 1)
of this polynomial that was asserted in the theorem.
We can now proceed directly to the proof of Theorem 5. Let (c 0' ••• ,
C n-I)
It will be sufficient to consider functions
f (z)
of the type men tioned in the
denote a boundary point of H~n l. H we treat H ~ n) as a set of points in 2n
theorem for which Jf is finite. As we know, such a function
dimensional Euclidean space with Cartesian coordinates xk' Yk (that is, ck =
everywhere on Iz I = 1, definite limiting values along nontangential paths inside
x k + iy k) and keep in mind the fact that it is convex, we conclude that through
Izi < 1.
the point (c 0' ••• , cn-I) we can draw a hyperplane I~~~ (akxk + (3kYk) where c
> 0, possessing the property (the support plane) that
=
in the portion of space defined by the inequalityI~~Io(akxk + (3kYk)
••• ,
Cn-I)' Consequently, this
Jj
HI
r-.l 0
0
< 1.
We
f (z)
= 2~ ~ \f (Q d~ I= 2~ ~ IP (Q + ~n Q ~) II d~ 1 ICI=1
is
ICI=I
lei - -2lt j IP(qIIIP(QI2_~nQ(QP(~HldU ICI =1
extremal and hence is of the form (7). This completes the proof of the theorem.
~ 2~ ~
Theorem 6. For any polynomial Co + ciz + ... + cn_Izn-II= 0 there exists a
unique polynomial of the form
ICI=l
+. "+ ~n_Y-IZn-H) (~n-.-l +...+~ozn-'-I), v-n-l
""""
8
_
1
-21t
Izl
such that the coefficients of zO, z, " ' , zn- I are equal respectively to co' cn-I' Out of all the functions fez) = Co + c lZ +"', that are regular in Iz\ < 1 with the same values of co' c p " ' , Cn-I> this polynomial minimizes the quantity 2,.
I P ~:) I ffi ( I P (~) 12 -
C j
ICI~l
(8)
,
ClO+'" +Cl.Z·::;i:O
c
2,.
H l' But since the function eai fez)
\I~:~ Ykq Is c. But equality can hold here for an arbitrary function fez) €
p (Z)=(ClO+' . .+Cl.Z·)2 (~o
2~
~ If(reiB)!dO='lim ~ \f(reiO)lde= ~ If(eiB)jdO.
If we now set fez) = P(z) + znQ(z), we see that Q(z) is regular in \zl have
belongs, for arbitrary real a, to H I whenever fez) does, we have corresponding to the boundary point (c 0'
sup
O
R(I~~IOYkCk) = c, where Yk = ak - i(3k' we conclude from our assumptions that
~HI~':~Ykck) ~ c; for all functions fez) €
2~
Jj =
Sc
(Minkowski). Since the equation for the hyperplane can be written in the form
has almost
We denote them by f(e''8 ). We have
c,
H~n 1 is contained
f (z)
But, on
1(1 =
~n Q(Q P (~)) I d~ I
1 C ( CnQ (C) P (q) IP(Qd~I-21t j ffi [P(')I Id~l·
(9)
ICI=I
1,
+." +Cl.~·)2 ~n-H I ~o + ... + ~n_._l~n-'-l 12
P (C) = (ClO
(10)
1"'"
Jj =
SUp
~ If(re iO ) j dO.
O
n We note that the question of inequalities dealing with the class HI has been investigated in detail by the author in [1946g]. (This second work of the author is directly associated with the results of the present section. It could not be included in the book because of technical reasons. )
and, consequently,
P (C) _ (;;0 + '" IP«()I-
la o
+ ;;X·)2 fn-'-l _
+'"
+apl2
1 ;;.
-C n- 1
+ ... +~oC'
ao+ '" +ap'
If we substitute this intO' the last integral in (9) and then evaluate it in accordance with the theorem of the mean, we see that it is equal to zero. Thus,
512
XI. SOME SUPPLEMENTARY INFORMATION
Jf
~ 2~ ~
1
P (C) dC
§7. ON THE CARATHEODORY-FEJER PROBLEM
form (8). Then,
= Jp (z).
I
1'1=1
n-I ' " 'Y C ~ Ik k k=O
Equality holds here only if
J ( I P (C) I~ - cnQ (C) P (C» almost everywhere on
1(1 =
=
0, i.e. J (CnQ (c) P (C»
= 0,
[!
IkCk
I~ i1t
= O.
.)
1(1 = 1
cn
J' I = I
..
(12)
jCI=1
that is,
I! IkCkl~
Since the function (II + IQ«(V(a o+ ... + a ll ,lI)2 can be represented in
< 1 in terms of its limiting values on
(c) P (C) dr
~ I!(C) P(C) dC I ~ 2~ ~ I P(C) de I,
n-I
'(I
C I j
ICI=I
k=O
Q (C)
J_ 21ti
n-I
written in the form
(O:o+··.+o:,C')
=
so that
1. In aFcordance with (10), this last relation can be
J (CHI
513
in accordance with Poisson's
~ l(ao+···+a.C·)(~o+"·+ + ~n_._lC'I-V-I) [91 de I.
1 2'[t
k =0
Iq=1
formula, this will also be true of the imaginary part. Also, since the limiting values are almost everywhere equal to zero, we have Q(z) '" 0 in
Iz I < 1;
(13)
that is,
(z) '" p(z). This shows that P(z) is an extremal function and, in fact, the only
one for the extremal property in question. The uniqueness of the extremal function now implies the uniqueness mentioned in the first part of the theorem. This com plete s the proof of the th eorem.
For equality to hold in this last relation for some function (z) E B, it is necessary and sufficient that 1) arg «(()P(OI
In the proof of Theorem 5, we considered the question of a bound for the sum l~nk--;;'~ Ykc k I with regard to functions in the class H I' as a result of which we
obtained the sharp inequality (5) and we exhibited the whole class of functions for which this least upper bound is attained. Theorem 6 makes it possible to investi
2)
I(()I
=
C- I ) = const
almost everywhere on ,,\
1 a1mosr everywhere on
1'1 =
=
1,
1.
Condirion 1), which is equivalent to the condition
arg 1(')(0:0 +.. ·+o:P)1 = canst almost everywhere on Yv
"I =
1,
gate the same question in an analogous manner for functions belonging to the class
leads, just as above, to the conclusion that ( , ) must be regular on
B.
Suppose that (z)
= ~:=oCk zk E B
and let YkJ k
= 0, 1, " ' , n - 1,
denote
arbitrary complex numbers not all zero. Keeping formula (2) in mind, we again
0:.
~
'Y
C __ 1
~,kk-2'[ti
k=O
~
I(C)
cn
('Y
In
+ +10'"rn-l)dr.... ...
+... +
!(C)=e
We now replace the polynomial in the integrand with the polynomial P«() that is represented in the
C'
~'+',,+~oC:. e=const. 0:0
,n-l
11 0
and from this we conclude immediately that (11)
/,=1
having the same coefficients for (0,', ... ,
~ 1. But
on !CI=I, arg I(C) (l1 o+ ... +o:,CV)=const _
obtain n-I
1'1
this condition can be represented in the form
+ ... +I1,C
Furthermore, from condition 2), we have
1(\
=
1. This is the only possible form
for a function for which equality is possible in (13) and, as one can easily see
---==::;;:::~-=-~;;:-~===~--=---=--~~~=---=-~~-=-==--==:..=;;;;-~==-==--=
S14
XI. SOME SUPPLEMENT ARY
-~-~--===:=-==-=~--=.,,;;.-~~-~-,_:::~-=;;:-~~-=~~~= ----------------------------------------
- ~~-~
~~~~=-~~-~~~~~~~~~- ~
~~O_=--=~~~~~~~
-----~
§s.
~FORMATlON
=_-___=_---~~~
~
==~--=-..:=;;;;;;;:-=~~~~ ~~~
SOME INEQUALITIES FOR BOUNDED FUNCTIONS
SIS
from (12), equality does indeed hold here. The inequality that we have obtained is
that will enable us to study questions in an extremely general and exhaustive
sharp.
form. This same method can be applied at the same time to the establishment of
We note that, in earlier works by various authors, a bound for the sums (11)
inequalities regarding functions in the classes Rand Be.
for the class B was made by replacing the polynomial on th.e right with the square
1°. We shall establish some general theorem£.
of another polynomial of degree not exceeding n - 1 that has the same ·coefficients
Theorem 1. Let m denote a nonnegative integer and let Yn (n
of
,0, "... ,
,n-l
as the original polynomial. However, the inequality obtained
could be sh~ only in random cases when the corresponding polynomial (8) is
aT;~--aperiect square. Thus, a sharp bound was obtained, in particular, for the finite sum Sn = Co + ... + c n - 1 (Landau 1), and also a sharp bound was obtained for the arithmetic mean of finite sums an = (s 0 + ... + S n-l )In. 2) However, the inequalities obtained for the absolute values of arbitrary functions in the class
B were not always sharp (Szasz). 3) This is explained by the fact that, in the
denote complex numbers such that Ym 1) I( m
=
-f.
°
=
m, m + 1, ... )
and !:=m \Yn I < 00. Then
0, a necessary and sufficient condition (or the inequality
In~ornCnl ~ 11 I
(1)
0
to hold (or all (unctions (z)
!;:'=ocnz n E B is that
=
00
latter case, the parenthetical expression in an integral of the type (11) was
m(_1 ~ rnCn) ~ ~
replaced by a polynomial that is not a polynomial of the type (8) for yn-l + ...
1'0
... + Yon-l • Equality holds in (1) only (or (z) §8. Some inequalities for bounded functions 4 )
satisfy the inequ ality 2) the class
I( (z) I :':: 1
and
_1_ 1'm
in that disk;
°
for
Izi < 1;
Ic 0 I =
where
(2)
1;
m-l
~ rncn +;::-~ ~ r~m-ncn I ~ 1 ~ 1'm ~
=
(3)
n=O
!;:'=ocnz n E B is that 00
3) the class Be of functions (z) that are regular in the disk
Izi < 1
mC~1
and
Our purpose is, first of all, to establish sharp inequalities regarding func tions belonging to the class B in which only functions of the form (z)
If I =
co'
n=m
to hold (or all (unctions (z)
assume in that disk only values belonging to a given closed convex region G.
where
=
00
IzI < 1
R of functions (z) that are regular in the disk Iz I < 1 and
satisfy the inequality jJH(z))~
I CI= 1.
2) I( m > 0, then a necessary and sufficient condition (or the inequality
We shall consider the following classes of analytic functions. 1) the class B of functions (z) that are regular in the disk
on
n=O
=
om,
1, can be extremal. Such inequalities dealing with the derivative
were examined by the author earlier. The basic tool was then the invariant form of the Schwarz lemma. 5) Here, we shall use an essentially different method, one 1) See, for example, Landau [1929a]. 2) Ibid. 3) Szasz [1920]. 4) Goluzin [1950]. 5) Goluzin [1945]. We note (hat by the same procedure Dieudonne had already n obtained (he following results [193Ia): If !(z)= !:=lcnz E B, then 1 in the dosed disk jzl:,:: Vi - 1 bur nor always in a larger disk (see also Caratheodory [1936]).
1t'(z)l:,::
!
n=m
rncn-m)~+
ICI=1.
on
Equality holds in (3) only (or (z)= cmz m , where
Icml
=
(4)
1.
(We note that, by virtue of the assumptions of the theorem, the series in (2) and (4) converge uniformly on the circle
1'1 =
converge for all (z) E B since we then have Proof-. Since the functions (z)
=
!:
=
1'1 =
Icnl:,:: 1
(n = 0, 1," .).)
oC nzn E B have limiting values along
radial paths almost everywhere on the circle values on
1 and the series in (1) and (3)
Iz I =
1, if we denote these limiting
1 by (z) and take the limit as r approaches 1 from below in the
formulas
cn=~l~ 21ti
!(C) dr Cn +1 ,",
Ie =r
~ I(C) CndC = Ie I=r
0
(n=O, 1, ...),
§s.
XI. SOME SUPPLEMENTARY INFORMATION
516
°<
which hold for
only when
< 1, we obtain the formulas
r
21tt ~
1('
where C is the circle
Cn+1
=
w",
(n=O,l, ...),
««)/ (m) = const
1 and arg
measure on the circle
~!(,)Cnd'=O
Cn=~ \ f(C)-/1'
I«()I =
1(1 =
1. But then
and this can be the case only if «z)
c
517
SOME INEQUALITIES FOR BOUNDED FUNCTIONS
=
on a set of points of positive
f«)j (m
=
const on the set mentioned,
eiaz m in the disk
Izi < 1.
Then equality
does indeed hold in (3) or (1). Thus, the sufficiency of the conditions of the
1 and the integrals are Lebesgue. On the basis of
this and the uniform convergence of the series l::'=mYn(n on the circle
1(1 =
theorem is proved.
1,
we have
Let us suppose now, conversely, that inequality (3), or inequality (l) if m = 0,
B. 1£ we apply this inequality to a function
holds for all «z) €
00
00
_ I \ f(C) Tn
C 1;;; n~mln n - 27t1 ~ cm+1 n~m lmcn-m d'
=_1 \' f(1:.l.27t1 J Cm+1 C
(~
£.
n=m
= _I. ~ f(C) m 1 27t1
C+
~
1nCm-n+
1m
2m (J... 1m
~
1m
'\'
'"
27ti
~ ill ~
2~1 \ {~91 [2m ~
Co
rnfmjn dl' ~m+l ~ '" n=m 1m 00
,m-n) d' _ _ I . ~ f(C) 27t1 Cm+!
1m
!
n=m
In,m-n) n-m
! lnc2m-n' 1m n=m+1
--J-
1Jd' -
i
laI9-2m.c~~aI9 00
f(C) [2m cm+l
=
(J... y In ,m-n) -l]d'. 1m ....... 'Y
1£ we transpose the term (5)
n_m
factor 1 -
la 1 , 2
la 12
where (=
1, we obtain from equation (5)
m-I
00
_I ~ 1m n=m £.
=
lncn
in~n-m-l
~l.
we obtain
+ (I -I a 1 9 )
1: ina n=m+ co
Tm
n- m - 1
19
1
~I.
00
+~ ~ 12m-ncn ~ -2~ ~ [2m (_1 ~ In,n-m) 1m ..... 1m ..... 7t
n=o
1Jdrp = 1,
n-m
e i ¢; that is, we obtain inequality
(3) and, by setting m = 0, inequality (1). Since the function in the square brackets in equation (5) is not identically
equal to zero on the circle
i n=m+1
to the right and divide both sides by the common
- 2m (_1 ~ inan- m ) 1m n=m+1 ..
0, the second summation on the left disappears. 1£ condition (4) is satis
fied on the circle ,(,
inan-m-ll~l
uLn-m)+ll----;~aI9
00
For m
i1 n- m- 1
lal<],
n=m+1
~
_I (1 9)
or, by squaring,
C
27tl
c n =(1
n=m+1
+~ I1 m-n n 1m n=O =_1
cm = - a,
0,
l _a+I--:;~aI9 !
2m
CL !
il
.. ,=cm_1 =
we obtain
m-I
lncn
=Cl =
(n=m+1, ...),
that is, 00
EB,
_
'\.1 !n ,n-m d' L. n=m 1m
00
=
(1_!aj9)i1 n - 1z n )
for which
00
~ In n=m
!
n=1
1nCm-n) dl' _ _ I
n=m
;z =Zrn(_I1+
!(z)=zm;
"1
00
~
1
1(1
=
1, equality holds in the relation (3) or in (1)
If we now let
la I approach
1, we obtain inequality (4) with (=
a,
Since
arg a is arbitrary, this proves the necessity of condition (4). This completes the proof of the theorem. Theorem 2. Let Yn
l::'=oIYnl
<00,
=
0, 1, . , , ) denote complex numbers such that YO';'
For the inequality
°
518
XI. SOME SUPPLEMENTARY INFORMATION
§s.
00
m(_I )' "10 '-I
InCn) ~
0
(6)
n=o
to hold for all functions fez)
=
SOME INEQUALITIES FOR BOUNDED FUNCTIONS
to hold for an arbitrary function fez) = L;::'=ocnz n € Be whenever the series L;::'= oYnc n converges, it is sufficient that the inequality ex>
m(_IYI~n»l"10'" n ~ 2
L;::'=oCnZ n E R, it is necessary and sufficient that
m(i!n=OIn~n)~+
1~1=1.
on
(7)
member of R. Just as above (formula (5) with m
where
t;;= re i ¢,
00
'-I
n=O
and 0
(i n~o
\t;;1 < 1.
write in the form
00
n=O
Itl=r
1] dr'"
~
1
(8)
2"
n InCnp ) =
ex>
1 (Q u (~) d~,
~=rei'P,
(13)
where
2~ ~ ffi (I (~)) [ 2ffi( "10 n~o'n Uf) - 1] d~. 1
(9)
(~) = 2~
II
m(~! InCnpn)~O, and this holds for arbitrary p in the interval 0 cOllverges. Consequently L;::'=oYnCnpn
< P < 1. But the series
Since, by condition (12) u (t;;) L~ =oYn c n
L;;'=oYnCn as p---> 1. Then from inequality (10) we obtain inequality (6). This proves the sufficiency of the condi tion in the theorem. --->
!
In
n=O
(~ f) -
1].
:?
0 on the circI e
It;;1 = r and since II ~ l=ruCO d¢ =
1, the integral on the right side of equation (13) can be regarded as the integral mean of fCO on the circle on the circle
1t;;1 = r
\t;;1 = r
with weight u Ct;;). Since the values of fet;;)
lie in a convex domain G, it follows that this mean also lies
in G; that is,L;::'=oYncnpn € G. It follows that, if the series L;::'=oYnCn con
To prove its necessity, we apply inequality (6) to the function
verge s, then, by taking the limit as p
ex>
I(z) = :
[2ffi C~
ex>
(10)
n=O
+:~= 1+2! anz E R, lal= 1.
----->
1,
we obtain condition (11). This com
pletes the proof of the theorem.
Tl
n=1
We note that the sufficiency of the assertions in Theorems 1 (for m = 0) and
We then obtain
2 follows from Theorem 3. 00
Theorem 4. Let
m( 1 + "I~ ~llnan)~ which yields condition (7) for
t;; = a.
Theorem 3. Suppose that Yn (n 0 and
~
< P < r < 1. Therefore,
ex>
-t
2{
1":n~oInCnpn =
Since the i!1tegrand is nonnegative, we have
Yo
< P < r < 1, we have formula (8), which we
0), we can prove the formula
---_~~
pn= ~ C n n 27tt"
00
m
=
f(~) [2m (_I ~ nc-n)_ ~ _ 10 .i.J InP
~ I c
hold in the disk
Proof. Again assuming that 0
Proof. Let us suppose that condition (7) is satisfied and that fCz) is a
(12)
n=O
<:t:!
_I 10
519
li~n--> v~1 00
:s 1.
\yAl)!
and
complex n umbers such that For the inequality
Yh
1
lyA 2 )j )
1= 0,
(n = 0, 1, ... ) denote two sequences of
Yh2 ) = 0,
and
L;::'=01 yP)1
+
L;::'=olyF)1 <
<Xl.
This completes the proof of the theorem. =
0, 1, ... ) are complex numbers such that
For the inclusion relation ex> 1 -1
'Y,
0
n=O
Incn
E a,
(11)
I.~o T~"·I + I.~o TA",·I.;; IT~"l to hold for an arbitrary function fez) = L;::'=oCnZ n E Be, it is necessary and
(141
sa.
XI. SO\lE SUPPLEMENTARY INFORMATION
520
SOME INEQUALITIES FOR BOUNDED FUNCTIONS
521
that is, to the inequality
sufficient that ~-
ro
ffi
i l'n~n-m ~ i
ro
(-~ ~ "l'(ll~n) - J..:::> )'(1) "'" In 2 0--
o n=O
I_I ~ "l'(~)~n I )'(1\ " "
n=O
0
In
I ~ I=
on
1.
I
(15)
n m InC -
I'
n=m
I
~
I 1= 1.
on
n=m+l
(18)
The same is true for condition (12). By an analogous transformation, we can show Equality holds in (14) only for functions f(z)
= cO' where \c 0 I = 1.
that condition (15) is equivalent to the condition
Proof. Inequality (14) is satisfied if and only if the inequality
In~o l'~)cn +
e
ia
n~/~~)Cn 1= r n~o(l'~) + eial'~) Cn r ~ 1l'~ll I
V
_I )'(1)
n=O
o
(16)
is satisfied for all real a. But, for fixed a, this last inequality is satisfied for
2 1/
all functions fez) € B if and only if (Theorem 1)
"'"
"l'((lcnli In
(1(11)
l: (l'~l + ei"l'~»Cn) ~ ~
on
o n=O
I ~ I=
Consequently, for inequality (16) to hold for all fez) E B, it is necf'ssary and
n=l
In
)'(t)
0--
0
"l'(llen
~ In n=!
I~ on
le1= 1.
(19)
2°. As an application of Theorems 1-4, we present Theorems 5-10. Theorem 5. 1) For all functions fez) = I;:'=ocnz n E B in the disk Izl..::: (m + 1) _ 1 but not always in a larger disk, we have
l
j t<m (z)
(17)
l.
1
)'(t) "'" 0
ro
ffi
_21_ ~ "l'(2)cnl~I_~ ~
+ n=! f (n + 1) (n + 2) ... (n + m) cm_nzn I~ mI.
(20)
= c m zm, where Ic m I = 1. f(z)= I;:'=mcnzn € B in the disk Izl..::: 2 t1 (m+1)_ I
with equality holding onl y for f (z)
sufficient that ine quality (17) hold for all real a and this is equivalent to the
2) For all
fun~tions
but not always in a larger disk, we have
condition
ffi
If(m) (z) I ~m!,
(J..~ "l'(I)~n) _I-~ V )'(1) "'" In )'(1) "'" In
"l'(21 Cn
o n=O
0
n=O
I
~2
on
I ~I= 1.
(21)
with the same extremal function. Proof. To prove 1), we need to investigate whether the condition of Theorem 1 holds. with
which is what we needed to show. In accordance with Theorem I. equality holds in (14) only when f(z)= co' where
\col
l'n=n(n-I) ... (n-m+I)zn-m
1. This completes the proof of the
=
theorem.
(n=m,
m+I •...).
In this cas e, condition (4) or condition (18), which is better. takes the form
With an eye to applying these general theorems to particular cases. we note that it is convenient to write condition (4) (condition (2) in case m
=
0) in a
1.-
different, equivalent form. Specifically. condition (4) is equivalent to the inequality
m dm
mJ dz
m ~ znCn- m I IJ..- d m ~ "'" :: ml dz "'" :::>,
n=m
n~m
znCn- m -
1
I
on
ICI=1.
But
~"l' cn-mi~
_I )'m "'" In
I
n=m
~
1_
I
co
~
-.y;; "'" rn~n-m n=m
I~
on
ICI=I,
m
co
co
I d ~ nCn-m _ I d m ~ ",I dz m £.J Z -",I dz m ~
n=m
n=O
nrn
z..
m
=
I d m C··m I "'! dzm 1- zf= -,,--
.,_~..
(22)
522
§s.
XI. SOME 3YPPLEMENTARY INFORMATION
Therefore, inequality (22) takes the form
1(I =
1 CI =
(23)
1.
1, we have
1 -I (1 - Cz)m+ 1 -
1 I~ 1 - «1
+ Iz )mH -
1) = 2 - (1
+ Iz I)m+1,
and this inequality is sharp (equality holds only when (= - e-ia,gz). Con sequently, inequality (23) is satisfied if and only if 2 - (1 + Izl)m +1 20, that is, if and only if Iz I :s 21/ (m +1) - 1. By Theorem 1, this proves 1) of Theorem 5, m including the uniqueness of the extremal function f(z) = cmz , where Icml = l. To prove 2), to the class
I)
we need only note that the function f(z) = l;::'=mcnzn belongs
closed disk Izl:s rm· Equality holds only for f(z) = co' where m, the number r m cannot be replaced with a larger number. 2) For all functions f(z) E
Proof. Let us apply Theorems 1-3 with Yn Yn = 0 for n
> m. Inequality
I~ml
l: (n+ 1) .. . (n+m)z1IC -11
zm- 1Cm - 1 1 on
ICI=
1.
(24)
1
where \(1 = 1, and its sharpness for even m (equality holds when (= - e-ia,gz)
II
imply that inequality (24) holds for all z for which 1 - Iz I
But this condition is identical to inequality (22) and is satisfied only for
But r m
< r m +1
(for m
2 Iz 1m 2 0, that is,
2, 3, ... ) since
=
1, which proves part 2). This completes the proof of the
2r:+1 +rm+I-1
theorem. In what follows, we denote by sm(z) and rm(z) the mth partial sum (begin n ing with m = 0) and the corresponding remainder of the series I;::'=ocnz : l;::'=mcnzn.
H lim m--> 00 r m
=
> 2r:.:/=1 + 2rm+I-1 = 0 and 2r: +rm-1 =0.
q < 1, then, for all m = 1, 2, ... , we have r m < q and, con
sequently, 2qm + q - 1
> O. By letting m approach
00
in this last inequality, we
see that q 2 1. Thus q = 1. This completes the proof of the theorem.
Theorem 6. Let m denote an integer greater than 1. Let r m' with 0 1, denot e the (unique) positive root of the equation 2r
-
for Izl:s rm. Here, when m is even, we cannot replace rm with a larger number. I) By virtue of Theorems 1-3, this proves all the assertions of the present theorem.
on ICI= 1.
=
on Icl=l,
1
11=0
l:~~cnzn, rm(z)
(18): which with m = 0 is equivalent to inequality
- zm- 1Cm- 1 m m m ~ 1 - I z 1 - I z I(l + I z 1 - I ) = 1 -I z 1- 21 z ,
OJ
=
and
The inequality
11=0
sm(z)
= 0, 1, ... , m - 1,
11 +zC+ ... +zm-lCm-ll~lzC+... +zm-lCm-ll
[I - cmz mI-I z 111
(n+ l) ... (n+m)zIlC Il I
Iz 1:s 21/ (m + 1) -
forn
1.
(2), is of the form
11 - cmzml ~ I z 111 -
OJ
~ I ~!
= zn
=
that is,
in the cl ass. B. By Theorem 1, this requires that we investigate the condition
l:
1. For even
the inequality ~Hsm(z)) 2 0 holds in the disk
3) For f(z) E Be, we have sm(z) E G in Izl:s rm. We note that r2 < r 3 < r 4 <"', that r 2 = 1/2, and that limm-->oor m
B if and only if f(z)/ zm E B. Therefore, the assertion in 2) regard
I n~o(n+l) ... (n+m)CnZIl
R,
Ie 01 =
Iz I :S r m' Here, r m cannot be replaced with a larger number.
ing the point z is e qui valent to the inequality
I~l
523
1) For all functions f(z) E B, the inequality ISm(z)!:S 1 is valid in the
1 ~ I (l - Cz)m+1 - 1 I on But, on the circle
SOME INEQUALITIES FOR BOUNDED FUNCTIONS
m
< rm <
+ r - 1 = O.
I) Inequality (2I) is obtained from inequality (20) by applying the latter to the func
tion f (z) = 1;::'=mC nzn E B. Howevet, the problem then is to find me greatest disk for which inequality (2I) is valid.
I) In the case of odd m, the corresponding gteatest numbet is obviou sly equal to the greatest r in the interval 0 < r < 1 for which min (11 - Cfflrm I' - r l /1 - cm-lrm-III) >: 0
Ie 1=1
and it is distinct from rm'
"""" ,
/
§s.
XI. SOME SUPPLEMENTARY INFORMATION
524
Theorem 7. For every m> 1, let r m denote the unique positive root of the equation 4r m + r - 1 = 0 and set r 1 = 1/3. For all functions fez) € B, we have
ISm (z) I+ I'm (z) I ~ 1, in the disk
Izl:s rm
with equality holding only for fez)
Proof. Let us apply Theorem 4 with y(1) =
n
0 for n 2 m;
y(2) =
n
0 for n
=
y(l) =
n
1
Ii - I
=
co' where
Icol =
1. For
lim'm=1.
00
zn for n
0, 1, ... , m - 1, and
=
y(2) =
lI=m
zn for n
1(1 =
> m.
I I
on
1 for all z in the disk
zn-m (for
Izi < 1.
1
which is satisfied on the circle
This completes the proof of the theorem.
Be, we set _sO(Z)+Sl(Z)+ ..• +Sm l(Z)
then am(z) €
IC11,
G for all m
=
CI = 1,
m
1, 2, ... and all z in the disk
Proof. Let us apply Theorem 3 with Yo
= m, Y1
=
'
Izi < 1.
m - 1, ... , Ym-1
= 1, and
Yn = 0 for n ~ m. Condition (12) takes the form
> 1,
ffi(m+(m-I)C+ ... +cm-l);?:~
the inequality
m
m
C~-l +
+ Iz J) -I Z /2 (1 + Iz I -l)2
41 z /m+l = (1 + I z J) (1 -I z 1_ 41 z 1
~ (1 -/ z I )2 - 21 z 1 (1
= 1 -I Z )2 -
m
41 z 1 -
m
e- ia rg z)
Izl- 41zl
inequality (26) holds for those z such that 1 -
m
C';-2 + ... + m C 1 + m + (m -
Izi :s
or, finally, by setting (= e
1(1
=
sin
1
(
The assertion of the theorem then follows.
Theorem 10. For all functions fez)
but not always in a larger disk. By virtue of
Theorem 4, this proves the assertion of the theorem. The supplementary remarks
1/1 (z) I
about the rm are proved in the same way as in Theorem 6. where
Theorem 8. For functions fez) E B, the inequality zm sm
(+) + r~~)
I~
1,
a ) ~o.
2.
= 11 - Cz Ii - 21 z 111 - Cz I;?: 11 - Cz I (1 - 3 Izl),
I
~ay ~
• \ Si l l
11 - Cmz m12- 21 z 111 - CZ I-I z 1211 - cm-lzm-112
Iz I :s 1/3,
1
i8 ,
r m' Here, for even m, the number r m cannot be replaced with a larger number.
1, we have analogously on the circle
ICI=
C~-l (1 +C+ ... + cm-l)2~0
imply that
~ 0, that is, for
1) C+ ... + Cm
-1;?: 0,
or
m)
and its sharpness for even m (equality holds for (= -
which ~ 0 in the disk
IC\=I,
on
which we can rewrite in the form
11 - Cmz m12 - 21 z jm 11 - Cz 1-_1 z 12 11 _ cm- 1z m-l Ii
=
I(zj:s 1,
"m (Z ) -
II-CmzmI2-2IzlmII-czl_lzI2II_cm-IZm-112~0, I C/=1. (26)
For m
I on
~ CIl-mzll-m
ll=m+1
Theorem 9. If, for fez) €
n -
cmz Cz - cmzm 12 2 1_ Cz ~ 1_ Cz
00
I ~ CIl-m zll-m I~ I
. which reduces to the obvious inequality
m-+oo
that is,
For m
=
n = m, m + 1, ... ). Then, condition (18) takes the form
0, 1 , " ' , m - 1, and
m
525
Equality holds only for
Proof. Here, we need to apply 2) of Theorem 1 for the case Yn
(25)
Inequality (19), which is equivalent to inequality (15), is now of the form
m 1 - cmz 1- Cz
Iz j < 1.
where m is an integer ~ 1, holds in the disk f(z) = cmz m , where Ic m I = 1.
even m, the number r m cannot be replaced with a larger number. We note that
'1<'i<'3<... u
SOME INEQUALITIES FOR BOUNDED FUNCTIONS
=
00
1l=0
B, the inequality
+ 1/2 (Z) I,,;:;;; 1,
-F ( ) _ ~ 21l+! Jl Z LJ C211+IZ ,
(27)
~:=ocnzn €
(28) 00
/2 (Z)
=
~ 11=0
C21lZ21l,
===~"""","~'
Iz I s V2 -
holds in the disk
(28) only for f(z) =
co'
1 but not always in a larger disk. Equality holds in
where
Icol
= 1.
Proof. Let us apply Theorem 4 to the case y<{,; = z 2 , l{,; + 1 = 0, Y2n+l = z2 n +l for n = 0, 1, ..•• Condition (19) then takes the form n
2
1 -I Z 14 -
21 z 111 -
I
or
But, on the circle
1 -I z I&-
IT_ I I
1 - 1Z 2 C2 I~ -
1~ I =
zCZiCi ~ 1 _Z2C2 I~ on ZiCi z~C~ I .~ 0 on
I CI=
y
z 9C9 1~ 1 -I Z 1& -
V2 -
=
1 only for 1-
constants, which invol ves an individual difficulty in each particular problem. Ibe author will use the example of a distortion theorem to illustrate this difficulty, indicating
1,
in a roundabout way the varied nature of extremal functions depending
on z. 10. We shall derive rwo variational formulas. Let us denote by
I C1= 1.
(29)
+ Iz 1)(1 9
21zl- Izl 2 ~ 0,
of functions f(z) defined by a Stie Itjes integral of the form b (1)
a
21 z 1-1 z
2
1 ).
that is, for
Izi S
where a and b are given 'real numbers, where g(z, t) is a given regular function defined in the region
A method of variations in the lheory of analYlic funclions
Iz \ < 1, a S; t S; b, and where a(t) ranges over all possible as; t S; b and satisfying the condition
functions nondecreasing in
b
1. By virtue of Theorem 4, the assertion of the theorem follows.
§ 9.
Eg the class
f(z)=~g(z, t)da.(t),
This inequality.1s sharp (equality holds for ~ = ie-iargz). Consequently, ine<pality
I~I
like and typically real functions. Here, as with many other variational methods,
21 z I(1 + Iz I~)
= (l
(29) holds on the circle
formulas to investigate a number of general extremal problems for classes of star the complete solution of the problem rests on the determination of certain unknown
= 0,
1,
21 z 111 -
527
§9. A METHOD OF VARIATIONS
XI. SOME SUPPLEMENTARY INFORMATION
526
~da.(t)=a.(b)-a.(a)=1.
(2)
a 1)
Among the various classes of analytic functions, we consider those in, which the functions have a parametric representation containing a Stieltjes integral of the form
Let us consider two types of variations of the function f(z) €
Eg that are
obtained by means of a suitable variation of the function a(t). 1) Let t 1 and t 2 denote any two numbers such that a S; t 1
< t 2 S; b. We leave < t < t 2 but
b
unchanged the values of the function a(t) outside the interval t 1
~ g(z, t) da. (t)
replace its values 'inside that interval once with the average (weighted) values of
a
a(t) and a(t
where a, b, and g(z, t) are given and a(t) ranges over all possible.monotonic functions of a given variation in the interval a S; t S; b. The solution of extremal problems in such classes reduces to determination of functions a(t) correspond, ing to an extremum. As is well known, the idea of variations as applied to certain specific questions of this nature has already been encountered in the literature. In particu
1
-
0) and once with the average of a(t) and a(t 2 + 0), that is, with
the values
+
(1 - A) a. (t) ArJ. (t 1 - 0), A=const, 0<,,< 1; (1 - A) rJ.(t) ArJ.(t~ 0).
+
(3)
+
lar, this method was used to give a complete solution to the rotation problem for the class of functions w = f(z) =
z
+
C
2z 2
+ ... that are regular and starlike in
\z I < 1. This solution will be given now in a somewhat different treatment. In the present section, this idea of variations w ill be applied to the deriva tion of convenient variational formulas. We shall then use the variational 1) Goluzin [1952].
Here, we take a(a - 0) = a(a) and a(b + 0) = a(b). The varied functions al(t) and a 2 (t) thus obtained obviously remain nondecreasing in the interval a S; t S; b and they continue to satisfy condition (2). They can be represented in the form
a.k (t)=rJ. (t)
+ A~k (t),
k = 1, 2,
where (Jk(t) = 0, k = 1, 2, outside the interval t 1
(4) and
528
§9. A METHOD OF VARIATIONS
XI. SOME SUPPLEMENTARY INFORMATION
~1 (t)
~~ (t)
=
(1
(t 1 -
0) - a(t) =
= a (t~ + 0) -
529
Formula (8) constitutes the second variational formula in the class Eg •
-I (t) - (t J - 0) I, a (t) = I (t~ + 0) - a (t) I, (1
(1
We note that, if the function g(z, t) satisfies the condition g(z, a)
(1
=
g(z, b),
it can be extended as a function of t to the entire real line as a periodic function of t with period b - a. If we extend a(t) according to the formula
inside that interval. This can be written in the single formula ~k(t)=(-l)kl(1(t)-ckl, ck=const, k=l, 2.
o.(t+ b - a)=a(t)
(5)
The corresponding varied functions in the class Eg are b
f k (z) = ~ g(z, t) do.k (t) =f(z)
we can replace the interval of integration a ::; t ::; b of the integral (1) with any other interval of the same length without changing the value of the integral. In
h
this case, the variational formulas (7) and (8) are also meaningful on the entire
+ 'A ~ g(z, t) d~k (t), k = 1, 2.
a
+ 1,
t-axis except when 0
~
t 1 ::; b - a.
2°. VIe denote by S· the class of functions Integration by parts yields, in view of the fact that (3k(t 1)
= (3k(t 2) = 0,
w=f(z)=z+C~z~+... ,
12
fk (z)=f(z) - (- l)k'A ~ g,(z, t) Io.(t) - Ck I dt, k= 1, 2.
(6)
11
that are regular in the disk
Iz I < 1
+ 'A ~ gi (z,
=
O. We know that a necessary and sufficient
condition for the function (1) to belong to the class
t) Ia. (t) - CI dt,
(7)
11
be satisfied in
where A is an arbitrary number in the interval - 1 ::; A::; 1, wher.e c is a constant
S· is that the inequality
m(zl'j(z)(Z)) > 0
12
f* (z)=f(z)
and that map that disk univalently onto domains
that are starlike about the point w
Formula (6) can be written more simply in the form
(1)
Iz I < 1,
and this in turn is equivalent to the integral representa
tion
with respect to t and A (except that c depends on the sign of A), and where formula in the clas s Eg'
< t 2 ::; b,
j eli
zda(t)
(2)
->t
2) If t 1 and t 2 are two points of discontinuity of the function a(t) such that a ::; t 1
C eit+z
z/' (z) _ j(z) -
( .(z), like ( (z), is a function in the class Eg' Formula (7) is the first variational
in
then, for suffic ien tly small real A, the function
(1* (t)
where a(t) is a nondecr.easing function in the interval - 11:S t ::; 11
such that a(11) - a(- 11) = 1. By virtue of this last condition, formula (2) can be reduced to the form
= a. (t) + 'A~ (t),
where a(t) is equal to zero outside the interval t 1
Izi < 1,
and equal to 1 inside
zj' (z) j
that interval and is defined in a suitable manner at the endpoints t 1 and t 2' is a nondecreasing function in the interval corresponding function in the class
a::; t
(z) -
1
=
z do. (t).
If we then divide by z and integrate with respect to z from 0 to z, we obtain in
Izi < 1
b
f** (z) =f(z) +'A ~g(z, t) d~ (t),
a formula equivalent to (2): 1C
a
f
which yields
f**(z)=f(z)+'A(g(z, tJ)-g(z, t~»
2z
j eft
-"
::; b that satisfies condition (2). The
E? is determined form the formula
C
(8)
(z)
= ze-~!
log (1- e-itz) da (I)
(3)
530
§9. A METIIOD OF VARIATIONS
XI. SOME SUPPLEMENTARY INFORMATION
Here, the logarithm refers to the branch that vanishes at z Relying on formula (3) and
§ 1,
3°.
= Q.
we shall present two types of variation of the
f* (z)€ S *
Let us give some applications of the variational formulas of subsection
2° to the solution of extremal problems in the class S*. Theorem 1. For a given entire function (w) and a given point z in the disk
of the function f(z) E So. Specifically, we denote the integral in (3) by ¢(z) and we apply it to the variational formula (7) of subsection 1°. The function
Izi
< 1,
the maximum value of each of the functionalsl}
corresponding to the varied function ¢*(z) is defined by the formula
. f * (z) =
'P (z)
fS 2ie-it z + AJ
it
l-e
ze
I u (t)
z
- c I dt
t1
ffi (
IAI< 1,
,
(log!~»)), I (log!;Z») I
f(z)
f* (z) =f(z)
+ A ~1 f(z)
12~-~~~ .. I<X (t) -
c Idt
+ 0 (A
2).
(4)
Z
t2)
(l-efUz)S ,
where a is real. Here, we are excluding from consideration the case in which '(log (f(z)/z))
Here, O() denotes the order of smallness with respect to ,\,; 0 (,\,2) is an analytic function of z in Izi
(1)
in the class S· is attained only by a function of the form
which, for small ,\" yields tg
531
=
0 at the extremum. Therefore, in Izi
< 1,
we have the
inequalities
< 1.
Analogously, if we apply to ¢(z) the variational formula (8) of subsection
ffi
1°, we obtain, for small real'\', the varied function
f** (z)=f(z)
+ AI (z)[log (1 -
e- itg z) - log (1 - e- it 1Z)]
+ 0 (A2).
-
t1
~
27T. In formula
functionals (1) gives at the same time an extremum to the functional
ffi
Returning to formula (3), we note that, if a(t) is a step function with discon ~
t1
< t 2 < ... < t n < 27T
(4)
obvious. Also, the function f(z) that gives an extremum to the second of the
(5), these numbers must be points of discontinuity of the function a(t).
tinuities at t l' t 2 , ••• ,t n such that - 7T
(3)
Proof. The existenc e of extremal functions of these extremal problems is
accordance with the remark made at the end of subsection 1°, we can take for t 1 and t 2 in formula (4) any two real n umbers such that 0
~iU Z)2))
I
(5)
Formulas (4) and (5) yield variations of two types in the class So. In
(1) (log !;Z»)) ~ rn:x ffi (
and if '\'k is the
saltus in a(t) at tk, that is, if
(e iT
with suitable y and this functional fits the type of the first of the fUllctionals (1). Therefore, it will be suffiCient to prove the theorem for the first of the func
n
Ak=a.(tk+O)-<x(tk-O),
k=l, ... n, ~ Ak =l,
tianals (1), which we now denote by If. Let f(z) denote one of the extremal
(6)
k=1
functions. Let us calculate the value of If for the varied function (4) of subsection 2°.
then
Since
f(z)
z
n
(7)
n
(I - e-itkZ) SAk
k=l
This function maps the disk Izi
<1
(log! ~) ) t2
+ ACP' (log!;Z») ~
-I
~~t:e? I<X (t) -
c Idt
+ 0 (A
2),
t1
onto the entire z-planewith n radial cuts.
1) Here the logarithms refer to those branches that are regular in Iz 1 vanish at z = o.
<1
and that
532
§9. A METHOD OF VARIATIONS
XI. SOME SUPPLEMENTARY INFORMATION
533
This condition shows that the function
we have tj
II*=If+Am(~ <1>' (IOg!;z)h ie-;~:tz la(t)-cldt)+O(A~).
m(<1>' (IOg!~)) log (1 -
(5)
e-it z))
tl
bas equal values at t
The extremality of the function f(z) requires that the coefficient of A vanish,
= t 1 and
t
= t 2• Consequently, its derivative with respect
to t, which is equal to F(t), must vanish at some point t 3 in the interior of the
that is, that
~
~
.
~1 m(<1>' (log! ;))
I
ie-;_:rz) Ia(t) - cIdt =
I
O.
(5 )
interval (t 1, t 2 ). But then the equation will have three distinct solutions in eit, which is impossible. This contradiction shows that a(t) has only one point of discon tinuity in -
2° with n
0
f.
But in the case in which
=
c
=
But then, in accordance with formula (7) of subsection
1, it must be of the form (2). This completes the proof of the theorem.
m(a(~Z)f +b)andla(!~Z)t +b[
(6)
'
then !a(t) - c I must be equal to zero in that interval, that is, a(t)
< 17.
A are constant, we conclude that the maximum of the functionals
equation
m(<1>' (log! z(Z)) l-e-itz ie-it z ) =
t
By applying Theorem 1 to the case when
Condition (5 ') shows that, if the interval (t l' t 2) contains no roots of the
F (t) =
=
17 ::;
in the class S· is attained only by the function (2). Since
const.
Iv
0, as we shall assume in what follows,
equation (6) is equivalent to the equation
m(<1>' (Iog!~)) ie-Itz(l -
eltz)) = 0,
(7)
the sharp inequality
that is, to a quadratic equation in e • Consequently, it has no more than cwo
Iv !(Z) -11 ~ I zI
solutions in e it • This leads to the conclusion that the function a(t) is a step -17
< t < 17
and that the corresponding values of t must satisfy equation (6).
(which was proved by the author earlier by a different procedure) holds in the
Let us suppose that there are cwo such points of discon tinuity t 1 and t 2' Let us construct the varied function
f • .
Theorem 2. For a given entire function
Iz I < 1,
section 2°. For this function, we have
+ ~m (<1>' (lOg! ~)) [log (1 -
class S·
in accordance with formula (5) of sub
11.. =11
e- its z) -log (1 - e- it1 z)])
m(<1>' (lOg! ~)) [log (l -
e-
z) - log (1 - e-it 1 z)]) = O.
the maximum of either of the functionals
m(<1> (log!, (z))) and l
+ 0 (A~),
(8)
in the class S· is attained only by a function of the form
which, by virtue of the extremality of f(z), yields itj
I z I,
for the function (2), by taking a = 1, b = - 1, and A = - 1/2, we conclude that
it
function with either one or two points of discontinuity in the interval
11 =
!(z) -
f(z) (7 )
z (I -
eia z)),j
where Al and A2 are real nonnegative numbers such that Al + A2
f3
are real. The case when
=
(9)
(1- el~ z)),s ' =
2 and a and
0 is excluded from consideration.
~~~-~=-~""'~-~~~~~~--~~~=::~'~~~=='.~--~-'----'--""'::-'-
534
'==~=-~'-:~~_-~~---='~----=O'~"'-~::-:-."'--'=_"'-:=:-:~~===_~~-=---~_~~_="'-=~_''==-""~~==-"""==="-'=''-''="=-
§9. A METHOD OF VARIATIONS
XI. SOME SUPPLEMENTARY INFORMATION
Proof. Again, it will be suffic ient to con sider the problem of maximiz ing the
the number of roots of the equation F(t)
535
0 in that interval would exceed 4, which,
=
from what was said above, is impossible. On the other hand, if there are no more
functional
than two, then; by virtue of equation (7) of subsection 2°, the function fez) must
If
= ffi ( (log/' (z»).
be of the form (9). This completes the proof of the theorem. In the same way we can prove the more general theorem:
The existence of extremal functions is obvious. Let f(z) denote one of these. Let us calculate the value of If for the varied function (4) of subsection
2°. We have Ig
log/; (z) = log!, (z)
+ A ~ f' ~z) ~ C:-;/ ~ zf(z») I
(.t
c Idt+O
(t) -
Theorem 3. For a given entire func tion (w) , a real constant k, and a given
point z in the disk
Izi < 1,
the maximum of either of the functionals
(z»)) I ( zk f' (Z») I m( ( log zkf' f (z)k , . log f (z)k
(A~)
11
in the class S· is attained only by a functwn of the form (9). The case when
and consequently
ff. = ff
+ Affi (<1>' (log!, (z) ~'. f' :z) ~ ~e~ :~}z; I
' (log zkf' (z) ) (.t
(t) - c Idt)
~-
+ 0 (A ~).
As a special case of this with k
The extremality of the function f(z) leads to the condition 19
C ffi (
d ie-it zf (Z»)'I e it z
dz I -
(t) _ (.t
C
-.
(12)
l· of functions F(~) = ~ + ao + a/ ~ + ... th~t are regular in the and that map the domain 1~1 > 1 onto
domain I~I > 1 excluding the pole ~=
in the class
00
(
dz
domains with complement that is starlike about ~= 0 is attained only by a func
ie-it zf (z) ) 1-- e- tt z = O.
(10)
tion of the form
But equation (10) leads to a fourth-degree algebraic equation in e it • This means
F (~) = ~ ( 1 -
that a (t) must be a step function with no more than four points of discon tinuity ~t
I (log F' (~» 1
ffi ( (log F' (C»),
containing no roots of the equation
11
> 1,
the maximum of either of the functionals
I dt - 0
It follows from this condition that a (t) = const in an arbitrary interval (t l' t 2)
in the interval -
2, we obtain
=
Theorem 4. For a given entire function (w) and given ~ such that I~I
11
F (t) = ffi
0
at an extremum is excluded from consideration.
11
J
(11)
< 71, which must be roots of the equation
F(e) =
ei~)A1 (
T
ei~)AI
1- T
' 1. 10
A~ ~ 0, Al
+ Ai =
2.
(13)
o.
On the other hand, application of the variational formula (5) of subsection 2°
The case when
' (log F' (Q)
in the present case shows that the quantity
= 0
at an extremum is excluded from consideration.
ffi
(
dz
e- it z» J
We note that, for 0
(10 ')
< Al < 2,
the function (13) maps the domain I~I
the w-plane with two rectilinear cuts issuing from w as a function of t has different values at all points of discontinuity, so that its derivative with respect to t, which is equal to F(e), also vanishes inside the iritervals between any two adjacent points of discontinuity. H there were more than two points of discontinuity of the function aCt) in the interval -
11
~t
< 11,
11.
=
>1
onto
0 at an angle other than
In a number of special cases of the type that we are considering, the extremal
functions (9) and (13) degenerate into functions with A2 is the case in questions of maximizing or minimizing
=
O. For example, this
!['(z)1
in the class S· or
""---,'~'~-------.-'-'''''''''''=="'
536
-
--m.,,;iii'iI.",:~
-...,
~_~_
_
...
~?7
~.. ",~,_,.;~J'!l'}t'L~~~~
Xl. SOME SUPPLEMENTARY INFORMATION
of minimizing IF'(~)1 in the class
I',
However, this degeneration does not
always occur, For exampl e, in the question of maximizing I' for the function F(~) = ~ + eictC1 + const, we have
IF' (t) 1= 1 whereas, for the functioo (9) with 0 as ~ approaches any point on w=o.
'(I =
§9, A METHOD OF VARIATIONS
IF '(~)I in the class
1
we see that obviously F '«()
subclass I; of odd functions F 2«() belongirig to the class have the representation
00
I P (t)
I',
IF '«()1 in the These functions
(I
1
t [ ,,;;;;; -,t-I
(I7)
I:
1('
> I, Let us set fez) = F(llz) liz = =lanzn, Then, , in accordance with (17), we have!f(z)\ < I in the disk Izi < I, Furthermore, holds in the domain
f(o)
= o.
But then, as we know, I)
IF(z)!,,;;;;;1
in
!zl
(18)
equality holding only in the case of a function of the form f(z)
where F(~) E I'. It follows from Theorem 3 for k function here isa function of the form
=
e;:
3/2 that the only extremal
If 1= 1.
=
(z, where
Inequality (18) yields
Iz~- F' ( ~ )
t
where Al and A2 are nonnegative constants such that Al + A2
z~ I~ 1 in
1Z 1
<112
1
or
1F' (t) =
and (3 are constants, Here we conclude just as above that, for we have 0
-->
y"p (t'A),
e;: t
(16)
,
author [1938] that the inequality
An analogous situation occurs in the problem of maximizing
Pi (t) = t (1 -
T
Proof. For the functions F«() mentioned in the theorem, it was shown by the
1 that corresponds to boundary points on
Pi (t) =
e i '"
where a is a real number.
+rtF'
< Al < 2,
P(t)=t+
537
I and where a
I(I
close to I,
< Al < I in the case of this extremal function. However, for
I(,
suf
ficiently large, one can show by a roundabout procedure that the function F«() = ( + e i a(-1 is an extremal function. This fact is included in the following distor tion theorem. Theorem 5. For a function F«()=(+atl~+a/(2+ ... EI' the sharp inequalities
1 I~
W
in
It I> 112 + 1.
This inequality leads to inequalities (14) and (IS). This completes the proof of the theorem. It is still an open question whether inequality (IS) is valid in the entire
I(I
> I under the conditions of Theorem 5. The corresponding question with regard to a bound for [arg f'(z) I in the class S· has an affirmative answer, domain
~" Th is wi 11 be shown below,
4°. Let us establish a theorem for the inverse of a starlike function. Although, as we have mentioned, the application of the variational method set forth here
hold for form
1(' >.J2 +
to the particular question of solving this problem is not new, we shall nonethe
1 + W'
(14)
• 1arg F' (t) I ~ arcsin W1
(IS)
IF' (t) I,,;;;;; 1
less give its solution, treating it as an application of the general Theorem 2 of subsection 3°. Theorem. For given z in the disk
Izi < 1,
the function arg f'(z) in the class
S· attains its maximum and minimum only in the case of functions of the f(lrm 1 with equality holding only in the case of a function of the
---l)~e
the author's article [1945] and also Theorem 5 of
§a,
Chapter Xl,
XI. SOME SUPPLEMENTARY INFORMATION
538
§9. A METHOD OF VARIATIONS
z f(z)=-,-, -_10_'. ,
corresponding to the numbers (1)
4
> O.
The problem
ffi
that we are considering is a special case of the problem of Theorem 2 of sub
t
(! Ck ) = !
COS
< 11 no more than two points of discontinuity, which must be roots of the
ffi (C) = ffi
equation e-i/ r
F (t)= ffi ( 1 ~ e
If
r
+C
e- it r (I _
e
it
)_
r)2
-
(this is equation (10) of subsec tion 3° with IlJ (w)
=
J..!!L
_
0, C-
± iw).
rj'(r)
(2)
rn (llog (1 --e- II. r) -
of the intervals (t I , t 2 ) and (t 2, we reduce it to the form
H there are two poin ts
ire-it
C 1- e it r
)
t
,=
e
,
~ ffi (i log (1
at
2r a)
rC
-
at (llog (1
+ Ce-
It
(1 - eit r)i) = 0
+ 2r + r a+ 2Cr - elf (r' + Cri» = O.
III
= O.
This equation must be satisfied by the numbers 'k = e it k , for k = 1, 2, 3, 4,
ffi (
e- it r
1 - e If r
)
=
r-cost
11 - e it r
-
e-itl r))
> at (llog (1 -
e- ita
12 '
r».
"
z
O~A~I,
we obtain
---l..!:Il.- =
C- r/'(r)
(4)
1~
Itj 12 1 "' r A l+e "' r +(l-A)i_e- r it2 l-e I l r
Therefore, by setting
we write this equation in the form of a fourth-degree
(1 + 3r' + C+ Crg) c + agC' + llaC +
e- it r»= _
< t < t 2• But then, since
(l-e "lz)2A(1_e-it2z)i(1-A) ,
Ck -
for t l
Furthermore, remembering that the extremal function r(z) is of the form
f(z) =
9
>r
we conclude that the first term in (3) decreases in the interval (tl' t 2 ), so that
algebraic equation I
r.
> r, k :-1, 2, 3, 4.
It follows, in particular, that cos t
=
ffi (e iU r - elf (1 + 3r 9 + C+ Cr'1) + 2r + r a+ 2Cr) = O. Then, by setting
+
COS k
H we now replace the first two terms with their 'complex conjugates, which does not change the real part, we obtain the equation
it
(JJ!L)::>: 1- r rj'(r) :O----1+r"
k=l
and we can then reduce this to the form
r - e- (1 + C+
we have
2 t ::>:~+3 r +1+r -3+ r+, (l-r)2(2+r»3+ ..:;.. cos k:o---- r r 11-r +r r (l r)
(3)
l/J'(t) has one zero inside each t l + 211). By ridding equation (2) of denominators,
ffi (e- lf (1 - e- it r) (1 - elt r)'
ffi (e-
Izi < 1,
This shows that
< t 2• Then, the function F(t)
it
+ 1 ~r2 ffi (C).
4
Let us suppose that there are two such points of discontinuity t I and t 2'
iit
3r
"'\"1
(d. (10') of subsection 3° wi th llJ(w) = iw) has the same value at the two points.
indexed so that t l
=++
Consequently,
of discontinuity, the function
~ (t) =
tk
k=1
On the other hand, since ~(f(z)jzf'(z) > 0 in
function a(t), correspo"nding to an extremal function r(z), has in the interval 11 .:::;
4
k=l
section 3° when IlJ (w) = ± iw. It follows from the proof of that theorem that the
-
mentioned above.
tk
We now have from (5)
where a is a real number.
Proof. Without loss of generality, we may assume that z = r
539
(5)
we have
1 +e-itkr l-e -itk r '
k= 1,2,
(6)
540
§9. A METHOD OF VARIATIONS
XI. SOME SUPPLEMENTARY INFORMATION
(C 1 ie~:ll~l r)- m(C_l_ie_~_it"II~:-1r-) -rn(tc( e- r _ e-1t'llr ))-l-rn(t l-e-1t1r l-e-It'llr - 2 -
where a«() is a real nondecreasing function in the interval 0 S () S
ffi
a(17) - a(o)
I C IS - = -23(AC.C~ _I C 1'11 3 (1 +e+lt!r
-
2
1
2
S
(z, t) da. (t),
(3)
where
_I C I'l 2r (l-r)(sint1 -sin t a)
l-e+it!r 'l-e-itSr -
~ -1
(1 - A) CiCg) =-2- 3 (C.C~)
its 1+e- r)
Il-i/1rll'll-e It'llrll
s (z, t) =
<°
1_
z 2tz
+
Z2
,
(4)
=
1. For arbitrary fixed t in the interval - 1
s(z, t) itself is typically real; furthermore, it is univalent
rn
ie- it1 r ) (C l-e It lr
< rn
(
ie-its r
)
C l-e-'sr ",.
we denote by Tr the class of all functions (1) that are regular and typically real
(7)
in
Izi < 1,
formula (3) is a parametric representation of that class. By relying on
formula (3), the author solved [1950a] a number of extremal problems for the class
> l/I(t 2 ), which con tradict s what was said above, namely, that the function 1/1 (t) has equal values By subtracting (7) from (6), we obrain the inequality 1/1 (t 1 )
Tr • In all those problems, the functions (4) proved to be the only extremal func tions. We now apply the idea of variation to the solution of more general problems.
at the two points of discontinuity t 1 and t 2• This contradiction shows that a(t) cannot have two points of discontinuity in the interval - 17 S t < 17. Consequently,
To do this, we write the variational formulas corresponding to formulas (7) and (8) of subsection 1° as applied to the integral (3). Specifically, we obtain two
the function f(z) must be of the form (1). This completes the proof of the theorem.
variations of the function (3) in the class Tr :
S°. The same method of variations can be applied to tht. solution of extremal
II
problems in the class of typically real functions. A function
!(z)=z+ C2Zg+ ...
f*(z)=I(z)+A~ s/(z, t)Ia.(t)-cldt,
Iz I < 1
is real on the diameter - 1
< z < 1 and if
is said to be typically real in that disk if it ~(f(z)) and ~(z) are always of the
same sign, that is, if for 3(z»0 <0 for ~(z)
any such function can be
f(z) =
"
~ S 1o
z 2 z cos 6
+
ZS
da. (fJ),
t1) -
S
(z, t~).
(6)
In formula (5), t 1 and t 2 are arbitrary numbers such that -1 S t 1 < t 2 S 1 and c
!\
is a constant in the interval t 1
< t < t 2• In formula (6), t 1 and t 2 are points of
discontinuity of the function a(t) in the same interval. In both formulas, A is an arbitrary sufficiently small real number.
3(/(z»>0
Izi < 1. The author showed [19S0a] represented in Iz I < 1 according to the formula
+ A(s (z,
Wi 'h
at all the remaining points of the disk
(5)
II
(1)
,
1** (z) =I(z)
that is regular in the disk
S t S 1 such that S t S 1, the function in the disk Iz I < 1. If
and a(t) is a real nondecreasing function in the interval - 1
'
a(l) - a(-I)
so that
such that
Formula (2) can be replaced by the equivalent formula
f (z) =
A) C~»
I C /'11
-
17
Conversely, every function f(z) defined by this formula is typi
Izi < 1.
C1-C'II ) AC1 +(1-A)C 1
+ (1 -
= 17.
cally real in
1t1
ICII =.-2-:.1 «C~ - C1) (AC.
541
Theorem. 1. For a given function
Izi < 1,
Iwl <
!Xl
and a
the maximum of each of the functionals
rn (tIl (I (z»)
that
<
or
ItIl (j (z» I
(7)
in the class T r is attained only by suitable functions of the form (2 )
1 (z) =
A.S (z, t.)
+ AgS (z,
t,),
Ai. Ag~ 0,
Ai
+ Ag=
1,
where t 1 and t 2 are numbers in the interval - 1 S t S 1. lI"e exclude from
(8)
"'-
-_.
-~
--
"- - ~ _.
-_.
""-
_::.
- ._-;;-~~ --
--':-~-;:'~~,~ --.----:-;;::-:--.~~
- - -:;;;: __,_____
•.• __ "__.__ •
..::Sf;',
__ .,:;;;:,:.F~~: ~ _.~=_=, ._~ ~~:~~;
~,..,~:;;~
542
XI. SOME SUPPLEMENTARY INFORMATION
§9. A METHOD OF VARIATIONS
consideration the case in which '(f(z)) = 0 at an extremum.
which is of the form (8) but with '\'1 and '\'2 positive. Then, the problem of
Proof. It will be sufficient to consider the problem of the maximum of the
minimizing
first of the functionals (7). The existence of extremal functions is obvious. By
(w)
applying formula (5), we can show, just as before, that, if f(z) is an extremal
Izi < 1,
function and (3) is its representation in
=
=
II w.
disk
Izi < 1,
±1
z~)
(1 - 2tz
the maximum of either of the functionals
ffi (If> (f' (z»)
and
0, which is a 2nd-degree algebraic equation:
ffi ( '(f(z)) (1 -
2. For a given entire function ¢ (w) and a given point z in the
O. Consequently, the only
possible points of discontinuity of the function a(t) are the endpoints the roots of the equation F(t)
=
If(z) I is equivalent to the problem of maximizing I (f(z))\ with
Theorem
then a(t) = const between any
two roots of the equation F(t) = ~('(f(z))«z, t))
543
+ Z2)~) =
in the class
0,
Tr
IIf> (/' (z» I
is attained only by functions of the form
(9)
so that the number of these discontinuities does not exceed 4. H there are more
and
3
f(z) =
~ AnS(Z, tn)' n=1
3
An~O,
~ An = 1, n=1
than two points of discontinuity, denote three of them by t 1, tz, and t 3 , indexed in such a way that - 1 $ t 1 < t 2 < t 3 $ 1 and F(t 2) = O. By applying formula (6),
where t n belongs to the interval - 1 $ t $ 1. If'e exclude from consideration the
let us show that R('({(Z))S(z, t)) has, as a function of t, equal values at the
case in which '{f'(z)) = O.
points t l' t 2> and t 3' But then its derivative has a root inside each of the inter vals (t1' t 2) and (t 2, t 3 ); that is, equation (9) has three distinct roots. This con tradiction shows that a(t) does not have more than two points of discontinuity in
f (z)
the interval - 1 $ t $ 1, and this means that the function
is of the form (8).
This completes the proof of the theorem. We shall now give an example in which the extremal function (8) does not degenerate into a function of the form s(z,
Izi < 1,
the maximum value of
If(z)1
d.
We know 1) that, for fixed z in
in the class
Tr is attained only by a fu"nc
tion of the form f(z) = s(z, t), where, for certain z, we have - 1 < t < 1. The function z 2/(1- z2)2f(z) belongs to T r whenever f(z) does. (This follows from the necessary and suHic ient condition for
f (z)
to belong to the class
T"
namely, that the inequality
ffi ( I /2 holds for class
1z
\
< 1.) Therefore, for
1z I
!(Z») >0
< 1, the minimum value of
If (z ) I in the
Tr is attained only by a function
f(z)
~
= (l-z2)2 S (z, t)
z (I +z)!
+ 2 (1 I-t
=-2- s (z, 1) Goluzin [195oa].
~
t) (l_z9)2 I
+t
1)+2- s (z, -1) E T"
The proof is analogous to that of Theorem 1.
THE SCIENTIFIC WORKS OF GENNADil MIHAILOVIC GOLUZIN (All in Russian) 1929
On certain inequalities dealing with functions executing a univalent con formal transformation of a disk, Mat. Sb. 36, 152-172.
1933
(with V. I. Krylov), A generalization of Carleman' s formula and its applica tion to analytic continuation of functions, Mat. Sb. 40, 144-149.
1934
The solution of the fundamental plane problems of mathematical physics for the case of Laplace's equation in multiply connected domains bounded by circles (the method of functional equations), Mat. Sb. 41, 246-276.
1954a Solution of the three-dimensional Dirichlet problem for Laplace's equation and for domains bounded by a finite number of spheres, Mit. Sb. 41, 277-283. 1935
On the theory of univalent conformal transformations, Mat. Sb. 42, 169-190.
1935a Solution of the plane problem of heat flow for multiply connected domains bounded by circles in the case of an isolating layer, Mat. Sb. 42, 191-198. 1935b On the majorization principle in the theory of functions, Mat. Sb. 42, 647-650. 1936
On distortion theorems in the theory of conformal mappings, Mat. Sb. 1 (43), 127-135.
1936a On the theory of conformal mappings, Mat. Sb. 1 (43), 273-282. 1936b On theorems of inverses in the theory of univalent functions, Mat. Sb. 1 (43), 293-296. 1937
On distortion theorems for conformal mapping of multiply connected do mains, Mat. Sb. 2 (44), 37-64.
1937a Some covering theorems for functions that are regular in a disk, Mat. Sb. 2 (44),617-619. 1937b Supplementary remarks to the article "On distortion theorems in the theory of conformal mappings", Mat. Sb. 2 (44), 685-688. 1937c On conformal mapping of doubly connected domains bounded by rectilinear and circular polygons in Conformal mapping of simply connected and mul tiply connected domains, ONTI, Moscow, pp. 90-97.
545
546
SCIENTIFIC WORKS OF GENNADII MIHAILOVIC GOLUZIN
1937d Conformal mapping of multiply connected domains onto a plane with cuts by the method of functional equations in Conformal mapping of simply connected and multiply connected domains, ONTI, Moscow, pp. 98-110. 1938
Some bounds on the coefficients of univalent functions, Mat. Sb. 3 (45), 321-330.
1938a On the continuitY.. method in the theory of conformal mappings of multiply connected domains, Mat. Sb. 4 (46), 3-8. 1939
Iterational processes for conformal mapping of multiply connected domains, Mat. Sb. 6 (48), 377-382.
1939a On limiting values of Cauchy integral, Leningrad State Univ. Ann. Math. Ser. 6, 43-47.
MR 2, 181.
1939b On complete systems of functions in the complex domain, Leningrad State Univ. Ann. Math. Ser. 6, 48-51. MR 2, 188. 1939c Zur Theorie der schlichten Funktionen, Mat. Sh. 6 (48), 383-388. MR 1, 308. 1939d Interior problems of the theory of univalent functions, Uspehi Mat. Nauk 6, 26-89.
MR 1, 49.
1940
Uber p-valente Funktionen, Mat. Sh. 8 (50), 277-284.
MR 2, 185.
1943
Uber Koeffizienten der schlichten Funktionen, Mat. Sb. 12 (54), 40-47.
v
v
19 (61), 183-202.
MR 8, 325.
Method of variations in the theory of conform representation. I, Mat. Sb.
19 (60, 203-236.
MR 8, 325.
Estimates for analytic functions with bounded mean of the modulus, Trudy
Mat. Inst. Stek1ov., 18, 1-87.
MR 8, 573.
The method of variations in conformal mapping, Leningrad. Nauen. Bju1!.
Univ. 9, 3-5.
Method of variations in the theory of conform representation. II, Mat. Sb.
21 (63), 83-117.
MR 9, 421.
Method of variations in the theory of conform representation. III, Mat. Sb.
21 (63), 119-132.
MR 9, 421.
Some covering theorems in the theory of analytic functions, Mat. Sb. 22
(64), 353-372.
MR 10, 241.
On the coefficients of univalent functions, Mat. Sb. 22 (64), 373-380.
MR 10, 186.
On distortion theorems and the coefficients of univalent functions, Mat. Sb.
23 (65) 353-360.
MR 10, 602.
On analytic functions with bounded mean modulus, Leningrad. Gos. Univ. Ucen. Zap. 111, 120-125.
1943b On the theory of the airfoil in the two-dimensional flow, Mat. Sh. 12 (54), 146-151. MR 5, 21.
Leningrad. Gos. Univ. Ueen. Zap. Ill, 126-134.
Some estimations of derivatives of bounded functions, Mat. Sb. 16 (58), 295-306.
1946
MR 7, 202.
On the theory of univalent functions, Mat. Sb. 18 (60), 167-179. MR 7,515.
1946a On the problem of Caratheodory-Fejer and similar problems, Mat. Sb. 18 (60), 213-226.
MR 8, 22.
1946b On some properties of polynomials, Mat. Sb. 18 (60), 227-236. MR 8, 22. 1946c On the distortion theorems for "schlicht" conform representation, Mat. Sb. 18 (60), 379-390. MR 8, 574. 1946d On the number of finite asymptotic values of integral functions of finite order, Mat. Sh. 18 (60), 391-396. MR 8, 23.
547
On distortion theorems and coefficients of univalent functions, Mat. Sb.
MR 5, 93. 1943a Zur Theorie der schlichten Funktionen, Mat. Sb. 12 (54), 48-55. MR 5, 93.
1945
"
SCIENTIFIC WORKS OF GENNADII MIHAILOVIC GOLUZIN
Variational derivation of a distortion theorem for univalent functions, Some inequalities for analytic functions, Izv. Akad. Nauk Kazah. SSR 60
Ser. Mat. Meh. 3, 101-105.
MR 13, 639.
Some questions of the theory of univalent functions, Trudy Mat. Inst. Stek
lov. 27.
MR 13, 123.
On mean values, Mat. Sb. 25 (67), 307-314; English transI., Amer. Math.
Soc. Transi. (1) 2 (1962), 275-283.
MR 11, 339.
Some estimates for bounded functions, Mat. Sb. 26 (68), 7-18. MR 11, 426.
On typically real functions, Mat. Sb. 27 (69), 201-218.
MR 12, 490.
On the theory of univalent functions, Mat. Sb. 28 (70), 351-358.
MR 13, 639. 1951a On the theory of univalent functions, Mat. Sb. 29 (71), 197-208. MR 13, 223.
548
v " SCIENTIFIC WORKS OF GENNADIIu MIHAILOVIC GOLUZIN
1951b On the majorants of subordinate analytic functions. I, Mat. Sb. 29 (71), 209-224. MR 13, 223. 1951c On subordinate univalent functions, Trudy Mat. Inst. Steklov. 38, 68-71. MR 13, 733. 1951d Variational method in conformal mapping. IV, Mat. Sb. 29 (71), 455-468. MR 13, 454.
BIBLIOGRAPHY L. AhHors
1930
1951e On the parametric representation of functions univalent in a ring, Mat. Sb. 29 (71), 469-476.
MR 13, 930.
1951£ On majorization of subordinate analytic functions. II, Mat. Sb. 29 (71), 593-602. MR 13, 454. 1951g On the problem of the coefficients of univalent functions, Dokl. Akad. Nauk SSSR 81, 721-723. 1952
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1952a Geometric theory of functions of a complex variable, GITTL, Moscow. MR 15, 712.
UntersuchuT'gen zur Theorie der konformen Abbildungen und ganzen Funktionen, Acta Soc. Sci. Fenn. 1, no. 9, 1-40.
1938
An extension of Schwarz' lemma, Trans. Amer. Math. Soc. 43,359-364.
1947
Bounded analytic functions, Duke Math. J. 14, 1-11.
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L. Ahlfors and H. Grunsky,
1937
Uber die Blochsche Konstante, Math. Z. 42, 671-673.
Ju. E. Alenicyn 1947
On mean p-valent functions, Mat. Sb. 20 (62), 113-124. (Russian) MR 9,23.
1950
On functions p-valent in the mean, Mat. Sb. 27 (69), 285-296. (Russian)
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1950a On bounded functions in multiply connected regions, Dokl. Akad. Nauk SSSR 73, 245-248. (Russian) 1951
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An extension of the principle of subordination to multiply connected regions, Trudy Mat. Inst. Steklov. 60, 5-21; English transl., Amer. Math. Soc. Transl. (2) 43 (1964),281-297.
1964
MR 25 #1281.
Conformal mappings of a multiply connected domain onto many-sheeted canonical surfaces, Izv. Akad. Nauk SSSR Ser. Mat. 28, 607-644. (Russian)
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1936
Zum Koeffizientenproblem der schlichten Funktionen, Mat. Sb. 1 (43), 221-228.
1936a Sur les theoremes de Koebe-Bieberbach, 1951
Mat. Sb. 1 (43), 283-292.
On distortion theorems and coefficients in univalent functions, Mat. Sb. 28 (70), 147-164. (Russian)
MR 12, 600.
549
~-
-....:...
--0..s;--
_---
"---"'"
550
._,;;:-"------ _ _- "
_
"'-~-,---~
~~....-~
.......' ~ - =
~~~",-~~~-"~;""..,,-~*~~~~~BS?~;;/;;~~~~~~~;,;;,.-,,;£, -;:;
BIBLIOGRAPHY
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Series trigonometriques et series de Taylor, Acta Math. 30, 335-400.
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Uber den Zusammenhang der Extremen von harmonischen Funktionen mit ihren Koeffizienten und iiber den Picard-Landau'schen Satz, Rend.
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Uber die gegenseitige Beziehung der Riinder bei der konformen Abbil
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dung des Innern einer lordanschen Kurve auf einen Kreis, Math. Ann.
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.'~>i.~,,;&:;[;'::Z~,k.---';';~~~~~~
BIBLIOGRAPHY
A. F. Bermant 1938
'ii__
--''''--"
Untersuchungen tiber die konforme Abbildung von festen und veriinderlichen Gebieten, Math. Ann. 72, 107-144.
Uber den transfiniten Durchmesser ebener Punktmengen. II, Math. Z.
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..
,~~"%-)~1
552
- _ . -
,"""''"~'''''?I''''w ~o:r"~,o)'''''''·,;;,I;<=t>,.~'''''*:!;(i!=,·,.,,~'14,}·(.);~w.~I.'\:>!:\1~_.:r,+:J,,1$~,,,._';O''''''.f.o\i>t!W:l'~·~''''1rJ<~
BIBLIOGRAPHY
BIBLIOGRAPHY
A. F. Filippov (39), 173-176. (Russian)
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MR 12, 628.
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H. Grunsky 1932
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Zwei Bemerkungen zur konformen Abbildung, Jber. Deutsch. Math. Verein. 43, 140-142.
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Uber die konforme Abbildung mehrfach zusammenhiingender Bereiche
On existence theorems of potential theory and conformal mapping,
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A proof of the Bieberbach conjecture for the fourth coefficient,
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Ber. Verh. Sachs. Akad. Wiss. Leipzig 81, 51-86. 1930 1931
Das Kreisschlitztheorem der konformen Abbildung schlichter Bereiche,
On sequences of analytic functions, Mat. Sb. 31, 147-151. (Russian)
G. Julia 1924 Lecons sur les fonctions uniformes , Gauthier-Villars, Paris.
apoint singulier essentiel isole,
M. V. Keldys 1939
Conformal mappings of multiply connected domains on canonical do mains, Vspehi Mat. Nauk 6, 90-119. (Russian)
Uber ein Variationsproblem der konformen Abbildung, Ber. Verh.
Sachs. Akad. Wiss. Leipzig 82, 251-263.
MR 26 #6382.
A. Ja. Hincin 1924
H. Grotzsch
On successive coefficients of univalent functions, J. London Math.
Soc. 38, 228-243.
1914-1915 Some remarks on conformal representation, Ann. of Math. (2) 16,
1928
Neue Abschatzungen zur konformen Abbildung ein- und mehrfach zusammenhiingender Bereiche, Schr. Math. Sem. und Inst. fiir Angew.
P. R. Garabedian and M. Schiffer 1949
553
1932a Uber die Verschiebung bei schlichter konformer Abbildung schlichter
An elementary proof of Jordan's theorem, Vspehi Mat. Nauk 5, no. 5
1950
_._-----
~
....~... ~:.!~;I'i\~ ..,'c~_.;~~llt~~,,~.,_='"'''''!I;;'"".&fl;'i.,~~
MR 1, 48.
M. V. Keldys and M. A. Lavrent' ev 1937
Ber. Verh. Sachs. Akad. Wiss. Leipzig 83, 238-253.
Sur la representation conforme des domaines limites par des courbes rectifiables, Ann. Ecole Norm. Sup. (3) 54, 1-38.
1931a Uber die Verzerrung bei schlichter konformer Abbildung mehrfach
P. Koebe
zusammenhiingender Bereiche. 111, Ber. Verh. Sachs. Akad. Wiss.
1907
Uber die Uniformisierung beliebiger analytischen Kurven, Nachr. Ges. Wiss. Gottingen 2, 191-210.
1910
Uber die Uniformisierung der algebraischen Kurven. 11, Math. Ann. 69, 1-81.
1912
Uber eine neue Methode der konformen Abbildung und Uniformisierung
Leipzig 83, 283-297. 1932
Uber das Parallelschlitztheorem der konformen Abbildung schlichter Bereiche, Ber. Verh. Sachs. Akad. Wiss. Leipzig 84, 15-36.
(Voranzeige), Nachr. Ges. Wiss. Gottingen, 879-886.
1915
Abhandlungen zur Theorie der konformen Abbildung. I,
J.
Reine Angew.
1929
Abhandlungen zur Theorie der konformen Abbildung. IV, Acta Math.41,
1929a Darstellung und Begriindung einiger neuerer Ergebnisse der Funk
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M. A. Lavrent'ev
1934
198-236.
Application of integral equations to the proof of certain theorems on conformal mapping, Mat. Sb. 4 (46), 9-30. (Russian)
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1943
On one-parameter families of analytic functions, Mat. Sb. 13 (55), 87-118. (Russian)
1946
1935
1930
Sur la representation conforme, C. R. Acad. Sci. Paris 191, 1426-1427.
1937
On certain properties of univalent functions, Mat. Sb. 2 (44), 319-326. (Russian)
MR 7, 201.
On integrals of a very simple differential equation with movable polar
N. A. Lebedev
singularity in the right-hand member, Tomsk. Gos. Univ. Ueen. Zap.
1951
N. A. Lebedev and I. M. Milin
1949
1947a On a special family ofone-sheeted domains, Tomsk. Gos. Univ. Ueen. Zap. no. 5, 22-36. (Russian)
Schwarz-Christoffel integral, DokI. Akad. Nauk SSSR 57,535-537. MR 9, 277.
1951
1947d On the theory
0
f univalent functions, DokI. Akad. Nauk SSSR 57,
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E. Landau
1904
1908
Memoire sur certaines inegalites dans la theorie des fonctions mono genes et sur quelques proprietes nouvelles de ces fonctions dans Ie voisinage d'un point singulier essentiel, Acta Soc. Sci. Fenn. 35, no. 7.
1915
Sur un principe general d'analyse et ses applications ala theorie de la representation conforme, Acta Soc. Sci. Fenn. 46, no. 4.
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1922
E. Lindelof
On a system of differential equations, Tomsk. Gos. Univ. Ueen. Zap. MR 11, 21.
Some remarks on the coefficients of schlicht functions, Proc. London Math. Soc. (2) 39, 467-480.
MR 9, 507.
no. 8,61-72. (Russian)
MR 13, 640.
V. I. Levin
1935 MR 9, 421.
MR 11, 339.
On the coefficients of certain classes of analytic functions, Mat. Sb. 28 (70), 359-400. (Russian)
1947c A remark on integrals of Lowner's equation, DokI. Akad. Nauk SSSR 57, 655-656. (Russian)
On the coefficients of certain classes of analytic functions, DokI. Akad. Nauk SSSR 67, 221-223. (Russian)
1947b On a method of numerical determination of the parameters in the (Russian)
Some estimates and problems on extrema in the theory of conformal mapping, Dissertation, Leningrad State Univ. (Russian)
A theorem on the solutions of a certain differential equation, Tomsk. Gos. Univ. Ueen. Zap. no. 5, 20-21. (Russian)
1948
Sur une classe de representations continues, Mat. Sb. 42, 407-424.
" M. A. Lavrent I ev and V. M. Sepelev
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1947
On the theory of conformal mappings, Trudy Mat. lost. Steklov. 5, 159-246. (Russian)
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1938
.. , Uber die Blochsche Konstante und zwei verwandte lr'eltkonstanten,
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1918
555
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=.= ...._'""~~~ _..... ~~=~. __-:~ ..
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1925 1944
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Lefons sur les fonctions univalentes ou multivalentes, Gauthier Villars, Paris.
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.0
1936
557
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1949
__
-'-'C"""'2~-=-~-=--=-"'=" "~~~,.==_,,_==
Lectures on the theory of functions, Oxford Univ. Press, London.
C. Loewner
1923
-=.- ---=-- -
(2) 23, 481-519.
don Math. Soc. 7, 167-169.
1919
_~=~_--~=
P. A. Montel
E. Littlewood and R. E. A. C. Paley
1917
~~~-
BIBLIOGRAPHY
MR 6, 26l.
1932
'-- -=__
_~._
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=-_-.::=-":"": .__
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1916
R. de Possel
R. M. Robinson
Zum Parallelschlitztheorem unendlich vielfach zusammenhiingender
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Cauchy's integral, Saratov. (Russian)
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On functions that provide a univalent conformal mapping, Mat. Sb. 31,
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On the coefficients of subordinate functions, Proc. London Math. Soc. (2) 48, 48-82. MR 5, 36.
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Variation of the Green function and theory of the p-valued functions,
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1948
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~~:,"""" ...::::-i§:;J"
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561
MR 21 #6498;
_.-
-=
!.
.-
-
SUPPLEMENT
METHODS AND RESULTS
OF THE GEOMETRIC THEORY OF FUNCTIONS
Since the publication of the first edition of Goluzin's monograph, there have been a large number of results in the geometric theory of functions by both Russian and foreign authors. Many of these results are directly related to the content of the present book and many of them have been obtained by applying Goluzin's ideas. Because inclusion of these results in the main text of the book would re quire considerable change in the author's text, it seemed expedient instead to supplement the present book with a separate survey of these results. This survey is by no means exhaustive. Some of the methods that have been developed at the present time and the results obtained by these methods could not very well be included with sufficient completeness within the framework of a sup plement to the book. Furthermore, the direction of the scientific interests of the authors is unavoidably reflected in the choice of material included in the supple ment.
§ 1 of and
this supplement was written by N. A. Lebedev,
§ 2 by
G. V. Kuz'mina,
§3 by Ju. E. Alenicyn.
In troduction As a preliminary, we Ust certain classes of functions that will be frequently encountered in what follows.
S
the class of functions f(z) valent in the disk
~
=
z + c 2z2 + •.. that are regular and uni
Izi < 1;
the class of functions F «()
= (
+ a 0 + a / ( + . .• that are meromorphic
and univalent in the domain
1(1
> 1;
S
= 1, 2,"') the class of functions f(z) € S that have an expansion of the form f(z) = z + c k + zk+l + c +! z2k+l + .•. ; 2k 1
563
-"'i.u
564
SUPPLEMENT
IO
the subclass of functions F((} in I
I(k) (k =
'+
the class of fuactions w = fez) € 5 that map the disk \zl =
<1
0;
Izl < 1
the class of functions w
=
f(')
€
I
that map the domain
that map the domain
> 0) the class of functions F((} € I such
1(1
1(1 =
If(z)1 < 1 for
> 1 onto a domain
If(z)1 < M for Izi < 1; that IF(')I > m for '(I> 1;
Izi < 1
and satisfy the inequality
Izi < 1.
the class of functions fez) that are regular in the disk
C
the class of functions fez) that are regular in the disk satisfy the inequality lR(f(z»
> 0 for
lzl < 1; Izi < 1 and
Izi < 1;
the class of functions fez) = z + c 2 z2 + ••• that are regular in the disk
Izi < 1; disk Izi < 1
and satisfy the inequality ~(f(z»~(z) > 0 for
the class of functions fez) that are regular in the
and that
satisfy the inequality fez I) fez 2) ,j 1 for arbitrary z I and z 2 in the disk
L
SO)
nSR.
Basic methods of the geometric theory of functions
,of the principle of areas leads to the proof of a "generalized theorem on areas". furthermore, such a theorem is the starting part for obtaining inequalities of :'various kinds. One of the first theorems of this nature was obtained by Lebedev i(and Milin [1951]. Lebedev [1961] gave a generalization and strengthening of this
~:result, which we shall now briefly expound. ':;" ~' Let ~ (00, al ,·· " a) denote the class of all systems lfk(z)l~ of functions ,~~ w = fk(z), for k = 0, 1,'" , n, that map the disk Izi < 1 conformally and univa
= 00 and fk(O) = a , for k Itk = 1, 2,'" , n, where aI' a ,··· ,a are fixed points. We denote by Bk(r), for n n 2 ~~k = 0, 1,"" n, the image of the disk Izi < r, where 0 < r ~ 1, under the function t: w = fk(k). We denote by B (r) the complement of the set U k= 0 Bk(r) with respect ~:'tO the extended w-plane. Suppose that a function Q (w) has a regular and single
'il:"lentlY onto pairwise disjoint domains such that fo(O)
lR
R
S~) is the class of function~ fez) €
565
proved the theorem on areas for p-valent functions. In more general cases, the use
0;
2
Izi < 1
BASIC METHODS
fashion by Prawitz [19271 and Grunsky [1939]. Later, Goluzin (see OJapter XI, §6)
> 1 onto
the class of functions fez) = az + a z2 +"', where a> 0, that are regular and univalent in the disk
T
_ =,;=;,~"."'"
...
By using the theorem on areas, Bieberbach proved the familiar theorems in the
onto a convex
SM (M ~ 1) the class of bounded functions fez) € 5: m
~',
)c1ass 5 and I (d. §4 of Chapter 11). The principle of areas was used in a finer
the class of functions in I
5(1)
~...:""," __"",.."""
used in its simplest form by Gronwall (d. the area theorem in §4 of Chapter II).
the class of functions fez) € 5 that map the disk
(m
.'_:
1°. The principle of areas. The principle of areas (area is nonnegative) was
onto a do
with convex complement;
I
,
of a complex variable
a domain whose complement is starlike about the point w
I
§ 1.
2' C 3' ... ;
domain;
I*
n SM' and
~-1f ..,
Other notation will be explained when introduced.
+...;
main that is starlike about the point w
S
functions f(z)€ S(k)
ak_/,k-l + a 2k _/,2k-1 the class of functions fez) = z + c 2 z2 + ... € 5 with real coefficients C
5*
1'1> 1;
that do not vanish for
1,2,"') the class of functions in IO that have an expansion of
the form F((} =
SR
§l.
7
Izi < 1;
i~valued derivative
in the domain B(r o )' where 0 < r < 1. The problem is to calcu o late the area a (r), where r 0 < r < 1, of the image of the domain B (r) under the lfunction ~ = Q (w):
a (r)
the class of functions f(z) that are regular in the disk satisfy the inequality fez I) fez )
~':'
f- -
Iz I < 1
1 for arbitrary z I and z 2 in that
disk;
'" '" R, L the subclasses of univalent functions in the classes Rand L respec
H I Q' (w) I~ dUl,
"here dw is the element of area. Consider some single-valued branch of the func tion Q (w). Then, in the annulus r 0 expansion
< Iz I < 1
Q (fdz)) = ~ ~~k)Z-·q
The classes of functions that are subclasses of several of these classes will
5<;)
is the class of
(1)
B(r)
OJ
tively.
be denoted by the corresponding indexes. For example,
=
and that
q=1
with appropriate cut, we have the
+ ~ b~k)zq + ~(klIog Z, OJ
(2)
q=o
in which only the coefficients b~k) and the branch of log z depend on the choice
566
§ 1.
SUPPLEMENT ,.,
n
11
00
11
00
k=Oq=1
k=Oq=1
q=1
k
2m ~ ~(k)(b~k) -1ti ~ ~(J)), k=O
I
in which the form of the summation with respect to j is associated with the par ticular choice of the branch of Q (w). It is easy to give various necessary and
where
Po> 1, under the function w
=
v'=1
2
q Ibq 1
q=1
•
! f 1~' q\
q=1
v'=1
q
(;~,)q r·
(6)
tbe theorem is proved. Oae can find the conditions under which equality will hold
,t<, in (5).
If F (0 belongs to I and if the function Q (w) is regular in the complement of
'(I> Po'
(5)
C.C.,-l
By using the preceding inequality, we obtain inequality (8) of § 2, Chapter IV, and
0,
we obtain the generalized theorem on areas presented by Lebedev and Milin [1951l the image of the domain then, by setting
C.C.,
-
v.v'=1
! ! 1~,bq (C~,)q I~ { !
q=1 =
'"'
The left-hand side member of inequ~lity (8) of §2, Chapter IV, is equal to
(3)
j=O
sufficient conditions for equality to hold in this conditional inequality. For n
2
.!. q I bq I ~ '""" Fe v' log -_,---.:...--
obtain in the limit the following inequality (generalizing the theorem on areas):
~ ~ q I b~k) 12 ~ ~ ~ q I~~k) 12 -
11
00
of branch of the function Q (w). If we compute u(r) and let r approach unity, we
567
BASIC METHODS
,:;.
F((),
All these generalized area theorems and the basic inequalities obtained from
[\ them can be written in integral form, which is also very convenient for obtaining i:.
'i",a number of inequalities. For example, inequality (4) can be written
00
bll-C~ +bo+! ~1I~1I, 1<1~I<po,
Q(F(Q)=! 11=1
we have
k
If ,yT.,
00
~~
Izl<1
11=1
Irp' (z) 12 do ~ ~ ~ I~' (z) 12 do, Izl<1
00
00
~ q I bq 12 ~ ~ q I ~q
1
2 •
(4)
q=1
q=1
k
~oo
We can consider the case of disjoint multiply connected domains. In this case, series of the form (2) are not applicable, but we can use the theory of orthogonal
Inequality (3) is the starting point for obtaining various kinds of inequalities (d. Lebedev and Milin [1951] and Lebedev [1951, 1961]). Success depends on
suitable choice of the function Q(w). In applying this method, one does not al ways obtain sharp inequalities (in connection with this, one can apply Bunjakov
functions (over a domain) introduced by Carleman [1922] and Bochner [1922] (see also Bergman [1950] and Meschkowski [1962]). The case of a sin~le function and simply or multiply connected domain was considered by Milin [1964, 1964a, 1964b]. A generalized theorem on areas in integral form was obtained by Lebedev
skil's inequality). However, this method sometimes makes it possible to obtain
[1966] in the case of disjoint multiply connected domains. (See also § 3 of the
certain inequalities even when other methods fail.
present supplement regarding theorems on areas for multiply connected domains.)
As an example of the illustration of the application of this method, let us prove Theorem 3 of § 2 of Chapter IV. If we set Q (w)
=
l~=l Yv log (w - F (()),
2°. The method of contour integration. This method is closely connected with the principle of areas and it consists basically in examining a double integral over the domain in question. Usually the integrand is nonnegative (in particular, the
we have
square of the modulus of the derivative of some function that is regular in the do
11
Q(F(Q)=
k
du is the element of area and where ¢(z) = lk=l bkz and l/J(z) = ~k=/3kz .
00
! Iv(IOgF(i-=-{(C')+IOg(~-~J) v=1
main in question). It is chosen in such a way that, in the contour integral obtained by use of Green's formula, one of the factors in the integrand has values on the
00
=
00
'"' b _I
'""" q=1
q cq
+b _ 0
'"' ( ~
q=1
n
1L)~q ' l.J qCIl v=1 v '\:1
contour of the region that are complex conjugates of the values of some other func tion and in such a way that the use of this relationship makes it possible to re place this integral with the integral of a meromorphic function and calculate it
and (4) takes the form
with the aid of the residue theorem. By such a procedure, we obtain an inequality. The success of the method depends on successful choice of the double integral.
SUPPLEMENT
568
§l.
The method of contour integration was first used to solve extremal problems
BASIC METIIODS
569
explicit inequalities but, as a rule, it does not ensure description of the extremal
in the geometric theory of functions by Grunsky [1932]. In §4 of Chapter V, we
functions or complete information regarding their uniqueness. This method has
shall present some results of the work of Garabedian and Schiffer [1949], in which
been extended to doubly connected domains. In this connection, see Komatu [1943],
this method is used systematically to obtain various identities in the theory of
Goluzin [1951e], Li En-pir [1953], Lebedev [1955c, 1955d], Kuvaev [1959a]. Fur
conformal mapping of multiply connected domains. The method of contour integra
thermore, Kuvaev and Kufarev [1955] have obtained a generalized equation of the
tion has been applied by numerous authors, for example, Goluzin [1937, 1946,
Loewner type for multiply connected domains. Kuvaev [1959] has obtained one for
1946eJ, Grunsky [1939], Bergman and Schiffer [1951], Meschkowski [1953], Nehari
automorphic functions. All these generalized equations are extremely complicated
[1953], Alenicyn [1956, 1961]. (With regard to certain results obtained in these
and few specific results have been obtained from their use.
works, see § 3 of this supplement.)
4°. Variational methods. We stop only for those variational methods that have
3°. The parametric method. This method consists in using Loewner's equation (see §2 of Chapter III) to solve extremal problems in the theory of functions. Loew ner used his equation to obtain the inequality z
+
C
2
Z
2
+
c 3 z3
Ie 3 1 S 3
for the functions f(z)
=
led to the solution of a number of extremal problems in the geometric theory of functions. The first of these methods is the variational-geometric method of La vrent'ev [1934]. By this method, some remarkable results have been obtained in
+ •.. € S. Peschl [1936] generalized Loewner's equation some
applied questions (see, for example, Lavrent' ev and Sabat [1958, Chapter IV]).
what. In a number of articles, Goluzin [1936, 1937b, 1939c, 1946c, 1946e, 1949c,
Problems in geometric theory that were solved originally by this method can now
1951, 1951a] used Loewner's equation and extended considerably the possibilities
be solved quite easily by other variational methods.
of its application. Solutions of Loewner's equation have been studied in detail by Kufarev [1946, 1947a, 1947b, 1947c, 1947d, 1948]. Furthermore, 'Kufarev [1943, 1947] gave a generalization of that equation. We shall present the formulation of one of his theorems without proof:
Suppose that a function p (w, t) is regular with respect to w in the disk Iw\
<1
for every t in the interval 0 S t
that interval for every w in
!R (p (w, t)) > 0 for
Iwl < 1.
Iwi < 1, 0 S t <
OQ.
<
OQ,
Then, the solution w
=
=
1 and
ferential equations for an extremal function, one for each extremal function. Later, of the method of interior variations and obtained formulas for the varied function under fewer assumptions. Beginning with these formulas, he arrived at the same
f(z, t), f(z, 0)
=
z
of the differential equation dw dt =-wp(w, t),
use of Schiffer's method in the solution of extremal problems leads to certain dif Goluzin [1946e, 1947, 1947a, 1951d] (see also Chapter III, §3), gave his variant
and measurable with respect to t m
Suppose also that p (0, t)
The next variational method in order of discovery is that of Schiffer. In his articles, he used both boundary [1938, 1938a] and interior variations [1943]. The
differential equations as Schiffer did, but by a simpler path. Goluzin's method frequently leads to the final solution of various extremal problems in the geometric theory of functions. However, in many cases, this method does not enable us to
O~t
is a function that, for every t in the interval 0 S t
<
OQ
regular and univalent in Izi < 1 and f(o, t) = 0, f:(o, t) Izi < 1, 0 S t < Furthermore, the function
carry the problem to its conclusion, and then it gives only a qualitative charac
is, as a function of z, =
e - I , If(z, t)1
<1
for
OQ,
f(z)=limi/(z, f)=z+cgzg+csz s + ... t ..... oo
belongs to the class S. The parametric method is constantly used in articles on the theory of univa
teristic of the extremal function and the domain onto which this function maps the unit disk. Frequently, this domain is the entire plane with cuts along a finite num ber of analytic curves. To a considerable degree, this information removes the difficulties involved in the use of Loewner's method. This suggests combining the parametric and variational methods. An attempt to do so was made by Lebedev [1951, 1951a]. Since he did not give the proofs in the article [1951], we shall ex pound certain ideas briefly here. If a function w
Iz I < 1
=
f(z) in the class S maps the
lent functions (d. Bazilevic [1951, 1951a, 1957], Kolbina [1952a], Lebedev
disk
[1955a, 1955b], Slionskil [1959]). It frequently leads to success in obtaining
the function f(z) (see Chapter III, § 3) can be represented in the form f(z)
onto the plane with cuts along a finite number of Jordan curves, then
g (f(z, t), t), for 0 S t
<
OQ,
'=
where f(z, t) is a solution of Loewner's equation
570
SUPPLEMENT
§ 1.
11 1 (t)
(d. (18), p.96) with some function k (t), for 0 ~ t < 00. Let
and 11/t) denote
Zk/' (Zk»)2 21 (Z)2
P(z, Zk)= ( 7(z0 l(z)-/(Zk) '
arbitrary functions that are continuous for 0 ~ t < 00 except at a finite number of
Q. (z, Zk' t) = (Z/~ (Zk, t»)2 (I (z) _ f' (z) ~ (z, t) 1 (Zk, t)
points of discontinuity of the first kind and suppose that they satisfy the condi tions
'11 1(t) etl
<,If and
111/ t )\
< ,If for 0
ciently small real A, the solution w
dw dt = --(I
+ AT/. (t»
is, for every t in the interval 0 ~ t the disk
Izi < 1
=
w
St
<
00,
where ,If
fez, t, A), fez, 0, A)
i
I+k(t)
A"l2(I)w iA"l (I)
1- k (t) e
2
w
,
> o. Then, for suffi
1 (Zk, t)
z of the equation
=
0:::;; t
<
Q~ (z,
Iz (z, t)
+
1 (z, t) \, 1 (Zk, t) --1 (z, t) )
Zk' t) = (Zl; (zk, t»)~ (I (z) _ I' (z) ~ (z, t) 1+1 (zk, t) 1 (z, t) ). 1 (Zk, t) fz (z, t) 1 - I (zk, t) 1 (z, t)
00,
If we set t
< 00, a regular and univalent function of z in
and satisfies the conditions
=
0 in formula (2), we obtain a formula that differs only insignificantly
from Goluzin's formula (24), of §3, Chapter III. If we apply it to a function
fez) € 5 that is extremal in some problem, we obtain a differential equation for I
f(O, t)=O,
571
BASIC METHODS
f; (0, t) =
e-I-A! "ld
Relying on this fact, we obtain the varied function [*(z) €
[(z) that contains the parameter t. We also obtain differential equations for
t ) dt
fez, t) and g (z, d, for the boundary of the image of the disk under the function w
5:
00
f*(z)=/(z)+A ~[L(Z, t) T/dt)-iN(z, t) T/~(t)] dt+A~R(z, t),
=
[(z), and for the function k (t) in Loewner's equation. By this procedure,
Lebedev [1951, 1951a] obtained only certain qualitative results for extremal func (1)
o
tions in the problem of coefficients for functions in the class S. With greater success, Kufarev [1951, 1954, 1956b] (see also [1963]) combined
where
Goluzin's variational method and Loewner's method of parametric representation. The variational-parametric method constructed by Kufarev for solution of extremal
L(z, t)=/(z)-I'(z) ~(z, t) I+k(t) I(z, t) fz(z, t) 1- k (t) I(z, t)'
problems also leads to a differential equation for the function g (z, t) (and conse ~. quently for fez) and fez, t)). Furthermore, by using this equation and Loewner's
N(z, t)=f'(z)/,(Z, t) 2k(t)/(z, t) f z (z, t) ( 1- k (t) 1 (z, t»2
equation, we reduce the problem of finding the function k (d in Loewner's equa tion to that of solving a boundary problem for a system of differential equations.
and R (z, t) is a function that is uniformly bounded with respect to t inside the disk 1 z 1 < 1. By a suitable choice of 11/t) and function [*(z) € 5: _m
11/d
in (1), we obtain the varied
mathematicians of the Tomsk school have solved many difficult extremal problems
f*(z)=/(z)+A ~ AkP(z, Zk)-A ~ AkQ.(z, zk,t)
in the geometric theory of functions (for some of thes e see § 2 of this supplement).
k=1
m
- A ~ AkQ~ (z, Zk' t) k=l
+ A~Rdz, t),
Goluzin's variational method has been extended
(2)
where the Ak are arbitrary complex numbers, the z k are points in the disk
Iz I < 1, R *(z, the disk
t) is a function that is uniformly bounded with respect to t inside
Izi < 1,
and
Loewner's equation reduces to determining the extremal function and the solution of the extremal problem. By means of Kufarev's variational-parametric method,
m
k=1
If from this system we succeed in finding the function k (t), then integration of
to
multiply connected do
mains (see Kufarev and Semuhina [1956], Aleksandrov [1963a], Gel'fer [1962]) :, and to multivalent functions (see Gel'fer [1954, 1956], and Goodman [1958, 1958a, 1958b, 1958c]). It is easy to obtain variational formulas of the type of Goluzin's formulas for classes of functions that can be represented by means of a Stieltjes integral (convex, starlike, typically real, etc.). However, it seems more conven ient to us in such cases to use other methods, in particular, other variational for mulas. We shall speak of such methods in subsection 8°.
§ 1. 572
BASIC DOMAINS
573
SUPPLEMENT
Here, these equations are expressed by means of suitable quadratic differentials,
5°. The method of extremal metrics. These methods rest on inequalities be
and, to investigate the problems of integrating these equations, we need in a num
tween the lengths of curves belonging to a family and the areas of the domains
ber of cases to have a qualitative picture of the trajectories corresponding to these
filled by them. Grotzsch [1928] was the first to use this method as a method in the
quadratic differentials.
theory of univalent functions and he called it the "method of strips". His method
This aroused interest in the behavior, local and global, of the trajectories of
is expounded in Chapter IV, §6. By means of it, Grotzsch solved numerous prob
a quadratic differential. The first systematic investigations in this direction were
lems for both simply connected and multiply connected domains. Many of these
the works of Schaeffer and Spencer [1950], Jenkins [1954a], and Jenkins and Spen
can now be solved more easily by other methods, in particular, by the method of
cer [1951]. Furthermore, the theory of the trajectories of quadratic differentials on
contour integration. The method of strips was perfected by AhHors [1930]. He
a finite oriented Riemann surface was developed by Jenkins [1958, 1960c]. In par
proved a very important inequality, known as the principle of length and area. We
~i.ticular, he obtained a very general "fundamental structural theorem" [1958,
shall give a formulation of this result.
Chapter III].
Let fez) denote a function that is regular in a domain B and let n (w) denote the number of roots of the equation fez) = w that lie in B. Define
"general theorem on coefficients" [1958, 1960a, 1963a]. This theorem deals witb
.
TJ
(p) =
1 21t
2"
~
1e
n (pe ) dB,
The fundamental result of the method of quadratic differentials is Jenkins's a set of univalent functions f/z) that are regular or meromorphic in disjoint sub
p>o.
domains /1 1 of a Riemann surface lR and that map these domains into disjoint
o
subdomains of lR. It is assumed that the domains /1
Let l (p) denote the total length of the level curves If(z)1
=
p in B and let a de
note the area of B. Then
tories of the quadratic differential Q (z) dz 2 on
1
are bounded by the trajec
!It. 1) Under the se conditions, one
can establish an inequality containing the coefficients in the expansions of the 00
~ l (p)2 dp b pP (p)
< 21ta.
The reader can find a proof of this result, certain of its applications, and a suitable bibliography in the book by Hayman [1958]. In 1946, AhHors and Beurling gave a new formulation of the method of extremal metrics and this formulation later enabled Jenkins [1958] to give a further develop
functions
,,~If Q (z) ~\:
f 1 (z)
and
Q(z)
in neighborhoods of each pole of
has in a neighborhood of z
~pole into the point
z
=
<Xl)
=
<Xl
Q(z)
of order m ~ 2.
(we use a local parameter that maps the
and expansion Q(z)
=
a/z 2 + terms
of higher degree in
z -1 (that is, a second-order pole of the quadratic differential), then an admissible
function f/ z ) must have an expansion f/z) = az + a + a/z + terms of higher de
o
gree in z-l. If Q(z) has in a neighborhood of z
=
<Xl
an expansion Q(z)
=
azm-4 +
ment of this method. The name "method of extremal metrics" is connected with
terms of lower degree in z, where m ~ 3, an admissible function flz) must satisfy
the fact that special metrics are introduced in certain of the studies in a domain
the condition f/z)
filled by a family of curves.
(Jenkins [1960a]). Jenkins's theorem on coefficients was supplemented by Tamra
6°. The method of quadratic differentials. The significant role of quadratic differentials in the solution of extremal problems was already disclosed in the familiar investigations of Grotzsch (see subsection 5°). In particular, in his theo
=
z + az -k + terms of higher degree in z -1, where k ~ ~ m - 2
zov [1965c] for the case in which the differential Q(z) dz 2 does not have poles of order higher than I. This very general theorem enables us to obtain systematically many familiar
rems on the existence of conformal mappings of multiply connected domains onto
results in the geometric theory of functions: with it, one can easily prove the dis
canonical domains, the latter are determined in a number of cases by the trajec
tortion theorems for functions that are univalent in the interior or exterior of the
tories of a quadratic differential. The role of quadratic differentials also showed up in the solution of extremal problems by the variational method. As we know, the method of interior variations usually leads to a differential equation for ex tremal functions and for the boundaries of the corresponding extremal domains.
1)
[1958].
With tegatd to the definitions in this subsection, see the monogtaph by Jenkins
574
SUPPLEMENT
§ 1.
BASIC DOMAINS
575
unit circle, one can study sets of values of these functions and their derivatives,
in that interval for every z € B (the kernel of the class), and where /l (t) is a
one can obtain a number of results regarding univalent functions without common
function that is nondecreasing on the interval [a, b] and that satisfies the condi
values, and one can prove existence theorems for mappings of multiply connected
tion /l (b) - /l (a) = I (the parameter of the class). In what follows, we shall de
domains onto canonical domains. With the aid of the "general theorem on coeffi
note the set of such functions /let) by M[a, b1.
"
cients", Jenkins posed and solved a number of difficult extremal problems. (With regard to some of these results, see
§ 2 of
this supplement.)
~{
For example, the class C of functions fez) such that f(o) = I is represented by the formula
,.
7°. The method of symmetrization. At the present time, we know several methods of symmetrization. We pause briefly for circular symmetrization proposed by Polya. For a simply connected domain
~
fez) =
B containing the point z = 0, one can
1 1
+ eitz it dp. (t), -e z
p.(t) E M [ - 'It, 'It].
(2)
-lC
easily prove that there exists a unique simply connected domain B. with the fol
The class of typically real functions fez) (the class T) is represented by the formula (see Chapter XI, §9) along arcs whose lengths add up to l(r), where 0::; l(r)::; 2rrr, then this circle in ,.
lowing property: if a circle
Iz I = r,
where
a < r < 00,
intersects the domain B
tersects the domain B. along a single arc of length l (r) with center at the point z
= -
r. The domain B. is symmetric about the real axis. If w
f.(z), where f(o)
=
f.(O)
=
=
Iwl
. If',
< 1, then
(:)J ,. ~ , If~ Y),L 19,
~
z •
n
.,.
-,.
fez) and w =
0, are respectively functions that map Band B. uni
valently and conformally onto the disk
f(z)=
dp. (t),
p. (t)
E M [- 'It,
'It].
(3)
In the solution of extremal problems and problems on finding the domains of the values of the functionals and systems of functionals in the class of functions that can be represented by means of a Stieltjes integral, we sometimes find the following theorem I} useful.
z E B.
There are other inequalities connecting certain quantities for Band B•. The method of symmetrization rests on an inequality of this sort. It is expounded rather
m
Suppose that the set of points (XI'···' X) in n-dimensional Euclidean space R n is represented by the formulas b
completely in the books by Hayman [1958] and Jenkins [1958].
Xk=~gk(t)dp.(t),
We note that this method, especially in combination with the method of ex
k=l, 2, ... , n,
(4)
a
tremal metrics, made it possible to solve a number of extremal problems that had
where the gk(t) are fixed functions that are continuous on the interval [a, b] and
not been solvable by other methods (see §2 of this supplement). A generalization
/l (t) is a function in the class M [a, b] (these functions being, in general, different
of the method of symmetrization to multiply connected domains has been given by
for different points of 311). Then 311 is the closed convex envelope of the set of points
Mitjuk [1964a] (see §3, subsection 2° of the supplement). 8°. The method of integral representations. The solution of extremal problems
Xk=gk(t),
k=l, 2, ... , n,
a~t~b,
is simplified in the classes of functions that have an integral parametric represen tation. We shall consider classes A of functions that have a representation by
and every point in 311 can be represented by formulas (4), where /l (t) is a piece
means of a Stieltjes integral
wise-constant function in M [a, b] with no more than n points of discontinuity.
This theorem and special cases of it have been repeatedly used by various
b
f(z)=~g(z, t) dp. (t),
(1)
authors (sometimes, unfortunately, without reference to Caratheodory).
a
where [a, b] is a finite interval, where g (z, t} is a fixed function that is regular with respect to
z in
some domain B for
a::; t .:S b
and continuous with respect to
I) This theorem follows easily from the results of Caratheodory [1911] (see also F. Riesz [1911]).
576
SUPPLEMENT
AN~ULUS
§2. UNIVALENT FUNCTIONS IN DISK AND
Caratheodory's theorem cannot by any means be applied to all extremal prob
577
lems in the class of functions that is represented by formulas (1). In such cases,
a combination of the method of extremal metrics with the variational method and
two variational formulas proposed by Goluzin (formulas (7) and (8) on p. 528) are
the method of symmetrization. With regard to the methods of the geometric theory
very useful. It is possible to give other analogous variational formulas in the
of functions, see also Kufarev [1956a, 1958] and Jenkins [1958, Introduction].
class A (see, for example, Zmorovic [1952]), and from one of these formulas we
§2. Univalent functions in a disk and in an annulus
can obtain variational formulas of the type of Goluzin's formulas in the class S
1°. Ranges of values of various functionals and systems of functionals and
(see Chapter III, §3). Goluzin's formulas and Zmorovic's formulas usually lead
inequal ities involving them. A significant role in the investigations in the geo
immediately to the fact that the extremal function is of the form
m
m
f(z) = ~ Akgk(z, t),
Ak~O,
metric theory of functions in recent years is played by the results dealing with
~ Ak = 1,
(5)
k=1
k=l
properties of functions in the class S and other classes and also the more general
where the t k are points in [a, b], and they indicate the value of m. We give yet another variational formula in the class A. Let fez) denote a member of A. Since g (z, r) € A for every
T
in a ~
T
problem of determining the ranges of values of these functionals and systems of functionals. It is primarily with such results that the study of functionals directly characterizing the distortion of the preimage deals.
~ b, the function
0<1..<1,
!*(z)=(I-A)!(z)+Ag(z, 't),
Goluzin's theorem on the distortion of chords (see Chapter IV,
b
fez) =!(z)
+ A ~ [g(z, 't) -
g(z, t)] dp. (t),
a
0<1..<1,
a~'t~b.
2
(6)
II
the extremal function is of the form (5).
for F «() €
In many cases, the class of functions can be represented with the aid of a Stieltjes integral. For example, the class S * of functions that are starlike in the
Izl < 1
is represented by the formula -2
J" log {I -
!(z)=ze -"
V I-~ I
2
1=11 / 1-. ~ m ... ,..
This formula, like Goluzin's formulas, usually leads immediately to the fact that
disk
which
the case of the class I m by use of Loewner's parametric representation: for arbi trary (1 and (2 in the domain 1(I > 1,
class A:
+ A(g(z, 't) -
§ 2),
has found numerous applications, has been strengthened by Bazilevic [195la] to
also belongs to this class and we have the following variational formula in the
f* (z) =!(z)
problems of obtaining bounds for different functionals characterizing the geometric
all
"I
-Tf'(0) 12
(1)
I 1------,--;;---;;
log F (~l)
+
Cl - ~2 C
F «(2) F(C l )-F(C 2 )
l+'2
'I
in the class I(2). From it, we obtain the sharp inequalities m
sent any difficulty in such cases.
methods (the variational-parametric method of Kufarev). Jenkins used successfully
j=l
.
m2
By the same method, Bazilevic [1951] obtained, for all (1 and (2 in the do 1'1 > 1, a bound for the functional
,p.(t)EM[-'lt, 'It].
spoke in subsection 4° about a combination of the variational and parametric
C2
'1 -
I
I m . Equality can hold in the first of these conditional inequalities for > 1 but it can hold for the second only when (1 = (2'
eltz} d", (I)
9°. Many different results can be obtained by combining various methods. We
I~ II r-V
F(C l ) -F(C 2 )
-. /
I'll = I' 21
main
Naturally, the use of Goluzin's variational formulas or formula (6) does not pre
~f
2
m
1
2
JJ
(
I
+ IF ('j) I I - 11fT
1-
1
1F7cJ) I 1+1EjT
~
I F (~l) F
+ F ('2)
F (C 2)
«(I) -
~II
I
2
j=1
'1 - (2 ~l
{
-H2
I m
I F (Cj) I
1+_1 I Cj I
m__ 1- 1_
I+V«j) I
l'jl
- ------------------------- ------------------------ ~~
-----------------
-
~---~
--
~
578
_....
~~
_.,..,...".. ~,._.-.,--""'""",.. -
.
-------~..,&i:t:&
.~-
--
...
-':'-~
_,~
By using the generalized area theorem, Milin (see Lebedev and Milin [1951]) established the following sharp inequalities for F(() €
I(I
q and for arbitrary (1 and (2 on the circle
=
llr
~(2), for arbitrary p and
we obtain a necessary and sufficient condition for a function to belong to the morphic functions.
2(lp+ql+lp-qj) 1
(1 +r2)
results for univalent functions without the boundedness condition. In particular, clas~ S, one that is equivalent to Grunsky's univalence condition [1939] for mero
> 1:
1
(l-r 2 )
579
§2. UNIVALENT FUNCTIONS IN DISK AND ANNULUS
SUPPLEMENT
-(Ip+ql-Lp-ql)
By means of Loewner's method of parametric representation, Slionski~ [1959]
I F (~l) - F (~2) IP
I ~l -(2I P
~ ~
obtained in 1956 results for bounded univalent functions that are directly connected with a number of theorems in §2 of Chapter IV. Thus, Slionski~ obtained a neces
2
+ r2) 2" ( I P + 1
X
(1
1~1+~2Iq
/F«(I)+F«(2)lq
~
q I - I P- q I)
sary and sufficient condition for a function to belong to the class ~ . These rem
1
suits also include a strengthening of Goluzin's Theorem 1 of §2, Chapter IV for
(1 _ r2) 2" ( I p + q I + I P- q I)
the case of the class ~m' In particular, inequality (1) follows. Suppose also In this particular case these results have a simple geometrical interpretation
l_f(X)f(y) M2 W(x,y)=log 1 -xy
(see § 2, subsection 2° of this supplement). General inequalities of the type of the distortion theorems, which constitute an application of formulas (2) on p.1l8, have been obtained by Kolbina [1952a). In particular, a generalization has been obtained of Theorems 3 and 4 of
§ 2, Chap
ter IV.
Let
f (z)
Nehari [1953] was the first to use systematically in the theory of univalent
n
~ I L..J
n'
~ ,
of his theorems.
'V, ...
and satisfies the inequality If(z)1
Iz I < I
in that disk. Let z l' z 2' .. "
and let a , a , " ' , a 1
2
n
Z
n
for k
~
I 1.
~v=1
a. a. logf (zv) - f (z",) v",
zv- z",
I~ ~
of a function fez) = a o + a z + a z 1 2
~v=1
+ ... that is regular in a neighborhood of the
n'
III'(2k) ( Jekyk zv'
Ivlv I
'=1
~
)
,-,
LJ
Ivl vI
00,
W(2k) ( , -') }1/2 Jekyll
Zv" Zv'
,
(
3)
0, 1, 2,'" .
I 1.
1.
'1'=1 ...'=1
main
1(1 > 1.
'.'v,' log
~ _~' v
I
v'
we obtain the corresponding
lJ
and ('lJ , in the do
This result strengthens Theorem 3 of §2, Chapter IV.
Another consequence of inequality (3) for functions fez) €
I
f' (Zl) - f' (Z2) 1 (f(zl)-f(Z2»2 -(ZI-Z2)2
Iz I < I. M approach
zv l
"', ... '==1
I {I ~
Together with the class of functions that we have considered above, let us also consider the class SM' By letting
n';
I
for arbitrary complex numbers y lJ and y'lJ " and arbitrary (
point, a necessary and sufficient condition for f(i) to be regular, univalent and
bounded in modulus by unity in the disk
=
(2)
Using this inequality, Nehari obtained, for the coefficients in the expansion 2
= 1,"',
and ~ m ,where m = 11M, Slionski'l obtained for F (() € ~ m bounds for n n' ~ ~ F(~)-F(~~I)
n
'\'1 a. rJ. log 1- f (zv) 77.i;,') 1. v'" l-z.z",·
n; v'
Then, by using the familiar relationship between functions in the classes S M
denote
denote complex constants such that
a 1 + a 2 + ... + an = O. Then
n
(where v = 1,""
n
~
has led to a number of new results in this category of questions. We present one
points in
Izi < 1
v=1 v'=1
gral for a harmonic function obtained by finding its singularities. His technique
Izl < I
,!xl
denote a function in SM' For arbitrary complex numbers Y lJ and Y~',
and arbitrary zlJ and z> in the disk n 2: 1, n' 2: I), we have
functions a method based on the classical minimal property of the Dirichlet inte
Let fez) denote a function that is regular and univalent in the disk
f(x)-f(y) xy
x-y f(x)f(y)'
and they have led to a number of covering theorems for the classes Sand S M
XI
SM is the inequality
1 If' (Z2) [21M2 (I-lzI12)2 -(1-lf(z2)j 2/M2 )2
1 (1-1 Z.12)2 -
If' (Zl) 121M2 (I-If (Zl) 121M2)'
I
I}1/2 •
SUPPLEMENT
580
§2. UNIVALENT FUNCTIONS IN DISK ANNULl/S
If we now set z I
1
=
z 2' we obtain the well-known inequality (Alenicyn [1956])
61 {I, z} I
+
M 2 1f'(z)12 /A,f!
1
J'I~\
1m
~
I -'-I- I
~
12\2
(4)
,
where {f, z I is Schwarz's invariant. On the one hand, inequality (4) gives a bound
581
D of the functional 1=log(zA[,(z)I-Alf(z)A) (where A is a given real number) defined on the class S. He obtained a differential equation for functions fez) € S corresponding to nonsingular boundary points 10 of the domain D, that is, points for which there exists an a¢: D such that II -
I
1
ai, where
I € D, attains its mini
for Schwarz's invariant in the case of bounded univalent functions; on the other
mum with
hand, it sharpens for such functions the well-known inequality of the hyper
ary of D. This equation contains a parameter that is a root of a sixth-degree alge
bolic metric. Inequality (4) can also be obtained in an elementary way (see Lebe dev [1961]) from the well-known inequality of Nehari [1949] for the absolute value of Schwarz's invariant in the class S:
{I, z} 1 ~ - "- ,
,
The set of nonsingular points is everywhere dense on the bound
braic equation. As a special case, we obtain the complete solution of the problem . of the maximum and minimum of
~ (J)
= log 1 zA
f f(Z) 1 -AIfez )1..1.
tion, Lebedev [1955a] determined the range of values of the system 1a, log (f(z); z)J
'0'.'
in the class
We note that if a function f(z) that is regular in the disk
Iz 1
< 1 and satis
S(l)
of functions f(t;) = a( + a 2 (2 + ... that are regular, univalent, 1(I < 1 and also the range of values
and bounded in modulus by unity in the disk
of the functionals log (z['(z)lf(z)), 10g(az 2 ['(z)/[2(z)) and loga['(z) in the
fies the inequality
I{f, z}I~~ ,
class S(1)[lf(z)\l of functions in ,~.
<.
in that disk, it is univalent in it (Nehari [1949]). Here, the constant 2 cannot be replaced with a smaller one (see Hille [1949]). PokornyJ: [1951] has obtained a generalization of this result of Nehari. A number of sharpenings of the familiar distortion theorems for the class S and also for a broader class of functions have been obtained by means of the method of symmetrization. We shall give some of these results below. The solution of extremal problems in the form of the obtaining of certain bounds on functionals is a special case of the more general problem of determin ing the ranges of values of the corresponding functionals and systems of func tionals in a given class: If we find the range of values of a given functional or system of functionals, from it we can obtain various inequalities. A considerable amount of study has been devoted to the obtaining of results in this more general posing of the problem. Grad [1950] has found the range of values of log ['(z) (where z is a fixed number in the disk
0
By using and developing further Loewner's method of parametric representa 6
I
=
1zl
< 1) in the class S by the variational method. The boundary
functions of this range are determined by a differential equation depending on a single real parameter and can be expressed in terms of elementary functions. By using Goluzin's variational method and certain other considerations, Le bedev [1955] solved the more general problem of determining the range of values
with given value of If(z)\. In this last
S(l)
problem, the range of values is not determined directly by an inequality obtained from Loewner's equation but is found as the convex envelope of a portion of the boundary of that set. In all cases, the boundary functions are found; here Loew ner's equation plays an essential role. A number of results in the problem of determining the ranges of values of functionals have been obtained with the aid of Kufarev's variational-parametric method. Thus, Aleksandrov £1958] has found the range of values of the functional
10g(zA[,(z)I-Alf(z)Alf(z)llL) (where A and Il are arbitrary real numbers and z is a fixed complex number in the disk Iz I < 1) on the class S( k), where k = 1, 2, •.•. Red/kov [1960, 1962, 1962a] solved an analogous problem for the class S( I) (a) of functions in S(1) with fixed a in the interval 0 < a < 1 and he found the ranges of values of the functional log (zA ['(z) 1 -A ['(0) v 1f(z)A If(z)I,u) (where A, Il' and v are real numbers) on the classes S( o[lf(z)\l and S( o' Furthermore,
Aleksandrov [1963] considered the problem of determining the range of values of the functional ](f(z), fez), ['(z), ] (WI' W ' W3' W 4) = ] 2
f'(;))
(Xl + iy I"" ,
partial derivatives of their arguments
X
on the class S(k), where
+ iy 4) has continuous first and second
4
X
k
, Yk'
for k
=
1, 2, 3, 4, and he determined
conditions from which we can find boundary functions of this range of values. Redfkov [1963] considered the problem of finding the range of values of the func -tional ](f(z), fez), ['(z), ['(z), ['(0)) on the class S(1)[lf(z)l]. Aleksandrov [1963c] studied the properties of boundary functions of the ranges of values of weakly
582
§2. UNIVALENT FUNCTIONS IN DISK AND ANN~LUS
SUPPLEMENT
583
differentiable functionals defined on the class S(zo) of functions f(z) that are
it aiso contains the points (X + tjJ, - Y, U, - V) and (U + tjJ, V, X, YL On the
< 1 and satisfy the conditions f(o) = 0, f(z 0) = zO' where z 0 is a fixed number in the disk Iz 0 I < 1. As an application, we determine the ranges of values of the functional J I('(z)/['(z) , ["'(z)/['(z)l,
boundary
defined on that class and thus on the class of all functions f(z) that are regular
and
regular and univalent in the disk
and univalent on the disk
Iz I < 1
Iz 1
and also of the functional j(f(z)' f(z)' f'(z), f'(z))
on the class S(zo) by indicating the method of finding the arguments of the func
r
of the set D, there are four points, namely, the points
+
) 1 Iz I ( log 1+1I Z /2 , 0, Iog 1_ Iz I ' 0 (0, -2 arc sin I z I, -log 1_11 z
12
'
2 arc sin I z I)
tions J (t l' t 2) and J (t l' t 2' t 3' t 4) corresponding to nonsingular boundary points
and the points symmetric to these through each of which there passes a definite
of the ranges of values in question. The logical conclusion of these investiga
one-parameter family of support hyperplanes of the set D. Furthermore,
tions is a consideration of infinite systems of weakly differentiable functionals.
tains a two-parameter family of rectilinear segments. The support hyperplanes of
Theorems of a general nature dealing with boundary functions of the ranges of
the set D at the remaining points of
values of such systems defined on the classes Sand
IO
are established in Alek
sandrov's article [19651.
points in common with
r.
r
are tangent to
For every point of
r
r
r
con
and they have no other
a specific procedure is shown for
constructing one of the functions in the class S that map this point into D.
As an example of the use of Jenkins's general theorem on coefficients, we point out that he obtained [1960] the most definitive results in the problem of the range of values of the functional
f (z)
for fixed z = re i e in the disk Iz I
< 1 on
the class SR. Suppose that f(z) = peilf;. He established, and one can easily show from a well-known result of Rogosinski [1932], that, for arbitrary fixed tjJ such that arg(re'·e /(1 + ret·e ) 2):s tjJ:s arg(re'' e /(1 -' e ret ) 2), the greatest value of p is
e, tjJ) = z/(I + 2tz + z2), where arg h (rei e, r, e, tjJ) = tjJ. With regard to a
In recent years, a number of investigations of both Russian and foreign authors have been devoted to the finding of bounds for various functionals in problems dealing with disjoint domains. Many extremal problems for basic classes of analytic functions reduce to these problems. Let us turn first to the problem of conformal radii of dis joint domains. Golu zin, using his variational method, considered the following problem: Find the
obtained only by the function h (z, r,
t € [-1, 1]
maximum of the functional rr:=llf~(o)1 out of all possible systems of functions
is found from the condition
lower bound
f)z) that are regular and univalent in the disk Iz I < 1, that map it onto dis joint
for p, for arbitrary tjJ in the interval indicated, the extremal function of this in equality is determined by the same investigating procedure in the terms of geome
domains, and that satisfy the condition f v (0) = a v ,where the a v (for v = 1, 2,' .. , n) are arbitrary given distinct finite points. For n = 2, the solution
try and the theory of quadratic differentials, which theoretically solves the prob
of this problem is the familiar result of Lavrent'ev [1934]. For n = 3, Goluzin
lem in question.
also found the complete solution of the problem (see §4, Chapter IV).
Using Goluzin's method of variations, Ulina [19601 studied the range of values
Applying Goluzin's variational method, Kolbina [1952, 1955] obtained a sharp
of the system !log({(z)/z), log ['(z)l on the class S for fixed z in Izi < 1, and
inequality for the products n~=llf~(O)\aV and rr~=llf~(o)lav for arbitrary given
she obtained a system of equations characterizing the boundary of that set. Using
positive numbers avo
Loewner's method of parametric representation, Popov [1965] obtained the com plete solution of this problem in a somewhat modified posing; specifically, he de termined the range D of values of the system of functionals
{log I/~Z)
I,
/(Z)
arg -z-'
I /(z) I log zl' (z) ,
With regard to the problem of the maximum of the product
in
the case of arbitrary n ~ 2, Alenicyn [1956] obtained the following result in this direction by extending the corresponding results of Nehari:
/(Z) }
If the functions fv(z), for v = 1, 2,"" n, map the disk I,z\ < 1 univalently
arg zj' (z)
onto dis joint domains and if z defined on the class S. The set D is closed, bounded, convex and symmetric in the sense that if it contains a point (X + tjJ, Y, U, V), where tjJ =-log
rrnV-I _ If'v (0)1 a v
2 ),
=
0 is a regular point of these functions, then, for
arbitrary real constants a v ;6 0 such that
InV=1 a v
= 0 we have the inequality
585
§2. UNIVALENT FUNCTIONS IN DISK ANNULUS
584
SUPPLEMENT
n n
n
2
If~(O)lay~
y=l
Ifk(O)-fy(O)I-~akay.
(5)
l~k
Equality holds here if and only if there are distinct numbers ale' k
=
1,2"" , n,
and a positive number p such that the curve 1
of all functions F (,) that are meromorphic and univalent in the domain It::"! > 1 and satisfy the condition F (00)
([(z), F(')) such that fez) For the class
n
n
Let mo denote the set of all functions fez) that are meromorphic and univa lent in the disk 1zj < 1 and satisfy the condition [(0) = O. Let 1Jll00 denote the set
(6)
W-ak lak=p
k=1
= [
n
I
v (z) of the equations
f-
k=1
F(,) for every z in Izi < 1 and every' in It::"! > 1.
Alenicyn [1956a] used Nehari's method to obtain the following
m C/. {
I
1
log
z2f' (z)f' (0)
i2 (z)
in
1'1
> 1, and arbitrary con
(C)} + 2C/.l~ log (1f-( ZF )(C) ) + C/.~ log F'F'(co) 2
~ - I C/.ll i log (1
I
z=e,p-~ (W, - a.) (n (W, - ak)ak)~,
00. Let 9J1 denote the set of ordered pairs
inequality for arbitrary z in Iz I < 1, arbitrary , stants a and a : l 2
partitions the w-plane into n simply connected domains. Extremal systems of functions are constituted only by the solutions w v
m,
=
-I
z I~) - I~ Ii log (1
-
l
I
C 12
).
(The values of the logarithms on the left-hand side are taken on suitable branches.)
k::j:.y
w,(O)=a" lel=l ('1=1,2, ''', n),
This result leads to inequalities involving various functionals both in the class of functions mentioned and for univalent mappings. In particular, Alenicyn
corresponding to the curves (6) that partition the plane into n simply connected domains. These solutions map the disk Izi
< 1 univalently onto the domains in
dicated.
obtained [1958] the following inequalities, which do not involve the derivatives of the functions in question: Let [l(z) and [/z) denote two functions that are meromorphic and have no
In the simplest case of n
=
2, Lebedev [1955b] posed and solved a more gen
eral problem. Specifically, let B denote a simply connected domain containing and a 2. If a k f- 00, for k = 1, 2, let w = [/z) denote a l function that satisfies the condition [k(O) = a k and that maps the disk Izi < 1 univalently and conformally onto a domain B k contained in B. If a 2 = 00, let
common values in the disk Izi
given distinct points a
[2(1;) denote a function satisfying the condition [/00) = 00 that maps the domain I" > 1 onto a domain B 2 contained in B. The domains B l and B 2 are disjoint. Let &, B (aI' a 2) denote the set of points M (x l' x), where x k = 1[~(0)1 if a k f- 00 and x 2 = 1[~(00)1 if a 2 = 00. Lebedev [1955b] obtained the result that, in the case in which B is the en tire plane (including w
=
00), the set &, B (0, 00) is the domain 0 < x 1 x 2 S 1,
< 1, and [/,) = Ot, for > 1 and 0 < t < 00, and the set &'B (aI' a 2) is the domain 0 < x l x 2 S lal-a212,
Xl> 0, with x l x 2 = 1 only when [1(') =
I"
t',
for It::"!
Xl> 0 with x l x 2 = la l - a212 only when [1(') = (alk l - a 2,)/(k l - ,), [2(') = (a l k 2 - a 2,)/(k 2 - ,), and Iklk21 = 1. Furthermore, by applying Goluzin's varia tional method, Lebedev determined the region &, B (aI' a 2 ) for cases when B is the plane with the point 00 excluded and when B is the disk Iwl < R. As a con sequence of these theorems, one can again obtain the results of Lavrent' ev [1934] and Kolbina [1952] (here B is the unextended plane) and those of Kufarev and Fales [1951] (here, B is a disk).
< 1. Suppose that [1(0) = 0 and [2(0) = 00. Then, < 1,
for arbitrary points z l' z 2 in the disk Izl ] log
(1 - j: ~::~) I~ -
Ij: ~::~ I~ I
Zl Z i
+
log (1 - I Zl
Ii) (1 - I z~ I~),
I/V(1 -I zll~) (1- I Z~ I~)·
These inequalities are sharp for arbitrary z 1 and z 2 such that Iz 11
=
I
z 21
=
1
with equality holding only in the case of univalent functions. These results lead immediately to a number of sharp inequalities for functions in the classes Rand L and functions associated with them (see Alenicyn [1956a, 1958]). For other results in this direction, in particular their extension to a cir cular annulus, see
§ 3,
subsection 2° of this supplement.
The same functional [1(zl)/[2(z2) in the class of functions indicated was studied by Lebedev [1957] by means of the variational method: for the class 1Jll, he found in explicit form the boundary of the range of the functional ~ =
[(zo)/F(,o) or, equivalently, of the functional ~= [(r)/F(p), where r= Izol and p = 1'0 I· The pairs of functions corresponding to given boundary points of this
range can be determined from the differential equations that we have obtained for
•._:
.• - -
-
.__...__.
-:_::-
.
_~_~
_..
_.
_._0_,:::.-
~.
__.
.....
~::_'"'_____
__.__
_:__.
__
_"'='
_"'0'_.
~
f.~
them. As a consequence, sharp bounds have been obtained for If(r)/ F (p)1 and Ilog (1 - fer)! F (p)) I, the radii of disks that prove to be the ranges of values of
Inequality (7) strengthens the previous result of Lebede; and Milin [19511: if
fez) €
R (or L), then
['(O)/F(p) and f(r)/F'(",,), the range of values of the system (If(r)l, 1/!F(p)\) in the class m, and also the range of values of the function fez) in the class R.
2"
i7' ~
o
The range of values of the system (1["(0)/['(0)1, I['(O)/F'(",,)\) in the class
mwas determined by Ulina [1960). maximizing the functional" J
=
If "(0)/[,(OW'-. 1['(0)/ F'(",,)!.B , a > 0,
f3 > 0
=
7fz, where 17f1
=
1.
In the problem of conformal radii of disjoint domains, Lebedev [19611 obtained
in
the following result:
Lebedev [19611 has obtained a number of results of extremely general nature in problems dealing with disjoint domains by beginning with the area principle.
a) of all systems Ifk(z)\~ of functions fk(z), for k = 0, 1," " n, that map the disk Izi < 1 conformally and univalently onto disjoint domains in such a way that fo (0) = "" and fk(O) = a k, k = 1" " , n, where a , " ' , an are fixed points. By using the generalized area theorem for He considered the class m("", a
1
functions in this class (see
§ 1,
1
n
IIlf';(O)IITkI2~ k=O
where f~(O)
m("", a 1 ,'" ,an)' in particu
2"
(8)
limz~o l/zfo(z). The cases in which equality holds in (8) were all
Inequality (8) generalizes to the case of complex Yk the inequality known A recent paper of Jenkins [19651 is closely related to this class of questions. eral inequality of the distortion-theorem type for the clas s 311, defined above, of
210
~ III (e ) I~ de· ~
Ifo (~i6)Ti de ~ 1,
i6
o
0
pairs of functions If(z), F«()l that depend on a large number of parameters. From that inequality, one can obtain a number of particular inequalities for the functions
fez) and F(,) and for functions in the classes Rand
with equality holding if and only if
I a I> I b I, II (z) =
"l =
ak - azl-2!R{TkTz},
In tbat paper, Jenkins started with the area principle and established a more gen
m("", 0), then
a fo(z)=-+b, z
j
l~k
earlier for real Yk'
The following results are among the applications of these general theorems.
;7'
=
II
= O. Then,
listed.
subsection 1 0 of this supplement), Lebedev ob
lar, inequalities of the same type as the distortion theorems of Goluzin in the If Ifo(z), f/z)\ €
Let Yk , for k = 1,'" , n, denote fixed numbers such that I~ =0 Yk for Ifk(z)}~ € m("", a 1 , · · · , a), we have
,""
tained a number of distortion theorems in the class
I.
I/(e i6) I dO ~ 1,
with equality holding if and only if fez)
This led to the solution of the problem of
that class.
class
587
§2. UNIVALENT FUNCTIONS IN DISK AND ANNULUS
SUPPLEMENT
586
canst,
I"ll =
2
L.
Furthermore, by means
of the extremal metric, Jenkins obtained a number of results for the functions fez)
2
(I a 1-I b1) 'lJZ a-bYjz '
and F (,) in question, though omitting the requirement that they be univalent, and for the classes Rand L.
1.
A number of theorems of the distortion theorem type have been established for
From this and the familiar result of Rogosinski (Theorem 1, §8, Chapter VIII), it
more general classes of functions, in particular, for functions that are p-valent in
follows that, for functions fez) in R (or L),
mean (in various senses of the word).
2~ ~ II (e ) I~ de ~ 1 i6
(7)
b
'lJZ
p ±
Y p2_1
=
lR lying above the w-plane such that the total angular measure of open arcs lying on lR over an arbitrary circle Iwl = p, where p> 0, does not ex ceed 21Tp. The function w = fez) is said to be p-valent in mean with respect to Riemann surface
with equality holding only for
f(z) =
fez) that is regular in the disk Iz\ < 1 is said to be p-valent in mean with respect to circumference in Izi < 1 if it maps the disk Izi < 1 onto a A function w
2"
i'IJz
p~l,
l"ll
=1.
area in the disk Izi < 1 if it maps the disk Izi < 1 onto a Riemann surface such
588
SUPPLEMENT
that the part of its area that lies over an arbitrary disk
Iwl :s p
theorem on the transfinite diameter (Theorem 3, Chapter VII,
Following Hayman, we shall say that a function f(z) that is regular in the
Izl
disk
f(z)
< 1 is weakly p-valent in that disk if, for every p> 0, the equation
w either has exactly p roots in \zl
=
< 1 for every w on the circle
or (2) has fewer than p roots in Iz I < 1 for some w on the circle
Iwl
=
Iwl
=
p
For weakly p-valent functions, Hayman [1951a, 1958] has obtained the follow
p. 1£ p is
ing generalization of Bieberbach's classical bounds on the modulus of a function and its derivative in the class S (see Chapter II, §4).
tion than the property of being p-valent in mean with respect to circumference, but
Let f(z)
= zP
+
being p-valent in mean with respect to area does not imply weak p-valence or vice
C + zP +1
+ ••• denote a function belonging to F'. Then,
p
i
P
ICP+ll~2p
versa. We denote by FO (respectively FDp or F'), where p is a positive integer, p p
Also, for
Izi
_z j( z) that are regular in the disk
P+ Cp+lZ P+l+
Iz\ < 1
rP cp+~z J+~+
... ,
II
and p-valent in mean with respect to circum !;Furthermore, w ~;
that disk.
in the disk
=
,
is regular in the disk
Izi < 1.
f(z) = w o + cpz P + •.. , where c p 1= 0 and P':::: 1
Let B = B
denote its range in the disk
Izi < 1.
f Suppose that the domain B* obtained from B by symmetrization about a straight
line or ray passing through the point
W
Suppose that a function w
o+
=
¢(z)
= W
o
lies in a simply connected domain B o'
<
and maps it onto the domain B
z + ... is regular and univalent in the
o.
Then
ICpl~lc;l·
lar function with the symmetrization results of Palya and Szego. For p> 1, the theorem was proved by Kobori and Abe [19591 This theorem has led to a number of distortion and covering theorems for regu lar functions, in particular, for functions in the classes FO and F'p (see, for ex p
,on~'.
Iwl < 4- P
exactly p times
Jz\ < 1. Equality holds in all these inequalities only for the function
fez) =
zP __ \21"
11
I e 1=1.
(10)
By combining the method of extremal metries that he developed and the re sults of the symmetrization method and by generalizing his own results for the class S [1953a], Jenkins obtained [1955, 1958], for f(z) € F? ' the least upper bound of If(r) I, where 0
< r 2 < 1, for a given value of If( - r 1) I, where r 1 is a fixed number in the interval 0 < r 1 < 1. In particular, we have the theorem: Suppose that f(z) € F?, that 0 < r1:S r 2 < 1, and that () is a real number. Then
1/(_rlei6)I+l/(r~eI6)1~(1~~r~)2
When p = 1, this theorem was obtained by Hayman [1951, 1958] by means of a combination of a bound that he obtained for the inner radius of the range of a regu
prP - 1 (l +r)
~\ I/(z) I ~"
11
f(z) assumes every value in the disk
A powerful tool in the theory of conformal mapping is the following symme
=
p (l +r)
(z) I ~ ~
trization principle. Suppose that a function w
fP - ~1/(z)l~ (l-r)~P'
-
ference (respectively p-valent in mean with respect to area, weakly p-valent) in
(9)
= r, where 0 < r < 1, we have the sharp inequalities
the class of functions of the form
Izi < 1
§ 3) for the case of
nonunivalent mappings 1) has also found applications in this class of equations.
a positive integer, the property of being weakly p-valent is a less stringent condi
disk
589
§2. UNIVALENT FUNCTIONS IN DISK AND ANNULUS
does not exceed TTpp2
If r 1
< r 2' equality holds only for f(z)
holds only for f(z)
=
=
+ (1+ r l)2' 1
z!(1 - e -je z)2. If r 1
= r 2'
equality
z!(1 ± e ie z)2.
As a consequence, we have the following inequality for a function f(z) € F?:
1/(-z)I+I/(z)l~
2r (l +r l ) 1 1 _ ..2\2'
Izl=r
(O
(11)
ample, Hayman [1958]). A generalization, also obtained by Hayman [1951a], of Fekete's familiar -----oMitjuk [I964a, 1965d] has obtained a further sharpening of Hayman's symmetriza tion principle and of his theorem on the transfinite diameter, and has extended them to multiply connected domains. With regard §3, subsection 3° of this supplement.
to
the first question, see §2, subsection 2° and
§2. UNIVALENT FUNCTIONS IN DISK AND AN~ULUS
SUPPLEMENT
590
Equality holds in (11) only in the case of functions fez) = z/(1 ± e i8 z)2, where
z = re i8 • In the case of univalent functions, this result was obtained by Goluzin
f
1) all the functions (that is,
fez) f,
0 for
€ ~ are regular and locally univalent in the disk Iz 1< 1
Izi < 1) and are of the form
[1946c1 without the assertion of uniqueness of the extremal functions.
fez) =z+CgZ~+ ... ;
By applying the same investigating technique Jenkins [19541 found a solution of Gronwall's problem in explicit form. This problem consists in finding, in the class S(c) of functions fez) = z + c z 2 + ••• € O:s
c :s 2,
where 0
2
S with given fixed c 2 = c, where
the least upper bound of the maximum of If(z)! for arbitrary fixed
< r < 1.
2) if fez) € ~ and
Izl < 1
¢ (z) is a fractional-linear transformation of the disk
onto itself, then
Iz\ =r,
f
This maximum m (r, c) is attained by a function with real coeffi
(If (z» - f (If (0)) If' (0) f' (If (0»
= z
+... E ~. A
cients. As r
-~
(1 -
591
,.
S, S, S , the class of functions that are locally univalent and p-valent in the disk 'Z I < 1, and also the Examples of linearly invariant families are the classes
1,
g r)g m (r, c) -+ 4a- exp (2 -
4a-t ),
a= 2 -
(2 -
C)l/ l •
class of all functions that are regular and locally univalent in the disk We note that we can obtain the sharp inequality
for f(z) € F~ from inequality (11) by an elementary procedure. Because of the simple relationship between the classes Fa and Fa, namely, that the relation
over (see Pommerenke [1964]) to linearly invariant families of locally univalent functions. Specifically, a number of inequalities involving only the order a of the family 53, where
p
fez) € F~ implies the relation [f(z)]l/p € F~, we easily obtain from (11) by a suitable choice of branch of the root the following sharp inequality in the class Fa: p
I~
d (~ -1)C~+tl~3.
From this inequality and inequality (9) we obtain, for fez) € Fa, the inequality p
2p9 +p,
As Spencer [19411 has shown, inequality (9) is also valid (and sharp) in the
FDp . However, as follows from an unpublished result of Schaeffer and Spen cer (see Jenkins [1957a], the class F~ includes functions for which 1c 1 > 3. 3 This shows that the result Jenkins obtained for the class F~ does not hold for
class
the entire class F~. We note that the last assertion in Hayman's theorem has been carried over by D
Fp
•
With re
gard to this question, see also Jenkins's article [1957a]. We cite yet another example of the extension of familiar distortion theorems for univalent functions to nonunivalent mappings. We give the following definition: A family ~ of functions fez) is said to be
linearly invariant if it satisfies the following two conditions:
sup
! E~
ICgl
have been obtained. (Here a:::: 1. Furthermore, a is finite if and only if
53 is a
In particular, for a
=
2, we obtain the classical distortion theorems for univalent
functions (see Chapter II, §4). We now stop for certain results of the distortion-theorem type for doubly con nected domains.
with equality holding for p ~ 1 only in the case of the functions (10).
Garabedian and Royden [1954] to the case of functions in the class
(J.=
normal family, and a = 1 if and only if all functions in the family ~ are convex.)
cp+g+ p
1 C p +9 I ~
A
number of results with which we are familiar for univalent functions can be carried (12)
\c 3 1:S 3
1
Iz I < 1.
In analogy with Loewner's parametric representation of univalent functions in a disk, Komatu [19431 gave a parametric representation of functions w = fez) that are regular and univalent in the annulus I onto the domain
Iwl>
< Iz[ < R
and that map that annulus
1 with cut along a Jordan curve. Here, the circle
is mapped onto the circle
Iwl
Izi
=1
= I and f(l) = 1
Goluzin [1951e1 gave a simpler variation of the solution of this rroblem in a different form. Let K (m/M) denote the class of functions w = f(z) that are regu
< Iz I < M, where < r 1 < r 2 < M, map the
> 0,
lar and univalent in the annulus m
m
that annulus, and that, for m
circle
of the image of the circle
Izl
fez) € K(m/M) for which f(l)
= =
r 2'
that do not vanish in
Izl
= r 1 into the interior
Let K (m/M, 1/1) denote the class of functions
1 and m:s 1 :sM. Following the method indicated
in the article by Goluzin cited above, Li En-pir [1953] considered the problem of
592
§2. UNIVALENT FUNCTIONS IN DISK AND ANNULUS
SUPPLEMENT
onto the unit disk with cut along the radius connecting the origin with the point
generalizing Loewner's parametric representation for functions in the class
K(m/M, 1/1) that map the annulus m < Izi < M onto the domain obtained from the w-plane by deleting two Jordan curves that terminate respectively at the points 0 and
00.
- f(b). When b
metric representation that he obtained makes it possible to obtain a number of in equalities in the class K (m/M) and its subclasses with a complete clarification of the question of extremal functions. As an example, we give the following re =
fl(z) and w
map the annulus m
f 2(z) denote functions in the class K(m/M, 1/1) that
=
< Iz I < M onto the entire w-plane with cuts along a segment of
the real axis with endpoint at w
=
0 and along the ray lying on the real axis ex
f 2 (z)
on the nega
tive half. For the functions fl(z) and f 2 (z), we have explicit expressions (see subsection 4° of this supplement). We have the following theorem (Lebedev
§ 2,
For fez) € K(m/M, 1/1) and Izi
< qo < Po < 1.
where 0
Let
J
r: q 0 :s z :s Po'
denote the class of functions w = fez) that are
regular and univalent in B and map B into the unit disk in such a way that the point z = 0 is mapped into w = 0 and the circle boundary of the image. Let the condition
1['(0)\
Je
Iz I = 1
is mapped into the outer
denote the subclass of functions in
J
that satisfy
= c, where c is a fixed positive number.
Tamrazov [I96Sc] posed and solved the problem of determining the quantities
Pf= sUPz€r If(z)1 and qf = infZ€r If(z)l in the class Je for various positive values of c. Let f o(z) denote a function in the class J that maps B onto the unit disk with cut along a circular arc with center at the coordinate origin and
=
the classes
with equality holding in the first (second) inequality only for the function (f (z )).
(13)
f l (z)
2
Inequalities (13) had been obtained earlier [1953] by Li En-pir without the
< Iz I < 1
For c € (0, 1], the only extremal functions are functions of the form fez)
=
where
iEl
disk Iwl
= 1 and w = /l(z) = cz + .•• maps the disk Izi < 1 univalently onto the < 1 with cut along a segment of positive radius. For c = 1, this cut is =
1. For c € (1, co), the qualitative nature of the ex
tremal mappings is different. The solution of the problem in this case depends on
and that map that annulus onto a domain contained
extensive use of the theory of quadratic differentials, the general theorem on Jen
=
in such a way that the circle
Izl
=
1 is mapped onto the circle
Iwl = 1. Let F 0 denote the class of functions in F that satisfy the condition fez) i 0 in B. Duren and Schiffer [I962] developed the method of variations in the class F and applied it to solve certain extremal problems in that class. Duren [1963] ap plied that method to the problem of maximizing and minimizing the quantity If'(b) I
< b < lout
o
consists only of functions of the form
fez) that are regular and univalent
Let F denote the class of functions w
<1
Jeo
f/l(-Z) in the case of the first problem and of the form fez) = f/l(Z) in the second,
!'.' constricted to the point w
assertion of uniqueness of the extremal functions. in the annulus B: r
Je
are empty and the class fez) = cfo(z), where lei = 1.
r, where m < r < M,
If1(- r) I ~ If(z) I ~ If~ (- r) I,
for fixed b in r
0 and is bounded by the unit circle and the radial cut
=
midpoint on the positive axis. Suppose that If~(o)1 = co' We know that, for c>c '
[I955c, 1955d]):
in the disk Iwl
b * (r), the extremal function changes its nature: the radial cut
Let B denote a doubly connected domain in the z-plane that includes the point z
tending in the opposite direction from the coordinate origin. Suppose that for f/z) the finite cut lies on the positive half of the real axis and for
=
forks at one end.
Lebedev [I955c] gave the complete solution to this problem. The para
sult. Let w
S93
of all functions in the class F o' The minimization
kins's coefficients, and Tamrazov's supplement [I96Sc] to that theorem. For arbi trary fixed c €
CI, co),
the maximum P f in the class
Je
is attained by all the func
= ffe(z), iEl = 1, where the functions ceiez + "', - &(c):s e ~ &(c), constitute a one-parameter family of mappings one of which (namely, w = fo(z)) is symmetric about the real axis and
tions (and only by these functions) fez)
w
=
fe(z)
=
for two of which the family is degenerate along one of the three arcs constituting the inner boundary of the image. The problem of minimizing qf in the classes
J e
problem is solved for all b € (r, 1) by mapping the domain B onto the unit disk
CI < c < co)
with cut along the radius connecting the origin and the point feb). The maximiza
complete analysis of the form of the extremal mappings and the question of their
tion problem was solved by Duren for every b :: b *(r) (the exact value of b *(r)
uniqueness in the problem of Duren mentioned above (see Tamrazov [196Sc]).
was also determined): the extremal mapping is the one that maps the domain B
has an analogous solution. By the same procedure, we can make a
In the investigation of questions associated in an essential way with classes
;;:-
594
SUPPLEMENT
""'";~
... __ --
....
..
...........-.r
__ ''!'=
_ '....
§2. UNIVALENT FUNCTIONS IN DISK AND ANNULUS
::.
595
.\
of mappings of a variable doubly connecred domain that are not compact in them
The extremal property of Koebe's function that we have mentioned leads to
selves, rhe basic method is clarification of the existence of finite bounds for cer
bounds on the area a(r) of the image of the disk Izj
tain functionals in these classes. A number of extremal problems of this type for
mapping by functions in the class S and its subclasses S(k), to bounds on the
bounded mappings are solved in the papers by Tamrazov [1962, 1962a, 1962b,
mean modulus of the function, and to other bounds (d. the articles mentioned
1963, 1965b]. In these articles bounds have been found for the values of the func
above). For example, in the class
5(2)
tionals in question that depend only on the modulus of the doubly connected domain. 2
lus. Let ~ denote the class of regular univalent mappings w
=
fez) of the annulus
B: r < Izl < 1 for which the bounded component of the complement of the domain Bf with respect to the entire plane includes the points w = 0, 1 and contains the image of the circle Izi = r. Let Pf denote the distance between the images of the circles
Izi
of finding
Izi = inf 1zl =l lf (z)\ rand
=
1 under the mapping in question. Whereas the problem in the class ~ can be solved in a simple manner by the
Bazilevic obtained the inequality
1+r 2
a (r) ~ 7tr (1 _ r 4)2
We present the following results dealing with unbounded mappings of an annu
:S r, where 0 < r < 1, under
+ c (r).
This inequality deviates from the sharp inequality by no more than c (r), which
approaches zero as r --> 1. For the mean modulus of a function fez) E proved the inequality
21t
in o ~
I/(rei'f') I dep
5, he
< 1 r r + 0.55,
2
method of circular symmetrization (the solution of this problem also follows from
which deviates from the sharp inequality by no more than an additional term.
the familiar results of Teichmiiller [1938]), solution of the problem of minimizing
A number of res ults obtained by Bazilevic in 1958 (d. his article [I961])
belong to the same class of problems.
Pf in the class ~ required the application of variational methods (Tamrazov [1965c]). Specifically, we have the following theorem:
These results are closely connected with the problem of finding bounds for
Suppose that a function w = g (z) maps B univalently onto the entire w-plane with cuts
-00
< w :S - t (where t = t{r) > 0) and O:S w :S 1. In the class
have P/2- t with equality holding if and only if fez) where
If I =
=
g(fZ) or fez)
=
~, we
Icnl<
1- g(fZ),
1.
which was not improved until 1964 (cf.
2°. Geometric properties of a univalent conformal mapping. Among the results obtained under this heading a significant place is occupied by covering theorems. The distortion theorems mentioned at the beginning of subsection 1° of
§ 2 of
this
supplement have led to a number of such results. Specifically, Bazilevic [1951] obtained the following sharp bound on the linear measure of a covering of the circle Iwl = P by the image B (r) of a disk of the form in the class
the coefficients. Thus, Bazilevic [1951] obtained for fez) E S the inequality
Izl < r < 1
under functions
fn+1.51,
§ 2,
subsection 3° of this supplement).
With regard to bounded functions, Bazilevic [1959] obtained bounds on the linear measure of a covering of the circle Iwl
=
P by the image of the disk
Iz I :S
r < 1, the area of that image, and also the mean modulus and its square in the class SM" We mention a few results dealing with a familiar class of problems on cover ing of segments and areas.
5:
If a function fez) E
5, then, for arbitrary
r
in the interval
0:S r < 1 and arbi
bitrary x ~ e 17 /e r, the intersection of the circle [wi = P with the domain B(r) has linear measure not exceeding the intersection of that circle with the domain B*(r) corresponding to the function f*(z) = z/(1 -
f
z)2, where
It: I =
1.
This result was obtained independently, with the method of areas, by Lebede v and Milin [1951] for p 2
> (2/v'3h/(1 - r 2 ).
Jenkins [1953] posed and solved the problem of maximizing the linear meas = p, where X < p < 1, that a func 5 does not assume in the disk Izi < 1. Specifically,
ure l/p) of the set of values on the circle Iwl tion w = fez) in the class
with the aid of Lindelof's principle and the principle of circular symmetrization, one can easily show that the function maximizing l/p) where
X < P < 1,
in the
class S is uniquely determined up to a rotation. One of these functions, which we denote by w = f o (z), maps the disk Iz I < 1 onto the entire w-plane from which an arc of the circle Iwl = p, symmetrically located about the real axis and
> __ ...
597
§2. UNIVALENT FUNCTIONS IN DISK AND ANNULUS SUPPLEMENT
596
intersecting its positive half, and the ray p find if (p), we obtain for f(z) €
o
:s w :s
"1/4 I . solving " / " ;Szego's ' Associated with the familiar theorem on V problem on
have been deleted. When we
00
the covering of n segments for the class S (d. p.174) is the following result of
S the inequality
If (p) ~ 2p arc cos (8
Xia Dao-xing [19581: let K p (where p is a fixed number ~ 1) denote a family of
Vil -
8p -
convex domains B in the w-plane that have the following properties: each domain
1).
B includes the point w = 0, and its interior conformal radius about w = 0 is equal
Sato [1955, 1955a] obtained the solution of the analogous problem for the
to 1; for every boundary point of the domain B, there exists a circle of radius p
class SM' where M> 1, b,y the same method. Goodman and Reich [1955] have ob
that passes through that point and encircles the domain B. Let w k ' for k= 1,'''' n,
tained a strengthening of Jenkins's result for a subclass of functions in S.
denote the boundary points of the domain B €
[1954] generalized the result of Jenkins given above to the case of functions that
Tn (p) = min
are regular and univalent in the annulus r < Iz I < 1 and that map it onto domains contained in circle
Iwl
=
Iwl > r
in such a way that the circle
Izj
B
= r is mapped into the
••••
max I Wk I.
wn ) k
~
iLdomain bounded by a regular n-sided circular polygon whose sides are arcs of a
transfinite diameter of the complement of the image of the domain 1{; 1 > 1 under a function in the class 2 is equal to the transfinite diameter of the continuum con
l:' circle of radius p. This enables us easily to determine the quantity T (pL If we
{
n
ri let p approach (
sisting of three segments connecting the coordinate origin with the points repre
00,
A
we get the solution of Szego's problem for the entire class S.
By the same method, Xia Dao-xing [1956] obtained certain theorems on the covering of intervals under a univalent conformal mapping of an annulus. Tamra
senting the cube roots of 4 (see also articles by Garabedian and Schiffer [1955],
zov [1965] obtained some very general theorems on the covering of curves under a
Reich and Schiffer [1964].
univalent conformal mapping that generalize the familiar results on the covering
However, the four intervals connecting the origin with the points representing the four fourth roots of 4 no longer form an extremal continuum maximizing the
of segments and provide us with more precise information. For example, he ob tained a precision of Rengel's theorem [1933] on the covering of n segments in
fourth transfinite diameter in the family of all continua whose outer conformal
the case of functions that are meromorphic and univalent in a disk, and the analo
radius is equal to 1 (Garabedian and Schiffer [1955]).
gous result for an annulus.
In this connection, we give the following result: by assuming symmetrization of a special form, Szego [1955] showed in a very simple way that the outer con
gion maximizing the conformal radius in the family of all simply connected regions
given and which issue from the coordinate origin at angles each equal to the next
containing the coordinate origin but not containing
is maximized when all these segments have the same length. It follows from this and from Lindelof's principle that, if the function w = f(z) €
=
tions associated with elliptic or automorphic functions. Koebe's theorem on the covering of the disk
f(z), we have
disk ... an I ~
I
Izi < 1
under a function w
=
f(z) €
Iwl < ~
by the image of the
S has been sharpened by Jenkins
[1960] to the case of the class SR. Let Bf denote the image of the disk
4 '
with equality holding for a function of the form f(z)
or a given system of points
case of a family of fundamental domains of groups of fractional-linear transforma
0 at equal
angles and belonging to the boundary of the image of the disk \ z I < 1 under the
I ala~
00
a l , · · · , am (for m ~ 1). This problem was solved by Gel/fer [1958, 1960] for the
S*, then, for n arbi
trary points aI' a 2 , · · · , a" lying on n rays issuing from the point w
Closely associated with questions on the covering of segments is the well known problem of Lavrent'ev (see Chapter IV, §4) on the determination of the re
formal radius of a system formed by n segments the product of whose lengths is
=
min
E Kp (W! •
,~. By use of the extremal metric, one can show that this minimum is attained by a
r.
Goluzin showed (see p. 144, Theorem 2) that the maximum value of the third
mapping w
K p that lie respectively on n arbi
trary rays issuing from w = 0 at equal angles:
By using the symmetrization results for doubly connected domains, Kubo
=
z/(1 + (Zn)2In
where 1(1
=
1.
n
f(z) and define D = fESR Bf" Applying the phrase "gen eral theorem on coefficients", Jenkins found the set D determining the boundary under the mapping w
,
Izl < 1
=
598
599
§2. UNIVALENT FUNCTIONS IN DISK AND ANNULUS
SUPPLEMENT
Rk
onto a convex domain and that such ~ function maps a
exceeding
basis of Lavrent' ev's results in the above-mentioned problem of maximizing the
disk with the same center but of radius no greater than R s = tanh (TTI4) onto a starlike domain (Chapter IV, §5) have been supplemented by Aleksandrov [1958a,
conformal radius in their simple particular case, Kuz'mina [1962] determined in explicit form the set D 1
=
nIE S [B I UBI]' where BI
1959, 1960] as follows:
is the domain symmetric to
Every disk of radius P:S 2 -
BI about the real axis. It follows from the form of the boundary functions of the
tcfl < 1,
set D 1 (they have real coefficients) that the sets D 1 and D coincide.
5 in the disk Izi
:s r,
where 0
Icfl < 1,
obtained by the Romanian mathematician Mocanu [1957, 1958]. Fel'dman [1963]
the mapping w
=
nIE S b I'
where b I denotes the image of the domain b under
fez), and he also solved the analogous problems for the class 5'.
Iz I < r,
where 0
< r < 1.
IBI
denote the family of doubly connected domains in the z-plane that contain the
point z
=
0, have the circle
Iz I =
1 as one of the ir boundary components, and
satisfy the following condition: under a univalent mapping of the domain B onto the disk Iwl
< 1 with cut along a segment of positive radius that maps the unit
circle and the coordinate origin into themselves, the point z the point w
=
=
1 is mapped into
1. Tamrazov [1962b] determined explicitly the majorant domain con
taining an interior boundary of an arbitrary domain in the family of domains in
IBI
with given fixed Riemannian modulus and pointed out a number of properties of
cf,
where
the non-Euclidean radius of which does not exceed TT/2 is mapped by an
Goluzin (see Chapter IV, § 5) introduced the concept of "generalized star
~:: likeness". The following theorem of Aleksandrov [1960] deals with his results on ~. the limits of generalized starlikeness in the class 5: ~.
Every non-Euclidean disk with non-Euclidean center at a point
Icf I < 1,
we give one result dealing with problems of this nature for an annulus. Let
where
result is sharp.
~
Cernikov [1962] found the set U IES bI in the case in which b is a disk
cf,
arbitrary function in the class 5 onto a domain that is starlike about f(cf). This
obtained more general results in the same direction. For example, for a domain b
domains U IE S b I and
with center at the point
is mapped by an arbitrary function in the class 5 onto a convex domain.
Every non-Euclidean disk with non-Euclidean center at a point
< r < I, have been
contained in the dis k Iz I < 1 and bounded by a closed Jordan curve defined by parametric equations in polar coordinates, he found in certain particular cases the
, V3 + Icf 12
The bound on p is sharp.
A number of generalizations of Koebe's theorem on the sharp bound for the modulus of the function fez) €
=
2-
J3
functions of that set in terms of the theory of quadratic differentials. On the
the non-Euclidean radius
0
cf,
where
f which does not exceed nTT/2 is mapped by an
.. arbitrary function in the class 5 onto a domain of the form D)f(cf)]. This result is sharp. In particular, every disk of radius not exceeding R nS = tanh (nTT/4) with center at the point z = 0 is mapped by every function in the class 5 onto the domain D n (0). Of special interest is the problem of the change in the curvature of the level curves under a univalent conformal mapping. We present the basic results that have been obtained in this direction. Miro~nicenko [1951] first showed that the following inequality holds, giving
Kr of a level curve L r (the image of in the class 5 for 2 - VI :s r < 1:
the greatest lower bound of the curvature
these domains. The behavior of the level curves (the images of concentric circles
Iz I = r < 1)
also shows clearly the degree of distortion under a univalent conformal mapping of a disk. This question has been well studied for sufficiently small values of r. For example, limits of convexity and starlikeness have been found (see Chapter IV, §5). It is also natural to study the behavior of functions in the class 5 on circles
the disk Izi
= r)
K
r~
1-4r+rl r
(11-r' +r)2
He also showed that the inequality
2k k Kr~ 1-2 (k+ 1) r +r (l +l)2/k r
(l-r7i)l
with a displaced center. The well-known results of Nevanlinna and Grunsky asserting that an arbitrary function in the class 5 maps a disk with center at the point z
.
=
gives the greatest lower bound in the class S( k)
,
where k 2 2, for
0 of radius not
rrni\lpr~;tv nf
W~~hinafnn
r i"-r~rv
600
SUPPLEMENT
J
k + 1 - k 2~2k :::; ,k < 1. These bounds are attained by the functions f(z) z/(l + z)2 and f(z) = z/(l + zk)2Ik, respectively.
601
§2. UNIVALENT FUNCTIONS IN DISK AND AN,NULUS
=
"more regular" in the geometrical sense than the curve L
r2
. However, Bazilevic Izl < 1 by a func
Beginning with Loewner's parametric representation, Koricki'l [1960] showed that these inequalities are valid in the entire unit disk.
and Koricki1 [19531 showed that, under a mapping of the disk
Up to the present time, the least upper bound of the curvature of the level curves in the class 5 has not been found.
the number of points of destruction of its starlikeness (points on the level curve
By using Theorem 4-of §3 of Chapter IV, Korickil [1957, 1960] found the fol lowing least upper bound for the curvature K p (the image of the circle for functions in the class k(k), where k ~ 1:
K
It;; 1= p> 1)
In certain subclasses of 5 and
k,
(t;;k +
~
S(k); where
k ~ 1, and in the
Izi < 1
in a larger special class of functions that are regular in the disk
and in the class 5(k)* , for k
~ 2;
cally with increasing '; that is, if , 1 <'2' it may happen that the level curve L'l
curves as ,
--->
Koricki'i [1955] did this by the same
method. Korickil [19551 also obtained the greatest lower and least upper bounds of K p in the ~lass k(k), for k> 1. Aleksandrov and Ceroikov [1963] first found the least upper bound of the curvature of the level curves in the class 5( k)*. For k = 1, this least upper bound
ctkR (r)
by the function f(z)
=
z/(l + zk)2!k; for
'0:::;' < 1,
<, :::; '0'
(ctk -
is not convex if , is sufficiently close to unity.
1£ an arc of a level curve L r of a function f(z) € 5 is contained in the annulus
it is necessarily starlike, but the same is not true of any of the wider annuli
(cts-e)R(r)O.
[(1 +zk)P.l (1_z'i)p.ap1k '
Bazilevi~ and Koricki'l found lower and upper bounds for the constants a k
< Ill'
11
2
< 1,
III + 11
2
=
1).
The investigation of the behavior of the level curves as we approach the boundary of the unit disk is a difficult problem because the behavior of the level curves as , --;. 1 may prove to be very complic~ted. For example, it is natural to '1
1< R(r), I z 1= r< 1,
it is attained by the function
where Ill' 112' and '0 depend only on k and , (with 0
assume that the level curve L
5 for which some arc of a level curve
e) R(r)< I/(z) 1< R(r), s> 0,
rJ.J«r)
it is attained
~, Izl=r
that is contained in the wider annulus
z
I(z)
< II (z) I < R(r), R(r)
it is convex, but there are functions f(z) €
is attained for all 0 <, < 1 by the function f(z) = zl(r + z)2. For k ~ 2, this bound has a different form for different values of ,: for 0
1. Specifically, Bazilevic and Korickil [1962] proved the existence
of absolute constants a k and as such that:
1£ an arc of a level curve L r of a function f(z) € 5 is contained in the annulus
by using the integral representations of these classes.
of the curvature of the level curves by the method of embedding of the class
*
in a specified direction) can change nonmonotoni
there will necessarily be a certain amount of regularity in the behavior of its level
In 1952, Bazilevic first found in the class 5 * the greatest lower bound 5
= ,
Nonetheless, for sufficiently rapid growth in the absolute value of a function,
1) 2! kit;;.
the problem has been solved completely.
the curvature of the level curves in the classes
\zl
curves can occur even for bounded functions in the class 5.
For example, Zmorovic [1952] obtained greatest lower and least upper bounds for class
moves around the circle
the level curve L r2 does. Furthermore, this paradoxical behavior of the level
p[p2k+2(k-l)pk+l]
P --= (pk -1) (pk + 1)21k
=
at which the direction of rotation of the radius vector changes as the point z
has more points of inflection or more points of destruction of starlikeness than
&
which is attained by the function F (t;;)
tion in the class 5, the number of points of inflection of the level curve Land r
embedded in the curve L
r2
(where,
1
< '2)
is
and as' They proved analogous theorems for the class
k.
With regard to func
tions in the class 5, in connection with the problem of finding the exact values of the constants a k and as' they posed the question: in what parts of the annu lus ,/(r + ,)2
< If(z)! < ,/(1 -
,)2 is an arbitrary arc of a level curve L
or starlike with respect to the origin for all f(z) €
5.
r
convex
~_~~~"
__ - __' _~='''~
,,_
_=-__ ~~
0 -
==
.__ _
602
--0._-'
-, - ._-- ---.'-.
~._=
-
~~"'==-- '--c- - -'-__ C'~""~;;~-;::~:=~;';=';'f!:
.------,-.~.
S(k),
where k
=
2, 3,···. With re
Let lR(B) denote the class o[ [unctions w
< Iz I < R,
regular in the annulus B: 1 domain
r !3(r)
The complete solution of the problem posed by Bazilevic and Korickil with regard to starlikeness of arcs of level curves was obtained by Aleksandrov and
trary function [(z) € S if and only if L r lies in the annulus
13 (r) < (f(z) I< ~s (r) 13 (r)
> lim r- 1 (3 s (r) = 1,
-
(r)
[(z) that are single-valued and
Iwl
=
Iwl = 1. Suppose 1, maps the annulus B univalently
I[ [(z) € lR(B) and i[ d f is the shortest distance [rom the origin to the inner B f' then d ? P with equality holding i[ and only i[ [(z) =
f[*(T/Z, R),
where
If \ = IT/I
f
=
1.
Kubo obtained an analogous inequality for bounded functions in lR (B): 1[(z)1 < M, where M? R, for z € B. For univalent functions, this last result had already been shown by Grotzsch [1928] and Komatu [19431. The method of symmetrization proposed by Szego [1955] for starlike domains has been generalized by Marcus [1964] for arbitrary sets. Let subset of the z-plane that does not include the point
R. (r) < If(z) 1< R (r).
a s (r) < lim r-1 a s (r) -
R), where [*(1, R) > 1 with cut w? P.
= [*(z,
., boundary o[ the domain
values a s (r) and (3 s (r)
have been found such that every arc of a level curve L r is starlike for an arbi
=
that map B onto a set B f contained in the
1, and that map the circle Izi = 1 onto the circle
onto the domain
for all functions in the class S.
< r < 1,
Iwl>
~ that a [unction w
where (3 is independent of r, in which the arcs of the level curves are starlike
rJ.s
603
transfinite diameter, Kubo [1958] obtained a number of theorems for functions that
[1963] showed that there is no interior annulus of starlikeness, that is, no annulus of the form
or in the annulus
~""~---~---.-------
are regular or meromorphic in an annulus. We present one of these.
gard to the behavior of the level curves for small values of 1[(z)l, Stepanova
Popov [1965]: For every r such that tanh (TT/4)
_"c~==~-~="",·~~_,_,="'''''''''~~_~=~~~''~"_,,·,_c~~
By applying the ideas presented in Hayman's article [1951a] to the hyperbolic
Stepanova [1963] obtained analogous bounds for the annulus of starlikeness
Here, {3 s (r)
'-._C"_.'_ -- c,__ ,- c=, --," -,·,. ,c__,-.C"
§2. UNIVALENT FUNCTIONS IN DISK AND ANNULUS
SUPPLEMENT
and the annulus of convexity for the classes
!:".~,-"",.-,-,,-=-
in
n.
For an arbitrary disk
Iz -
trary ¢ in the interval 0:5 ¢
= a s = 0.109'" .
Stepanova [1965] established an analogous relationship between starlikeness
< p,
Lp(q:»=
Together with a more profound study of the geometrical properties of conformal
and let Zo denote a point
where p> 0, contained in
n
and for arbi
let us consider the set E of points z €
such that Iz - zol = r > p and arg(z - zo)
of arcs of the level curves and the values of 1f'(z)l. mapping by univalent functions, a number of results of the covering-theorem type
zol
< 2TT,
00
n denote an open
J\ dr r'
=
n
¢. Let us define
R(cp)=pexpLp(cp).
E
-Furthermore, for arbitrary n
=
2, 3," . , we define
have been established for general classes of functions. For functions that are regular in the disk Izi eralization of Koebe's theorem on Let [(z)
=
< 1,
n-I
~ L~n)(cp)=n "'"
there is the following gen
I
X' .
k=O
n-I
z + c 2 z 2 + ••• denote a member o[ lR and let d f denote the linear
measure o[ the set o[ all positive p [or which the circle
Iwl =
the image B o[ the disk Izi < 1 under the mapping w f equality holding only [or [unctions o[ the [orm
[(z). Then d ?
Z
f(z)='1_~Z)9
=
f
X',
II
RI/n
(q:>
+ 2:k) =
p exp
L~n) (q:».
Obviously, R(n) (¢) is independent of p. We shall refer to the transformation of the set
,lsl=I.
(q:» =
21tk) ' +n
k=O
with
This result was obtained by Hayman [1951a] by use of the geometric proper ties of the transfinite diameter.
R(n)
p is contained in
( Lp cP
n
and 0:5 ¢
into the set of points z such that z - Zo
< 2TT,
as the Sn transformation of the set
=
re i ¢, where 0:5 r
< R(n)(¢)
n with center at the point z O.
Under the Sn -transformation, an arbitrary domain is mapped either onto the .plane or onto a domain that is starlike about the point
Z0
and possesses for n? 2,
604
SUPPLEMENT
§2. UNIVALENT FUNCTIONS IN DISK AND ANNULUS
605
,
n-fold symmetry of rotation about that point. As Marcus has shown, such symme trization does not increase the interior radius of the domain. By using this last result, Marcus obtained the following theorem:
As a supplement to the results of Goluzin and Bermant regarding a covering performed by regular functions in a disk (see Chapter IV, §6), we present the fol lowing theorem (Jenkins [1951]).
Suppose that f(z) = z + c z2 ••. is a member of ~. Suppose that the S 2
n
transformation with center at the point w R(II)
(If)~
=0
-:/1 V 4'
is applied to the domain Br Then,
In connection with these results, we have the following theorem of Antonjuk
with equality holding for a function of the form
Ie I=
[1958] for functions that are regular in an annulus.
1.
Suppose that the function w
This theorem shows that at least one ray of an arbitrary system of n rays issuing from the point w
=
If f(z) E ~, then in the image B f of the disk Izl < 1 under the mapping = f(z) there exist n straight line segments issuing from w = 0 at equal angles
the sum of the lengths of which is arbitrarily close to n/r*, where p* = 1"* 2 is the area of the preimage of the star of the domain B!,
0 ~ If< 21t,
j(z)= (l-:ZII)I/II ,
w
0 at equal angles to each other intersects the domain
and satisfies the conditions
=
f(z) is regular in the annulus B: 1_ < Izi
If(z)1 2:
1 and
(Ill 17 i) Ie (f'(z)/ f(z» dz ?: 1, where C
is a contour in B that is not homologous to O. Let B; denote the star of a finite doubly connected Riemann surface Bf onto which the annulus B is mapped by the
B f along a set whose linear measure is no less than \11/4. A number of results characterizing the geometric properties of conformal map
function w
=
f(z) with respect to a system of rays issuing from the point w
=
0
ping were obtained by Mitjuk [1964a, 1965a-d] by successive application of sym
(the existence of such a star is proved). We have the following inequality con
metrization methods. With regard to this question, see also § 3, subsection 3° of
necting the area P* of the star
this supplement. We point out that, by confining himself to the case of a disk,
B;
with the area p* of its preimage:
(p* + 1t) (P* +1t)~1t2R~
Mitjuk obtained [1965a] the following sharpening of Hayman's symmetrization principle: if f(z)
=
w
o+
cpz P
+ ••• E
at,
where p
2: 1, then, in the notation used
n IZkI Pk,
k~
Izi < 1
and p , p , ' ' ' , Pk"" 1
2
z, where
kI =
1.
A number of results dealing with the problem of finding a lower bound for the area of the image of a disk and an annulus and also of the star of that image have been obtained by Mitjuk [1961b, 1965a-d).
1
where z l' z 2' • " , Z k' ••• are the nonzero roots of the equation f(z) the disk
= f
Hazalija [1958] had already obtained in 1951 the inequality p*2. 17(R2 - 1).
in the theorem of subsection lC.>,
ICpl~IC;1
with equality holding only for a function of the form f(z)
o = 0 in are their multiplicities. Mitjuk [1965d] W
also obtained an analogous strengthening of Hayman's theorem on ~ that we have given and of Marcus's theorem.
In connection with the above-mentioned result of Kubo [1958], we mention that Mitjuk [I965b] obtained a more general covering theorem for functions that are regular in an annulus. Specifically, he considered a broader class of functions and studied the multiplicity of the covering produced by them. We mention also that Mitjuk [1965c, d] extended Hayman's theorem of subsection 1° to the case of functions that are weakly p-valent in an annulus. For functions of this last class, he obtained a theorem on the linear measure of the covering of arcs of a circle, thus generalizing the theorem of Kubo [1954] mentioned above.
3°. Bounds on the coefficients of univalent functions. The problem of the coefficients in the expansions of univalent functions has played a significant role in recent years in the general turn taken by the geometric theory of functions. As we know, for functions f(z) mining for every n
Ic 2' •••
, Cn
I
=
z + c 2z2 + ••. in the class S, it consists in deter
2. 2 the ranges of values of the system of coefficients
of functions in that class. A particular case of this problem is the
problem of finding sharp bounds for the coefficients. Success in this search would lead to proof or refutation of a well-known conjecture of Bieberbach.
In recent years, a number of problems associated with this class of questions have been solved. Schaeffer and Spencer [1950] used their own form of the variational method to determine explicitly the range of values V 3 of the system of coefficients
Ie 2' c 3 1
606
SUPPLEMENT
§2. UNIVALENT FUNCTIONS IN DISK AND ANNULUS
in the class S. Although V 3 is a body in 4-dimensional real Euclidean space, the
2
IlXnl~ n+l'
property of symmetry with respect to rotation admits a simple description of this set if we determine its three-dimensional cross-sections for which, for example, = 0 (c = a + ib , where a k and b k are real). The boundary of the k k k 3 range of values of each of the systems {a 2, a , b J, {a 2, b 2, a31 is expressed in 3 3 explicit form in terms of elementary functions.
b 2 = 0 or b
In the class 1 of functions F(,)
=
X + e -6. of
,-3
l
a
3
~
1
This proves that the bound found by Goluzin [1949c] for the coefficient
for odd functions in the class
1 is sharp throughout that class.
In 1955, Garabedian and Schiffer [1955a] first proved the validity of Bieber bach's conjecture for the fourth coefficient. This proof uses both the method of variations and Loewner's parametric method. The ranges of values of the systems of initial coefficients in classes of
(1)
Let M denote the class of functions f(z) meromorphic and univalent in the disk
Izi < 1
" function fez)
that satisfy the conditions f(o) =
z + 1~-2 J-
C .
J
zi E M, and if
'+ a o + a 1'-1 +"', Garabedian and
Schiffer [1955] obtained by the method of variations the sharp inequality
607
C .
= =
0 and ['(0) 0 for J'
J
=
1. If
n>
2, if the
< (n + 1)/2, then
2 n-l'
ICnl~
(2)
Equality holds in (1) and (2) only for the functions F(,; n + 1, a) and
F- 1 (z-l; n - 1, a) respectively, where F(,; n, a)
=,
(1 + e ina ,-n)2/n.
Inequalities (1) and (2) were obtained by Goluzin for the classes
1 0 and S
t respectively as special cases of results that are valid for p-valent functions (see Chapter XI, § 6). If n > 0 and 00
bounded functions in the classes
S(k)
and
l(k)
[1957] by means of Loewner's parametric representation method. Thus, in the class S<j;) of functions fez) = z + c k +1z k +1 + c2k+1z2k+1 +"', where c Vk +1 = a vk +1 + ib vk +1 for k = 1, 2,"" Bazilevic determined in explicit form the range of values of the system
{Ie k +1 1, Ie 2k +1 1J and,
b 2k +1 = 0, the range of values of the system
under the assumption that b k +1
la k +1' a2k +11
=
and he exhibited all
their boundary functions. Bazilevic posed and solved the same problems for the
1<,/:) and also for the classes of functions inverse to the functions in S~k) and 1<,/:) , respectively.
class
Gharzynski and Janowski [1959] determined the range of values of the system
Ic 2'
C 31
in the class SM by a method analogous to that used by Schaeffer and
Spencer [1950]. Interesting bounds on the coefficients of univalent functions, bounds which from a new standpoint explain how the vanishing of some of the initial coefficients affects the growth of the remaining ones, were obtained by Jenkins [1906b], by extending his general theorem on coefficients to the case of quadratic differentials of a suitable form [1960a]. We present some of his results.
'+
Let n denote a positive integer and let F (,) denote the function F (,)
1~=0 (a/,i) E 1, where a i
=
'\1 f}.J F(Q=C+ i.J CJ
were investigated by Bazilevic
0 for j ~ (n - 1)/2. Then,
=
E 1;,
J=n then, for arbitrary real
t/J,
we have the sharp inequality
m{- e-i"'iClgn+l + oe-"'l(l.n - ~ ne-i"'l(l.~} 3
~ 8(n+l)
oi
1
4(n+l)
ai log48 + n+l' 1
For n = 0, inequality (3) remains valid for all F(,) E extremal functions are indicated.
O~o~4.
(3)
1 when 0 < 0 ~ 4. All the
A consequence of inequality (3) for functions of the type indicated is the
inequality
l~n+ll~ n~l
[1 +2exp (-
2n~2)J.
(4)
From inequality (3) we get an analogous result for functions that are univalent in a disk. In particular, we have the following sharp inequality: Let n denote an integer greater than 2 and let f(z) denote the function f(z) z + 1':"_ c.z i E M. Then J-n J
I C2n_ll
~
nI I [1 +
2 exp (--'- 2
nn2) J.
=
(5)
608
SUPPLEMENT
§2. UNIVALENT FUNCTIONS IN DISK AND
Inequalities (4) and (5) had been obtained earlier by Goluzin [1949cJ for func tions in the classes I(n +1) and s(n -1) respectively.
A necessary and sufficient condition for a function f(z) is regular in the disk
In particular, inequality (4) shows that the function I
I
Fn(~)=~ 1 -r (
which maximizes
la11
and
l
)2/(n+l)
Cn+1
a 21
2 _n
=~+ n+1 ~
I
in the class
all odd-subscript coefficients a 2n +1' n
+...,
Ia n I in that class
ential equations for the function maximizing
is not extremal for
I for which 1 a n 1 > n -I +3 , where 0 is a positive con-
stant, for an infinite sequence of values of n. This function maps the domain "I
At the same time, the inequalities \ r),n
are valid in the class I simple manner for n
I~
n
In
f(z)-~(C)
n = 0, 1, 2, •.. ,
Suppose that f(z)
= ,
+ a
1/' ,2
The only extremal function is the function e -1 F n (e'), where
!
=
F (J') = J' + a + a / J' + • •• € I * for n = 0 1 2 ••• '"
0
1
'"
,
+ a2/
Ie 1=
+ .•. €
I.
Ic 4 :s 4, which shows 1
than Loewner's result proof of the inequality
Ic 3 1:s 1
"
mn
are determined by the expansion
amnzm~n, 1z I< 1, I ~I < 1.
=i- (C' -
2CgCa
+ :~ C:),
C:)-152C:+lgcgl~211Ig+~.
(8)
For the function .
In both cases, the proof is based on an inequality
F(~)
III
I Cgl
4 is obtained by means of
We shall present the first of these proofs of the inequality
g
+ 31 Ca -
so that
simple inequalities that follow from the area theorem.
Ic 4 1:s 4
in a some
(+)
EL
the area theorem yields
for suitably chosen combination of the coefficients c 2' c 3' c 4' of a function in the
Ic 4 1 :s
(7)
I)
that this inequality is actually more elementary
class S. From this inequality, the inequality
ll
I ( Ca-4Cg 3 ata=aal=2 ,
IC,-2(C g-l)(ca-:
I. By using the
3. These authors also obtained [1960a] a geometrical
Ie 4 :s 4.
~ -nIXnl ..i..J
we obtain I) from (7)
In 1960, Charzynski and Schiffer [1960] gave a much simpler proof of the inequality
is that the inequality
= z + c 2z2 + .•. € S. Then, f 2 (z) = jj(z2) € S(2) and in
same idea of proof, Pommerenke [1962a] obtained inequalities (6) for all functions '"
z 2 + •• , that
such that the right-hand member of this in 00
aU=2 C"b
(6)
Specifically, Clunie [1959] proved inequalities (6) in a
2: 1 and for functions F (,)
C2
m,n=O
aaa
*.
z +
n=1
Ix nI
I
+I '
1
=
609
equality (7) is satisfied for it. If in (7) we set xl = l, x 2 = 0, x 3 = 1, x 4 =
x 5 = ••• = 0 and remember that
> 1 onto a domain bounded by a Jordan curve. 2
I
amnxmxfl ~
equality converges, where the coefficients a
2: 1. In connection with this, we note that
Izl < 1
< 1 to be univalent in
be satisfied for every sequence
Clunie [1959a] constructed along the lines of Littlewood's well-known construc tion a function in the class
Izi
~ ..i..J Im,n=
and satisfies for all n the differ
AN~ULUS
I ca -
: c:
g
:
c~ I ~ 4,
I~ ~a. -V4 -
g 1 Cg I ,
(9)
what modified form. The proof rests on Grunsky's familiar condition [1939] for univalence (see also Chapter IV,
§ 2).
We present this result in the special case
of functions that are regular in the disk
1
z 1 < 1: I)
1) The article by Garabedian, Ross and Schiffer [1965], gives an elementary proof of this theorem that is based on the area method.
Without loss of generality, we can assume in the problem that we are con sidering that c 4 2: o. Then, from (8) we have
1) Inequality (8) can also be obtained directly from inequality (12) of Chapter IV, §2, with n = 3 and the same values for x l' Xz, and x3'
610
SUPPLEMENT
611
§2. UNIVALENT FUNCTIONS IN DISK AND ANNULUS "
Ct ~ 21/12 + ; +ffi {2 (C2 -I) (CS -
:
The Grunsky-Nehari inequalities, which give necessary and sufficient condi
5 12 C~ _/2C2}'
C~) +
tions for univalence and boundedness of a function in the disk
and, in view of (9), this leads to the inequality
Ct~21/12+ ~
+
-:a IC2-/IV4-lc212+ffi{152c~-f2c2}'
If we now set
1=2xe
_i.'L
=
O~x~
rp,
2
1,
I
in the interval 0
fl) (b
I)
of functions f(z)
< b I < 1,
Isin(3¢/2)1,
.
For example, for 0
by a function w
bIZ + b 2 z 2 + ... € 5
fl)
2: O. Schiffer and Tammi [1965] examined this
< b i < 1/11,
fo(z) that maps the disk
=
the maximum value of b 4 is attained
Izi < 1
onto the unit disk with cut
< 1, the f I (z) of the disk Iz \ < 1 onto
lying on the negative radius and only by such a function; for 19/34:S b I maximum value of b 4 is attained by a mapping w
=
the unit disk with three radial cuts of equal length making equal angles each with
and, consequently,
the next. In these statements, the constant 1/11 cannot be replaced with a larger
2 + 8X2- -14XS - (8 X2- -4X 3)Y 2+ -----=-X 8 '/-1 Ct
3
constant nor the constant 19/34 with a smaller. With regard to a bound for
ya
3
By taking the maximum of the right-hand member of this expression with tespect to y, we arrive at the inequality
Ct~~+ 8x 2 _ 3
.!.i x 3 + 4 (l-x 2) 6-x
3
48 -IOx-132x2+ I08xs -14x' ~O
O~X~ I,
w =
fo(z)
is extremal for this last problem for some interval 0 =
fI(z) has the same property for all 19/34:S b i
< 1.
~ a and the
This last fact
With regard to bounds on other initial coefficients of the functions in the for
0 ~x~ 1
5, we mention the following results. Duren and Schiffer [1962/63] found infinite quadratic forms S n giving second variations - 8 2 ?Rc n for the coefficients
class
in the expansion of Koebe's function [*(z) (48+86x-8x2)(I-X)2+6xS(l-x)~O,
O~x~
1.
Obviously, this last inequality is valid and equality holds in it only when x This means that 1
=
z/(1 - z)2. These formulas point out
the significant difference between the coefficients c n with even and odd sub =
1.
scripts. Duren and Schiffer were able to show that the quadratic forms 5 n are positive-definite for n:S 9, and this means that Koebe's function gives a strict local extremum for
Ic 4 ~ 4
Ic n I at these
values of n for all the variations in question.
Later, Garabedian, Ross and Schiffer [1965] established an analogous extremal = z
+
2, that is, for a function of the form
j(z)
< bI
the idea in the paper by Charzynski and Schiffer [1960].
or
with equality holding only for a function f(z)
1
in the entire class S(l) (b I)' Schiffer and Tammi showed [1965a] that the function mapping w
'
Ib 4
was obtained by the use of the Grunsky-Nehari conditions and a generalization of
and it will be sufficient to show that
where IE 1= 1.
=
and this led to sharp bounds for b 4 under
tions of this problem. The extremal functions have different forms for different
12} 14 x s +4 S2 3 xy I C2 - II = 2xy, ro{5a 1.1\ T2 C2 C2 = - 3
=
(see sub
problem by using the variational method and they found all possible extremal func
3
2 COS
values of b l we obtain, using the notation y
Ic 21
5
fourth coefficient in the clas s with fixed b
the additional assumption that b 2
c2=2xe iep ,
Iz \ < 1
section 1° of this section) were used by Singh [1962] to obtain a bound for the
C 2z 2
+ •.. € 5 such that
property of Koebe's function for all even n ond variation
z (l-ez)2 ,
=
2m by means of a bound on its sec
8 2 ?Rc 2m' Proof is bas~~ on use of Grunsky's conditions in matrix = /f(z2), where fez) = [*(z) + of(z). Corresponding z/(1 - z2) is the unique matrix lam) of Grunsky's coeffi
form for the odd function f/z) to the function f;(z)
=
cients. Specifically, they proved the following theorem:
For every m
=
1, 2,""
there exists an
Em
> 0 such that if the condition
!JNOI!!\!!'!'
~.J!!;,-:..
r.m-~'i!_,!!~:~~_iZ£PZ::: ........_
...._~
_~~-
P2";~$-~'?~
_",,"~-;.,~-~.~~
_.. _.?"-~:m:'..~~.~;;.~,.
"",,,
=.='~~~~!3.~,!'!'!!._~_!!.!_,1~"'~m!!
,
.. ~_, __...rC!~~'f
"",,"'I'J"'''''
-r,~" .""
SUPPLEMENT
612
613
§2. UNIVALENT FUNCTIONS IN DISK AND ANNULUS
2,~:2Ick - k1 ::; f m is satisfied for f(z) = z + c 2 z 2 + ... € 5, then ~c2m:S 2m with equality holding only for the function [*(z). 2
For odd functions, it is well known that
\ b~n_l I ~ A,
Furthermore, the problem of global proof of Bieberbach's conjecture for the where
in five independent variables. In connection with this last, we point out that
a uniform inequality of the form
::;
6 in the class
5 for
> 1,
A is an absolute constant. It has also been shown that A > 1. Furthermore, Ib2 n -11 ::; 1 for n 2: N is impos sible (see Chapter IV, § 7). The exact value of the constant A is unknown. By using the bound for
sixth coefficient is reduced to the problem of proving a trigonometric inequality Bombieri [1963] had alre~dy proved the inequality ~c 6
n
Iz I ::; r < 1
values of ~c 2 sufficiently close to 2 and Ozawa [1965] used Grunsky's condi
the area of the image of the disk
tions (7) to prove this inequality under the assumption that 0 S c 2 ::; 2.
(see the article by Lebedev and Milin [1951]), Gong Sheng (Kung Sun) [1955] ob
With regard to the question of finding bounds for the coefficients for all
n> 4,
tained the inequality
I bin_ 1 \ < 2.55, n ~ 1.
the best result in this direction that has so far been obtained is the inequality found by Milin [19651:
under a function in the clas s S( 2)
Let us denote by 5 (a) and 5( 2) (a) respectively the subclasses of functions
Ic n I < 1.243n,
> 4.
This result was obtained by investigation of univalent functions based on use of
in th~ classes Sand 5(2) for which (10) holds with fixed a = a, where 0
extremal properties of certain systems of functions.
the plane with an analytic cut. The class 5(2)(a) is defined analogously.
for n
"-
We point out a few results characterizing the asymptotic behavior of the coef
Hayman [1955] showed the close connection between the maximum modulus
n
Bazilevic [1965] showed that the value of a in the class S (a) determines not only the limit relation (10) but also the closeness of the coefficients in the
ficients in the expansion of univalent functions.
M(r, = max I =r If(z)l, where 0 < r < 1, of the function f(z) 1z ... € S and the coe fficients of the two functions f(z) and
"
=
expansion of the function
z + c 2z2 + ...
ep (z) = In f (z) z
=
2 (liZ
+ l~z'1 +...),
(11)
"
f'1 (z) =
V f(z'1)=z
where f(z) € S (a), to the corresponding coefficients in the expansion of the function
+b3 z 3 + ... E S('1),
namely, the limits
lim (1 T-+
1
00
r)'1 M (r, /)= lim I en , = lim I b'1n_ll~ =a.J ~ 1. n-+co
n
(10)
where a = 1, exist only for f(z) = z/(l - e z)2. f One result of this is the fact that, for every function f(z) € 5, there exists a number N for which Bieberbach's inequality ICn! holds for n:::: N . However, it f f still remains an open question whether there exists a number N independent of the function f(z) in the class S beginning with which Bieberbach's inequality is satisfied. Hayman [1958a] proved the following:
Ic n I
k=\
n-+-oo
ie
If Cn is the least upper bound of
I k ep * (Z )- - l n (I _ 1 Z2) --2~ ."- Ii Z under a suitable rotation of the unit disk. He also proved certain other relation ships qualifying in this sense the spread of the coefficients in the expansion of functions in the class
If f(z) €
5(a),
5(a).
and
Specifically, he obtained the following theorem:
f(e ieO ) = 00,
then, for arbitrary real A, the equation
00
~
."-
kiTk _~e-lkOol'1,~k-(A_l)ln_l_ k I _,2
,,~\
for all f(z) € S, then the limit
lim Cn=rJ. n-+oo n exists. (This limit is equal to 1 if Bieberbach's conjecture is true.)
+(~-A)lnrJ.+EA(')' where fA(r) ---> 0 as r --'> 1, holds for coefficient5 in the expansion (11) of the function 4J (z).
(2)
614
§2. UNIVALENT FUNCTIONS IN DISK AND ANNULUS
SUPPLEMENT
Recently, Hayman [1963] obtained a bound on the growth of adjacent coeffi
It follows, in particular, from (12) that co
V k '-
cients that is asymptotically sharp not only for the class 5 but also for the class F~.
Ilk-Te-JkOO 1 . l'i =T1n-;-, 1 1
If a function f(z)
=
k=1
and f(e iflO ), then, for arbitrary real A and
T,
In the particular case in which
the inequality
such that
1
+i
k=1
where g (T) tion $(z) the disk
=
-->
10k Ii r ik ~ (
)41
:2~1~ I
(l -.:
_1)2
=
1:S n V+I
(l
+ g(r)),
:S K,
= 0 for a sequence n = n v' for
V=
1, 2,' .. ,
we have
n ~ no'
I Cn I.:;;;;;; AK,
(15)
Inequality (15) cannot be improved as regards order of dependence on K, as is shown by the functions CX)
1 + G1z + G 2z 2 +"', which is regular in
fa (z) =
Izi < 1.
V
Z
1
'l_
~~~ A
I
_2
sinn6
= '- sin 6
n Z •
n=1 'V
'V
.
mean square modulus in the classes 5(a) and 5(2)(a), the inequality giving a lower bound on the distance between adjacent coefficients in the expansion of the
In particular, if
mean in a joint paper by Bazilevic and Lebedev [1966] with the aid of methods ex
TTlK, where K is even, then for n = K(m + Y2L
C
n
=
0 for n
inequalities regarding regular functions, we also use the following finer result: Let f(z) denote a function belonging to F~. Let z 1 with
T
1
<1
and T2
< 1. If If(zl)1
. fI
=
Tie'
1
and z 2
.fI
=
T2e ' 2
~ If(z2)1, then A
14
\!(Zl) 1 / If(Zj) \3/4.:;;;;;; (l-r )1/2(l-r )1/2I z2
pounded in the monograph of Hayman [1958l.
Km, whereas
=
In the proof of inequality (14), in addition to the familiar and frequently used
Some of the results that we have given regarding the spread of the coefficients are strengthened and generalized for the case of functions that are p-valent in
e=
Icnl = \cosece I> KITT
V
function f 2(z) = f(z 2) and various other inequalities characterizing the spread of the coefficients of univalent functions.
1
zll'
(16)
2
where A is an absolute constant.
The problem of investigating the interrelationships between the order of growth of sequences of coefficients is related in a natural way to the question of the asymptotic behavior of the coefficients in the expansions of univalent func tions. The first systematic results in this direction belong to Goluzin (Chapter
IV, §8).
For f(z) E 5, the inequality obtained from (16) by repl~cing its left-hand member with If(z2)1 was obtained by Biernacki [1956] from an inequality of Golu zin [1946e, 1949c]. F or functions in the class 5, this proof of inequality (14) is simplified only
Goluzin also examined the problem of finding bounds for the growth of adjacent coefficients in the expansions of functions in the class
5 (ibid.). For functions in
5*, he obtained [1946e, 1949cJ the inequality f
to an inconsiderable extent. In connection with this, it is appropriate to cite Hay man's assertion that it is difficult to prove a result of an asymptotic nature for univalent functions by a method that could not at the same time be applied to func tions that are p-valent in mean.
I Cn+l I-I cn II < A
A is an absolute constant less than 100), which is sharp with regard
the order of dependence on n.
nv
(13)
Special cases of inequality (13) are the inequalities giving a bound on the
(where
-
Cn
2
0 as T --> 1, holds for the coefficients in the expansion of the func
[(f(z)jz)(1 + e- iflO z)2A]T
(14)
where A is an absolute constant.
Milin [1965] is used, this theorem leads to the following result:
S(a)
z + 2~=2 C nzn E F~, then
ricn+1 1- I cn II ~ A, n ~ 1,
When an inequality for regular functions that was obtained by Lebedev and
If f(z) E
615
to
Lucas (d. Hayman [1965]) extended the result given above to functions that are p-valent in mean:
616
SUPPLEMENT
If f(z) =
+
Co
C 1Z
+ ."
§2. UNIVALENT FUNCTIONS IN DISK AND ANNULUS
is regular and p-valent in mean with respect to area
obtained the following result:
in the disk Izi < 1, then
!lcn :_ll-\c n l[=O(n
gp
If f(z) = z +
-'.l), p~l,
1~=z cnz n € 5
(17)
I
1 /
(1S)
Inequality (I7) cannot be improved and, for p < 1, it is no longer true. For
Furthermore, for a <
Y2,
p = Y2, inequality (1S) is true for a function f(.,[;), where f(z) = z + b 3 z3 + ••. •.• E 5(Z) (see Hayman [19651). This leads to the relation t
II bgn + 11-[ bin_ill =
and If(z)1 < M/(1 - Izl)a. in Izi < 1, then
1 I cn I~ M . A (a) n-1/ s (log n)l/~, a=2' 1 ICn I ~M· A (a)n- s, a<2'
and, for ~ < P < 1,
II cn+II-1 cn II =O(n2 (p-VP).
617
Ic n l=o(n- 1/ s)
as n_oo.
Analogous relations hold for functions that are p-valent in mean with respect to
0 (n - Y2),
area. As Pommerenke [1962a] has shown, for starlike univalent functions, the which improves the earlier result of Goluzin (see Chapter IV, §S). For the coefficients in the expansions of univalent functions that are repre
condition M(r)=O((I-r)-a.), where O
sented by lacunary power series with rapidly widening gaps, we have a stronger
I Cn I= o (n- 1 +
result than inequality (15). Specifically, Pommerenke [1964a] obtained the follow
for all a in the interval 0 ~ a ~ 2.
ing theorem:
If a function f(z) = z + and if nv+/n v
11 )
2q
for all
1:=1 cnvz Il,
nv
is locally univalent in the disk Izi < 1
where q is a constant greater than 1, then
Clunie and Pommerenke [1966] proved this same result for functions that are close to convex. With regard to the problem of coefficients for special classes of univalent
ICIII=O(~).
functions, we mention the following results: In the class 5 * , the problem of the coefficienrs has been solved. Specifically, Hummel [1957] determined in explicit
This result cannot be improved without supplementary restrictions on f(z), as is Let l17k I denote a sequence that approaches 0 as k so that n k + Ink 1
-->
00
as k
-->
00
and 1
oo
_ k -1
17
nk
-->
00.
Let us choose n k
< 1. Then, the function
00
!(z)=z+ ~
n'/lInkZnk
k=1
is bounded, univalent, and starlike in the disk
Izl
< 1
For functions f(z) = z + 1~=zcnzn E 5, for which If(z)\ < M/(I-lzl)a. in the disk Izi < 1, we have, for the coefficients
form the set V*n in (n - I)-dimensional complex Euclidean space consisting of all points (c z' .•.
shown by the following example:
a> Y2, the well-known sharp bound on the increase in
I cn I< M· A (Cl)n- I +
ll ,
where A (a) depends only on a. (See, for example, Hayman [195S] 3.3; an analo gous inequality is valid for p-valent functions.) For a ~
Y2,
Pommerenke [1961/62]
, c)
that are put in correspondence with at least one function
f(z) = z + czz Z + ... + c zn + ... € 5 * . Here, V * is determined from its twon
dimensional cross section
*
en'
for which c z'
n
... , C n -1
are fixed. If (c z,' .. ,c n -1)
* -1' then en* is a disk whose radius does not exceed is an interior point of Vn 2/(n - 1). This is a sharp bound since it is atrained when c Z = c 3 = •.. = c n-I = 0 by a function of the form f(z) = Z/(1 - (Zn-I)2/(n-l), where If\ = 1. In particular, the boundaries of the cross-sections C*Z and C*3 are determined by the equations C
z = 2e
i8
, and c 3 =3c;/4+ e i8 (4-l c l)z/4. z
In the class K (see p.619) of functions f(z) = z + to convex in the disk Izl < 1, Bieberbach's conjecture
C
zzz + ••• that are close
Ie n l ~n is valid, as Reade has shown [1955]. For functions F (() = ( + a
o
+ a / ( + ...
618
1(1 > 1, Pommerenke [1962] has
that are meromorphic and close to convex in
The structural formula for the class C also leads easily to integral represen
ob
tations of the classes
tained
I I= (Xn
that is, functions [(z) that are regular in the disk
z + I~=2cnzn E K, then
and regular and convex in
t. 0
°2 m{l J C
is sharp with respect to the order of dependence on n. In particular, for all func tions in 5* except the functions
z
=
I cn +ll-1 cn \ =
o(n > 0 0
This class was introduced by Kaplan [19521.
Izi < 1
and [(z) is close to convex
+zr (Z)}dO>-1t
f' (z)
61.
°
< 01 < 2 :::; 17 and z
ie
re , where r < 1. The class of close-to-convex functions coincides (see Lewandowski [1958,1960])
for arbitrary 01 and 02 such that -17
Izi < 1
with the class of functions that map the disk
(l-ei&I Z) (l-i0 2z) ,
where 01 and 02 are real, there exists a 0
and satisfy there the
and
n"?= 1.
Pommerenke also indicated the form of all functions for which this inequality
f(z)=
Izi < 1.
Every close-to-convex function is univalent in if and only if f'(z)
II cn +ll-1 cn II < : e~,
Izi < 1
condition ~1f'(z)/g'(Z)j > 0, where g(z) is some function that is independent of [
cients for univalent functions, Pommerenke [1963] proved the following theorem: =
5* (see Chapter IX, §9) and -- S. By this procedure, one can
find an integral representation for the class of functions that are close to convex,
0 ( ~ )•
Supplementary to the results given above, on the growth of adjacent coeffi
I[ [(z)
619
§2. UNIVALENT FUNCTIONS IN DISK AND ANNULUS
SUPPLEMENT
=
onto domains linearly acces
sible from the outside, which Biernacki [19361 had first investigated. We denote by K the class of functions [(z)
such that
and close to convex in the disk
(~5)'
=
z + c 2z2 + ••• that are regular
Izl < 1.
It is obvious from the definition of the class K that 5* eKe S.
4°. Structural formulas for various classes of analytic functions. A large num
Ju. D. Maksimov [1955] introduced into consideration the class C(p) of (
ber of results have been obtained for classes of functions that can be represented
locally-convex functions and the class S/p) of (-locally-starlike functions [(z)'
by some "structural formula" or other, in particular, by aStieltjes integral. It is
with [(0)
well known that the class C of functions [(z) the disk
Izl < 1
=
1+
C 1z
+ ••• that are regular in
O2
sented by the Stieltjes integral
"
~
(1)
=
Robertson [1935] and Goluzin [1950a] used this formula to obtain an integral repre sentation for the class T (see Chapter XI, §9). Beginning with this representa tion, Goluzin [1950a1 obtained in the class T a number of distortion and rotation r
< 1.
JC\rn {1
dO> - e,
(z) } _ + zj" f' (z) dO -
2p 1t
-It
O2
f' (z)
on the
"
}dO> - e ,
01
1.
On the basis of a simple connection between functions in the classes C and T,
=
+ I' (z)
~ m{Z~'(~i
where /1(t) is a nondecreasing function in [-17,17] such that /1(17) - /1(-17)
Izi
z 1 < 1 and that satisfy the conditions
for [(z) E C(p), and
eit+z e' -z
- . t - dfl- (t).
theorems and also bounds on the mean moduli of [(z) and
I
11
'zr (Z)}
&1
-11
circle
0, that are regular in the disk
C { ~ m1
and satisfy the condition ~[(z) > 0 in that disk can be repre
f(z)=
=
=
I(z)
-It
for [(z) E S(p) in the annulus 0:::; p
z
Cm{ zl' (z) }dO = 2p1t
J
< Izi < 1, where
p depends on the function
·e re ' ,-17 < 01 < 02 :::; 17, and (> O. Let C(p, q) and S(p, q) denote subclasses of functions in C(p) and S(p)
that are of the form
fez) =
zq
+
q c q +1Z +t
+...
.__
~
~:_ ~ _
!!!
•
_~.:_
_'!.- .-
620
--
'~.:.::
·§rtiW:~"'..ii"
;OJ!!! - - - .:....
_~
:n.;,'J§.!'_~"'"
_if,-;;:;;,-"i2;~,,;;""
ii5lGiliiiiZi'""o"'t-"~c:;;;:';';~'~-.>"--::'.~_ ~""'
*,,~~i~~,"::7i"""-:'~
""o.'
~~~~-""-li::;"
-=
-"..._~:.,-
621
§2. UNIVALENT FUNCTIONS IN DISK AND ANNULUS
SUPPLEMENT
-~""':1'~~~--
These classes contain many well-known classes of functions. In particular,
Loewner-Kufarev equation in quadrature in a special case, Bazilevic [1964] gen
C1/1, 1) is the class K defined above, the class S11(I) contains functions rhat
eralized his previous results by proving the following theorem: If two functions po(z) and Pl(z) are regular in the disk Izi
are starlike with respect to direction (see Robertson [1936]), and the classes
positive real parts in that disk, then the function
Co(p) and So(p) were examined by Goodman [1950l. Ju. D. Maksimov [1955, 1955a] has obtain~d an integral representation of the
z
ing max~l
o
0
w (z) =
out of the class of functions C{(p, q) the zeros of whose derivatives are fixed, For po(O)
= p/O) = 1,
Z
'It
'It
-2~a.~2'
Rahmanov also investigated [1953] the question of membership in S of the arithmetic mean of a finite number of functions in certain subclasses of S. The classes of functions that he found are characterized by the corresponding geo metric properties of a conformal mapping performed by them. Certain of Rahmanov's
where f* (z) €
(z) =
\ f. t(l) pdt) dt,
(3)
S*. If, in particular, PI (z) = zf~ (z)/ f/ z ), then w = f* (z); that is,
the class (3) contains the class S*. One can easily show that the class (3) is broader than the class S*. As Krzyz [1964] noted, the class (3) is a subclass of the class K. In connection with this, we note that membership of a function f(z) in the class S does not imply that the function z
results have been generalized in papers by Zmorovic [1954, 1959, 1959a]. Thus
fl!l(Z) =
Zmorovic [1959] introduced the class of functions, regular and univalent in the
< 1,
l z I< 1.
b
belongs to the class S.
disk \zl
(2)
z UI
+ eiazf (z)],
t
ds-Ipo (0)
we obtain from (2) the class of functions
then the function
<Pa (z) = 1 ~ela [f(z)
+ ...,
+c~z~
Rahmanov [1951] proved the following theorem:
S;
I
Po (0) dt)
belongs to the class S if by w (z) we mean that branch that has the expansion
<1
and obtained bounds for If'(z)! and argf'(z) in that class.
~
PI (0) .,
variation of functionals (see Chapter XI, §9) to solve a number of extremal prob lems for these classes. In particular, he solved theoretically the problem of find
s
W(z) = [-Po (O~ CPi (8) SPo (0)- 1 exp (C Po (t) -
classes C{(p, q) and S{(p, q). On the basis of it, he used Goluzin's method of
If f(z) €
< 1 and have
that map it onto domains that are convex in the upper and lower
halfplanes respectively in the direction of a and (3, and he obtained its integral representation. This class isa natural broadening of certain classes of Rahmanov [1953] and of the class of functions that are convex in a given direction, -which was first studied by Robertson [1936]. The article by Lozovik [1963] deals with this class of questions. Loewner's parametric representation in this class of questions has played the following role: Kufarev's differential equations [1947], which generalize the familiar equations of Loewner, have made it possible to obtain an integral repre sentation of a rather broad subclass of functions that are univalent in a disk. This class includes, in particular, known representations of conformal mappings of the unit disk onto starlike and helical domains. Specifically, by solving the
~ f~t)
p(t)dt,
o
where p (z)
=
1 + c z + ••• € 1
C, belongs to the class S. This is shown by an ex
ample constructed in 1963 by Bazilevic (see also the article by Krzyz and Lewan dowski [1963]). The well-known integral representation (1) for functions with positive real part in a disk has been extended to the case of an annulus. Specifically, Zmoro vic [1953] obtained a structural formula for the class C (q, 1) of functions f(z) that are regular in the annulus q for q
< \z\ < 1, that satisfy the conditions 'Rf(z) > 0
< Iz I < 1, and that are normalized by the condition _1 211:1
\ n~ ~ z Izl=r
dz = 1, q
< r < 1.
(4)
w
"'iiiiiiiliiiiiiiiiiiiiiEl_iiiiiiiiiiiiiiiiiiiiiiiiiiiiii_ _i iiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiilii&iiiiiliIiiiliSiiiiiiiiiiiiiiiiiiiiEiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiil iijiiiiiji j ii F. &ijii;' Pi 4 ; H"-'U'
622
iiiiiiiiiiiiiliilii_ _r5iiiiiiD&iiliiJ"=_~ie·iii!~iilliiii·iii·~Ei:l·· &Bi&f.!~iJ1· :lJ~~_iiiilli· E"""i5a!'li,,!!i!!ll~i§i'2'~Ri':AJ,4ii!m;;;;4 ;;Z",,";:;;;;iliii!i!i4 ~I£J #i¥i@"'" . . '. . :"" . . "
SUPPLEMENT
, hiiiil:m:iliMll"
#
623
§2. UNIVALENT FUNCTIONS IN DISK AND ANNULUS
By means of this representation, one can easily obtain (see Zmorovic [}953])
satisfying appropriate conditions. These last results carryover to an annulus the
structural formulas for the class U*(q, 1) of functions that are regular and starlike
familiar representations of analogous classes of functions in a disk.
in the annulus q
< Iz I < 1
and for the class of functions that are regular and con
vex in that annulus. An integral representation for functions that are regular and
With regard to an n-connected circular domain K n , where n ? 3, Zmorovic [1958] generalized to the case of such a domain the familiar formula of Schwarz
starlike in an annulus was obtained independently by Li En-pir [1953b]. He also
for a disk (in a form different from Meschkowski's representation [1954]). By using
obtained an integral representation for the class of functions that are regular and
this result, Zmorovic [1958] obtained structural formulas for the class of fWlctions
typically real in an annulus (d. Li En-pir [1953a]).
Kn and for the more general class of functions that are regular in the domain K n . Maksimov [r961] that are regular and have positive real part in the domain
From the structural formula for tbe class of starlike functions we get, in par ticular, a representation for functions that map the annulus q
< Iz I < 1
onto
used the method employed by Goluzin [1934] in solving problems of the Dirichlet
doubly connected domains the boundary of each of which consists of n segments issuing from the origin and of m rays the reverse continuations of which pass
and Neumann type to generalize to the case of the domain K n the structural for mula for the class of regular functions that are univalent and convex in an annulus.
through the origin and which have no points in common with the segments. Func tions of this kind are extremal in a number of problems. For m
=
n = 1, for func
tions of this form that belong to the class U*(q, 1), we obtain
1
C~1ti In (ze - tlJ);
A generalization of the class C of functions that are regular and have posi tive real part in the disk
Iz I < 1
is the class C (r), for 0
1T,
of functions that
can be represented by a Stieltjes integral
,
f(Z)=CZ[:"(~l*--:,);q)]g .
(5)
q)
\ eit +z f (z) = ~ etl z dp. (t), -, where /let) is a nondecreasing function on [- r, d. By using the results of
(6)
Ahiezer and Krein [1938], Monastyrskil [1959] gave an actual solution of the prob
Here,
lemof finding necessary and sufficient conditions for the existence of a function
iil(2~ilnz; q)=-iq-1/4VZOi(2~ilnz;q),
fez) in the class C (r), for 0 < r ~
'tt/,;
where 0 0 ('; q) and q) are Jacobi's theta functions with periods 1 and r, where r= (i/IT)ln(l/q). By letting q approach zero in (5), we obtain Koebe's
n-I
+ .i./'z, V ' co>o. ""2 ,=1
and which assumes a given value w at a given point Zo
Cz (l -
e-ICXz)1 •
the role of Koebe's function for a disk. For example, the function (5) with a
=
and only such a function attains extrema of If(z)\ for all z such that
q < Izi < 1 in the class U*(q, 1) (and in the broader class of functions that are
< Izi < 1
[see subsection 10 of §2]) and also extrema
of argf(z) for arbitrary z in the annulus q
f.
0 in the disk
Izl
< l.
In the article by Andreeva, Lebedev and Stovbun [1961], problems are posed
The function (5) plays in many problems a role for an annulus analogous to
univalent in the annulus q
whose Maclaurin-series expansions begin
Co
fWlction
f3 ± IT
IT,
with a given polynomial
fez)
Uiiiii.J.iii!
< Izi < 1 in the class U*(q, 1) (see
Dunducenko [1956]). We mention that Zmorovic [1956] obtained an integral representation for yet more general classes of fWlctions that are regular in an annulus with real part
and solved regarding functions in the class C (r) for which the problem formulated above, of Monastyrskil, is a special case. In that article, Lebedev solves the fol lowing general problem: Let M denote the class of functions w
Izl < 1
=
fez) that are regular in the disk
and let G denote an arbitrary simply connected domain in the w-plane that
does not include the point
00
and some other given point. We shall say that the
function fez) € M is subject to the domain G if, for arbitrary z in the disk Izi
< 1, the point
w = fez) € G. We denote by Me the class of functions that are
subject to the domain G. Let z 0
=
0 and let z k (k
=
1", • , m) denote distinct
624 points in the domain 0
< Izi < I,
numbers, and let w~v) (k We use the notation x
=
=
let a k (k = 0, I,""
0, I,""
m; v = I, 2,""
Alenicyn [1962] has obtained results of a very general nature regarding the
m) denote given natural a ) denote give~ numbers. k
-
-
(...,(1)
",.,(110) '4'0' ••• , UVO I " "
",.,(1)
",.,(l1
U'lm' ••• , U'lmm
))
in it by a sum of Stieltjes integrals, and he obtained their boundary functions:
•
Let G denote a finitely connected bounded circular domain in the z-plane and
Let M (x, a, X) denote the class of functions f(z) € M that satisfy the conditions ('I
11)Ij
ranges of arbitrary finite systems of Laurent coefficients of functions that are regular in an arbitrary given finitely connected domain and that can be represented
x (zk) = (z , " ' , Z ), a = a (a ) = (a , " ' , a ), k o m O m
x --X (",.,(Y» Wile
let
F denote a class of functions that can be represented in G by a formula of
the form
k=O, I,
"'J
m, '1=1, 2, ... ,
m
r].k'
./J
f (z) = ~ ~ gk (z, t) dp./i (t), k=la
Let MG (x, a, X) denote the class of functions f(z) that belong both to the class
MG and to the class M (x, a, X). Suppose that the class MG is given. Suppose that a given system cI> (f, MG) of functionals
<'P~){j)=(''1
625
§2. UNIVALENT FUNCTIONS IN DISK AND ANNULUS
SUPPLEMENT
where the gk(z, t) are suitable fixed functions and the p./t) are members of
M [a, b] for k
I, 2,'" , m. Let Zv (v
=
=
I"" , s) denote distinct fixed points
each of which either belongs to the domain G or is the center of one of the circles 1 l)I!(H) (Zk)' k=O, I, .. "
m, '1=1, 2, ... ,
r].k'
(7)
serving as part of its boundary. Suppose that the expansions 00
!
is defined on the class MG'
M G (x, a, X). The boundary functions of this range of values are uniquely deter mined by a consideration of the variable class N (A) = MG(A) (x, a, X), with A' A:S A' , where x, a, and X are fixed and G(A) is a variable domain that is con tinuous with respect to A on the interval A' A:S A" and possesses the property
:s
:s
:s Al < A2 :s A" , we have
G(A 1) C G(A 2 ) and G(\) f- G(A 2 ). For this class, N(A') is empty and the class N (A') is nonempty. A number of theorems for functions that are regular and have positive real part and also for regular typically real functions in a disk or annulus have been obtained by Komatu [1957, 1958, 1958a]. The ranges of values of arbitrary finite systems of coeffic ients in the expansions of functions in these classes that are defined in an annulus (or disk) have been studied by Nishimiya [1957, 1959]. The ranges of values of functionals that are the sum and product of Stieltjes integrals have been studied by Remizova [1959]. The results that she obtained generalize to the case of functionals of the form indicated certain of the theorems in the paper by Asnevic and Ulina [1955].
c~) (z -
z.)n, (8)
00
cI> (f, MG) of functionals (7) defined on the class MG ' that is, find the set of points X in n-dimensional complex Euclidean space R n , where n = 2:=1 a k , such that, for every point X = X (w~ v» in this set there exists a function f(z) €
and A2 such that A'
~ n=-oo
The following problem is solved: find the range of values of the system
that, for any two numbers \
(z) =
gk(Z, t)= ~
a~.k(t)(z-zyt,v=l,... ,s, k=l, ... ,m,tEra,bJ,
n=-oo
are valid in a neighborhood of the point Zv in the first case or in a circular annu lus with center at z v in the second. Furthermore, let
I C l' •••
,
C N I denote an
arbitrary fixed system of coefficients C~V) considered as a point
P [n
=
R N and let P[gk(z,t)] I, 2,'" , m) analogous to it defined in accordance with (8). Since the range of values of the system IC 1'" • , CNI in the class F is the range of values of the functional P[J] on that class, this range is the convex IC 1 , · · · ,
CNI inN-dimensional complex Euclidean space
denote the point of RN (k
=
envelope of the union of the sets U a:S t:S b P [g k (z, t)] for k = 1,' •• ,m. As an application in the class T of the functions f(z) = z + c 2z2 + •.• , one can find the range of values n(T) of the system l[(zo)' c21, where Zo is a nonzero fixed point of the disk Iz I F or ~ z 0
f-
< I:
0, the range of values of n (T) is a body in three-dimensional real
space bounded by two conical surfaces
Iw where f(zo) = W
(1
= Xl
~o ZO)I l~ ~ (Zo + zip) ---:- (X~ -+- 2) ~ { (I ~o Zg)1 W}, +
iX
2
and c = x • 2
3
._,~.".,;_~~,__ ~.:!t:,._~"'.'f~~~~~~.~"'-"!,~:""":5i.'!'J.E,~,n,,,"Z,,~~
.,=""",",=~-,=,,--=-...;;:::.-=-",,~~~~~""""=~===
626
-,>,~~~;P~'S;...s;;"':"1'i,",,-"="""·'~·~~f~;.~&;'~:,,3.c':"'?""-'_"'}"'i'f~'",.... _,,_ .•. ..:."',:<;~~0:~~-s.~~F~,.,~9iiFiEi~mi'fi{~~~~:!~~j.~
,
For ~zo = 0, the set D.(T) is a plane domain bounded by an arc of the hyper
fixed coefficients c 2' ... ,
C n'
The article by Nosenko [1963] deals with the same
bola x = Zo + 1/zo - l/ x 1 where Z/(l + zO)2 S Xl SO Z/(1 - zO)2, and the chord 3 subtending the endpoints of that arc. Corresponding to every boundary point
question.
Iw,
are regular and satisfy the inequality ~f(z) >
c 21 of the range of value s D. (T) is a unique function in the class T. Let us denote by T (c) the class of all functions f(z)
=
z +
C
2z 2 + ..• in
Now, let C denote the clas s of functions of the form
For ~zo 1= 0, the range of values of D.(T(c 2)) is a circular lune with vertices Z/(1 - c 2z0 + z~) and zo(1 + c 2z0 + zO)2/(1 - z~)2, one of whose boundary cir cles passes through the point z/(1 - zO)2 and the other through the point
z/(1 + zO)2. For ~zo = 0, the range D.(T(c)) is the segment of the real axis connecting the first two points mentioned. By using this result, Goluzina [1962] obtained a number of sharp inequalities in the class T (c 2)' Some of these results strengthen the corresponding theorems of Goluzin [1950a] and Asnevic and Ulina [1955]. Following the method used in the article by Andreeva, Lebedev and Stovbun [1961], Goluzina obtained [1965] necessary and sufficient conditions for the exis
~
k=1
Iz I < 1
the range of values of the system of functionals, U(z), c 2'
n::::
C
and she determined
3' ••• ,
C
n l,
where
2, in the class T. As an example, we present the following theorem, which is
1
Z + ... that
and let
eN
de
k
.
I-ze-'~k'
where the ¢k are arbitrary real numbers and the Ak are arbitrary nonnegative num bers satisfying the equation 11.1 + 11. 2 + ... + AN = 1. Robertson [1962, 1963] used a variational method that he constructed to show that a number of extremal prob lems in the class C are satisfied by functions belonging to the class C 2' Zmoro vic proved [1965] in a very simple way the following theorem, which contains, in particular, the corresponding theorem of Robertson:
Let <1'«(; w) denote a real function of the complex variables ( and w that is single-valued and bounded in the domain lR( > 0,
Iwl <
00.
Suppose that, for
arbitrary fixed ( in ~( > 0, this function attains its greatest lower bound or its least upper bound in any disk \w the quantity
J = extr
tence of functions f(z) in the class T that have given initial segments of their Taylor expansions at different given points of the disk
Iz I < 1
C
N
°
= c 2' where - 2 < c 2 < 2, we obtain the result:
in the disk
1+
f(Z)=~A l+ze-i~k
function f(z) ranges over the class T (c 2)' When we look at the intersection of 3
°
f (z) =
note the subclass of functions in C of the form
T with fixed coefficient c . Let D.(T(c 2)) denote the range of values of f(zo) 2 under the condition that Zo is a fixed point in the domain < I Zo 1 < 1 and the the domain D. (T) with the plane x
627
§2. UNIVALENT FUNCTIONS IN DISK AND ANNULUS
SUPPLEMENT
wol S R
on the boundary of that disk. Then,
extr
fEG Iz/=r<1
(where the abbreviation extr denotes the greatest lower or least upper bound) is attained in the class C 2' and
J
directly related to the result given above for the class T (c). Let T (c 2' c 3) de
= extr extr
w),
flI
note the subclass of all functions in T with fixed coefficients c 2 and c 3' The range of values D. (T (c 2' c 3)) of the functional f(z) for fixed z in the disk
Iz I < 1
in the class T (c 2' c 3) when ~ z 1=
°
is a convex lune bounded by
arcs of two circles (these are explicitly defined). In the case ~z
=
0, the range
of D.(T(c , c )) degenerates into a segment of the real axis. 2 3 In connection with the results given above, we mention that Lozovik [r963a] presented a geometrical method of determining the range of values of the system of functionals that can be represented by a Stieltjes integral and also the subset of that set that satisfies given additional conditions. As an illustration, let uS find the range of values of the functional ['(z) (where z is a fixed point in the disk Izi
< 1) defined on the class of functions f(z) = z + c 2z2 + ••. €S with
as ( varies in the disk I( - at S p, where a = (I + r 2)/(1 - r 2 ) and p and w varies over the circle Iw - ~ «( 2 - 1)1 = ~ (p2 - I( _ aI 2).
= 2r/(I-r2),
This theorem is easily generalized to the case in which we are given a real function <1' «(, w 1 ,"', w n ) of complex variables (, WI"'" w n that is defined in the domain lR( > 0, Iwkl < 00 for k = 1, 2,"" n and one studies its extrema on a disk 1z\ = r
< 1 for (= f(z) € C, w k = zkf{k)(z) for k = 1, 2,"" n.
As an application of Zmorovic's theorem presented above, we can find the greatest upper bound of the curvature of the level curves in the subclass S~)* of the functions f(z) in the class S{k)* satisfying the condition ~(z['(z)/f(z)) > a in the disk Izi
< 1, where
°S a < 1. For a
=
0, we then obtain a theorem proved
628
§3. FUNCTIONS ANALYTIC IN MULTIPLY CONNECTED DOMAINS
SUPPLEMENT
by Aleksandrov and Cernikov [1963] by a different procedure (see §2, subsection
regular in the disk Iz I < 1 and satisfy for arbitrary r in the interval 0
2° of this supplement).
condition
629
< r < 1 the
20:
Aleksandrov and GutljanskiJ: [1966] proved a number of results regarding
~; ~ If(reI9)I~d9:o::;:; 1, . ~
boundary functions of ranges of values of functionals of a general form that are de fined on the class K and its subclasses and also on the class C. They used in ternal variations in these classes. As an application, they found in explicit form
A parametric representation of the class H s follows from the representation found
the range of values of the functional ['(z) on the class K. This range of values
by Smirnov [19321 for the broader class of functions that are regular in the disk
had been found earlier by Krzyz in 1965, who started directly with an integral rep
Iz I < 1
and satis fy for arbitrary r in the interval 0
resentation of the class K. Aleksandrov and Gutljanskil [1965] obtained other
the condition
2"
(2~ ~
quite general results regarding boundary functions of ranges of values of weakly
If(rei9)16d9t8
o
differentiable functionals and systems of functionals defined on classes of func tions that can be represented as a sum of Stieltjes integrals. They did so by using
These authors used the variational formulas that they obtained in the class H s of
variational formulas in these classes of the same type as the variational formulas
the same type as Goluzin's variational formulas (see Chapter XI, §9) to find the
of Goluzin (see Chapter XI, §9). We present one of the results obtained by these
greatest values of the functionals ~I(log([(z)/B(z)))1 and ~I(Iogf'(z))l defined
authors for the class C. Let z 1 ' ••• , z p (for p
on that class where z is a fixed number in the disk 1 zl
the disk Izj
< 1.
and j = 1, 2,""
Define u
mj
=
f(m) (z') and v J
=
mj
=
1, 2, ...) denote fixed points in Ii (where m = 0 1 " ' , S. mj
" J
I(f)=J(uOl>
'VOl> ••• ,
us1 l'VS1 t; ... ;
llop,
'Vop ,
.,., llspp,
not vanish in the disk Izi
'Vspp )
z
The only boundary functions in the range of values of the functional 1 fined on the clas sCare functions in the clas seN' where N 2p - 1.
(n
de
:s s 1 + ... + S P +
=
l([) = J (f(z), fez), ['(z), ['(z» defined on the class C are functions in the class eN for N:s 2.
In particular, for functions fez) € H S that do
they obtained sharp bounds on I['(z)! for given
re i8 , where r < 1. The inequalities giving these bounds depend on the rela
[1962] by a different procedure. These include a number of results for the class H S [1961, 1962, 19631. defined on the class A M (see Chapter IX, § 3) of functions fez) are regular in the disk
o :S r < 1
c 1 z + ..• that
~ log+ If(re i9 ) \d9,;::;;:;M. o
metric representation of these classes of functions. Gel'fer [1965] applied Golu
§ 3.
and y is real).
+ C 1z + ••• that are regular in the disk Iz I < 1 and satisfy there the conditions If(z)\ < 1 and f(z) t- O. Using its simple connection with the class C, they arrived at a Dumber of inequalities in that class. Gel' fer and Kresnjakova [1965] con sidered the class H s (where [; > 0) of functions fez) = Co + C 1 z + ... that are
=
and that satisfy for arbitrary r in the interval
2"
zin's variational formulas in the class T to the problem of finding the maximum in Galperin [1965] obtained an integral representation of the class of functions fez) '"
Iz I < 1
the condition
We mention some of the other recent results that are based directly on a para
Iz I < 1
< 1,
Gel' fer [1966] used the same method to find the extrema of certain functionals
It follows, in particular, that the only boundary functions of the functional
that class of ~ Ie it' zf'(z)j f(z)l for fixed z and y (where
is a given entire func
tionships between 0 and r. One of them had already been found by Kresnjakova
denote a weakly differentiable function defined on the class C.
Co
< 1.
depending on its zeros in 1t;;"1
p). Let
< 1, (w)
tion, and B (t;;") is an appropriate multiplier in the parametric representation of f(t;;")
1
0
•
Functions that are analytic in multiply connected domains
Conformal mappings of multiply connected domains onto canonical sur
faces. In Chapter V we presented the basic extremal properties of conformal map pings of a multiply connected domain onto certain very simple canonical domains: onto the plane with parallel rectilinear cuts, onto the plane with cuts along arcs of logarithmic spirals all of the same inclination, and so forth. For the theory of
630
§3. FUNCTIONS ANALYTIC IN MUL TIPLY CONNECTED DOMAINS
SUPPLEMENT
631
been considered earlier (see Garabedian and Schiffer [1949]).
functions that are analytic in multiply connected domains, it will be of interest to answer the question about the existence and extremal properties of conformal map
Thus the function P (z) defined in terms of a suitable function S{z; (, a),
pings of a multiply connected domain onto canonical domains of a more complicated
and only this function minimizes the area of the image of the domain B in the class of all functions that are regular in that domain and that have fixed initial segments
type. We mention certain results associated with this question.
-in their Taylor expansions at given points of B (segments such that this class
Grunsky's article [1939] initiated the study of conformal mappings of a given multiply connected domain onto many-sheeted canonical surfaces. In that article,
does not include a constant). This provides a generalization both of the theorem
he studied a function that is regular in a multiply connected domain except at a
of minimization of area that was given in Chapter V, §4 and of Bergman's theorem
pole of multiplicity m and that maps that domain onto a Riemann surface with
[1950], which he obtained by a different method and in a different form, on the
parallel rectilinear cuts. For mappings that are defined on a multiply connected
minimization of area in the class of all functions that are regular in the domain B
domain and that possess the geometric property mentioned, though with singular
with fixed initial segment of their Taylor expansions at a given point of the domain. We define the outer area A (f) of a function f{z) in a domain B as the quan
parts of the mapping functions of a more general type, Alenicyn [1964] obtained
tity (finite or not)
the following results by the method of contour integration: Let B denote a bounded finitely connected domain in the z-plane with bound ary C consisting of simple closed analytic curves Cl'
... ,
Cm' Let (
l' ••• ,
where 5 ~ 1, denote arbitrary distinct points in the domain B. Let a oI)., a 1 ,}. , ' " ..• , a .. , where j = 1,' .. , 5, denote arbitrary coefficients (not all zero) such P )'1
that
A if) =
(5'
S (z;
~,
/ (z)j' (z)dz,
c(,)
interior of the domain B as v
--7 00.
Then the function
Q(z), and it alone, maxi
:, mizes the external area in the class of all functions of the form f(z) s
Il) =
~i ~
, where the C( v) are the boundaries of the domains B( V), which approximate the i'
Define
lim ...... 00
i :
I;=l aO,i = O.
-
!
}=1
p.
[1:
k=1
(z
~'C~)k + Gto.
j
log (z -
~j)]
=
S(z; (, a) +
g (z)' where S{z; (, a) is a fixed function of the singularities and g (z) is an
arbitrary function that is regular in the domain B. The properties of the function e{z) can be used to obtain theorems
(a function of the singularities). Under these conditions, we have the theorem: For an arbitrary given angle
e in the interval
-17/2
< e ~ 17/2 and an arbitrary
given function of the singularities S (z; (, a), there exists a function, unique up to an additive constant, e(z)
=
S(z; (, a) + Fiz) where Fe(z) is regularl) in
the closure of B, that possesses the following property: on every boundary curve
of the distortion-theorem type for functions that are meromorphic and p
valent in mean with respect to area in a multiply connected domain. A very
special case of the corresponding theorem on the range of values of a certain
functional defined on the class of functions that are meromorphic and p-valent in
mean with respect to area in a given multiply connected domain is the following:
CJ.I- of the domain B, every branch of the function e-iee(z) has a constant im aginary part.
In the class of all functions f(z) regular in the domain Izi
The functions
of a fixed point z 0 IeI
P (z) = 2 [ID~ (z) 2
I
+ $0 (z)]
=
00
with cut in a neighborhood.
of that domain and that have a representation of the form
= f{zo) + (f(p) {zo)/p!)(z - zo)P + " ' , the range of values of the functional
log (r(p) (zo)fpt) is the disk
I w I~ For p
In what follows, when we speak of regular and meromorphic functions, we mean single-valued regular and meromorphic functions.
zP (I + a/z + •••) that are p-valent and
f{z)
solve more general extremal problems than do functions of the same kind that had
1)
00
=
> 1 except for a pole at z
Izi > 1
=
P log (1 -. I
z: 12 ) •
1, this result is valid for an arbitrary fixed point Zo of the domain
and it is the special case, with n
=
1, of Theorem 1 of Chapter IV, §3.
§3. FUNCTIONS ANALYTIC IN MULTIPLY CONNECTED DOMAINS
SUPPLEMENT
632
Komatu and Ozawa [1951, 1952] established the extremal properties of the same type as the theorems on the distortion of univalent conformal mappings of a given multiply connected domain onto a plane with cuts along two mutually per pendicular systems of parallel segments, along two systems of radial segments and circular arcs, etc., and they showed that proof of the existence of such map pings can be reduced to a consideration of a domain of considerably lower-order connectedness than the given domain. Jenkins [1963] used a slight modification of the extremal-metric method to find a number of more general extremal properties of certain analogous mappings including some that are not necessarily univalent but with a unique fixed simple pole in the domain in question. He was the first to obtain theorems of this type for functions that are regular in a domain. We present one of his theorems. Let B denote a finite domain in the z-plane that is bounded by closed analy tic curves C j for j = 1" .. , n + 1. Let PI' P 2' P 3' and P 4 denote arbitrary points on C 1 numbered in the order in which they are encountered as one moves nonnegative integers such that m + p + q
=
~o
B. Let m, p, and q denote n. Let !'(B; m, p) denote the class of
around that curve in the positive direction relative
633
corresponding to the functions l, q, and f respectively. We define A = L ': areaf(B). Under these conditions, we have the following theorem:
:v
For arbitrary {3 and no~nnegative y such that {3 + y = 1, the sum A + ({3-y) L is maximized in the class F (B; m, p) only by the function {3l + yq and this maxi . mum is equal to {32 L 1 - y2 L 2' If m = p = 0, then th~e function {3l + yq is univa :i lent and the preceding result is valid for the class !'(B; m, p). In this last case, the function {3l + yq maps B onto a domain for which each finite supplementary ,continuum has in common with an arbitrary horizontal or vertical straight line no more than a single segment (or point). Alenicyn [1965] established a number of properties of the functions (fl e(z),
P (z), and Q(z) mentioned above for analogously defined functions in the case of mappings of a finitely connected domain onto many-sheeted Riemann surfaces with cuts along two mutually perpendicular systems of parallel segments. In particular, he ohtained a generalization of certain of Jenkins's results [1963] that deal with meromorphic functions. Various area theorems are immediately associated with conformal mappings onto canonical domains. Meschkowski [1952, 1953, 1954] obtained an area theo
all functions fez) that are regular and univalent in the domain B and that map
rem generalizing to functions that are univalent in a multiply connected domain
that domain onto the domain bounded from without by the contour of some rectangle
the Gronwall-Bieberbach theorems (see Chapter 11, §4) and also a number of dis
0< 'R fez) < L, a < ~f(z) < 1 under the condition that C1 is mapped into that con
tortion theorems on functions that are meromorphic in a multiply connected domain.
tour, the points' PI' P 2' P 3 and P 4 are mapped respectively into the points 0,
Abe [1958] particularized Meschkowski's area theorem for the case in which the
L, L + i, and i, the curves C., for j = 2", . , m + 1, are mapped into horizontal ] cuts, and the curves C., for j = m + 2, ..• , m + p + 1, are mapped into vertical ] cuts. Finally, suppose that F(B; m, p) denote the class of all functions fez) that are regular in B and that possess the following properties: The mapping w = fez) maps C 1 into the contour of some rectangle of the type indicated with the same correspondence between the points PI' P 2' P 3' and P 4 and the vertices of the rectangle; the functions fez) are regular on C. for j = 2"" , n + 1; on each of ] these boundary components 0< 'Rf(z) < L, and a < ~f(z) < 1; on C., for j = ] 2, •.• , m + 1, the functions f(z) have a constant imaginary part; on C., for j = ] m + 2, .•. , m + p + 1, they have a constant real. part. lt follows from Jenkins's reo sults [1957] that there exists a function l (z) fez; B, m, p, 0) E !'(B; m, p) such that the mapping w = l (z) maps the boundary curves C., for j = m + p + 2, ... ] ... , n + 1 into the horizontal cuts and a unique function q (z) fez; B, m, p, "/2) to !'(B; m, p) such that in the mapping w == q(z) these boundary curves correspond to the vertical cuts. Let us denote by L l' L 2' and L the values of the length
domain in question is a circular annulus. Alenicyn [1964] generalized Meschkow
~
=
=
~
ski's area theorem to functions of a more general form, namely, to functions with singular part S(z; (, a) and with nonnegative exterior area in a given multiply connected domain. In particular, he obtained an area theorem for functions that are meromorphic and p-valent in a disk and have the property that the sum of the orders of the poles in that disk is equal to p, and this gives a generalization of Goluzin's familiar area theorem (see Chapter Xl, §6) for p-valent functions. Milin [1964a, 1964b] discovered a yet more general proposition of the area theorem type. Specifically, let B denote a finitely connected domain in the ex tended z-plane with boundary C consisting of closed analytic Jordan curves and suppose that"" E B. Suppose that a so-called Taylor system of functions lei> n (z)l, where n
=
1, 2,' .• , is constructed satisfying the following conditions: The func
tions el>n(z) are regular in the closure of B and they have an expansion of the form eI> n (z) == !,OOk_- Jl. a n kZ -k , where a n n > a' for n == 1 , 2 , " ' , in a neighborhood of
634 z
=
SUPPLEMENT
635
§3. FUNCTIONS ANALYTIC IN MULTIPLY CONNECTED DOMAINS
00; the derivatives ¢'n (z) constitute an orthonormal system in the domain B.
~~~
tpk (z)
tp~ (z) do =
8k n> k, n = 1, 2, •.• ,
B
that is complete in the class L~ (B) of all functions that are regular and square integrable with a unique inverse in B. Let 1
=
1, 2,"', denote a
system of functions that are regular in the closure of B except for a pole of order n at the point at infinity, in a neighborhood of which they have a Laurent expan
sion without constant term, and that satisfy on every boundary curve of the domain
B relations of the form
=
=
¢ n (z) + const. The system of pairs of the form
1, 2,""
is called a "Laurent system of functions" for
the domain B. The set of all points of B for which the Green's function g B(z, 00) of the domain B satisfies the condition g B(z, 00) < log R is called the boundary annulus set B1,R' where 1 < R. Milin showed that
If a function ljJ (z) is regular on the boundary annulus set B 1, R' then
1f'(O)lexp
K
r!~\
In particular 1['(0)1 ::; form f(z)
00
00
n=l
2)
+n=l ~ Antp~ (z);
00
2 A (~) =1t (~I An 1 -
n_1
where
A(ljJ)
00
~ 1An 12),
n=1
is defined by the same limit as the quantity A (f) mentioned above.
Thus, we have the following generalization of the area theorem: Let ljJ(z) denote a function that is regular on the boundary annulus set of a domain B. If A(ljJ) is nonnegative, then 00
00
~ I An 12 ~ ~ \.A n j2 n=l n=l
VA/TT,
with equality holding only for a function of the
az, where a is a constant.
=
From this we get the following theorem.
1) ljJ'(z) can be expanded on that set in a Laurent series
~' (z) = ~ An~ (z)
2~ ~ ~ 1F'(Z)12dXdY~~.
n
Let
denote a bounded domain in the z-plane that includes z = O. Let F
denote the class of all functions f(z) that have the following two properties: 1) f(z) is regular and univalent in
all z €
n.
domain
n
n;
2) f(O)
=
0, ['(0)
> 0,
and If(z)!
<1
for
Then, a function w = ¢(z) such that ¢'(O) = maxfEF['(O) maps the onto the unit disk with concentric circular cuts.
2°. Extremal properties of analytiC functions in multiply connected domains. The necessary and sufficient conditions found by Grunsky [1939] for univalence of a function that is regular in a multiply connected domain except for a simple pole at z
=
00
were then obtained by Bergman and Schiffer [195lJ (in a somewhat
different form) by a considerably simpler procedure, one that starts from the theory or orthonormal families. They developed a theory of kernels of the first and sec
T
A number of works have been devoted to the question of the existence and extremal properties of conformal mappings of infinitely connected domains onto canonical domains. Grotzsch [1955/56J gave a proof of the existence and unique ness of a univalent conformal mapping of an infinitely connected domain onto the unit disk with concentric circular cuts that did not involve the condensation prin
ond kind of a given domain, they established a relationship between these ker nels and univalent conformal mappings of a multiply connected domain onto canon ical domains, and they obtained a number of new extremal properties of functions that are univalent in a multiply connected domain. The more fundamental of these results are treated in the article by Schiffer [1953J. Bergman and Schiffer chose as their point of departure for finding these
ciple. Reich and Warschawski [1960J obtained a simple proof of the existence of
necessary and sufficient conditions for univalence of a function an inequality
a conformal mapping of an arbitrary bounded domain onto a disk with concentric
which they obtained, a simple special case of which is the inequality
circular cuts and, if that domain has at least one nondegenerate boundary
1{f (z),
z}
+
67';[ (z,
z) I ~ 61tK (z, z),
z
E B,
636
SUPPLEMENT
§ 3.
637
FUNCTIONS ANALYTIC IN MUL TlPL Y CONNECTED DOMAINS 1
where fez) is a funcrion that is analytic and univalent in a given finitely con nected domain B, !fez), zl is the Schwarz derivative of that function, and K(z, () and l (z, () are Bergman's kernels of the first and second kind of the domain B with respect to the class L 2 (B) of all functions that are regular and square integrable in B. It follows immediately from the results of Bergman and Schiffer (1951] that this inequality remains valid if we replace the function l (z, z) and K (z, Z-) respectively with the f~nctions lo(z, z) and Ko(z, Z-), where lo(z, () and Ko(z, () are Bergman's kernels with respect to the subclass L~ (B) (see above). Milin ob
holds for all functions F(z)€'1(B), For xk=Yk and zk=t k (for k=I,···,N), this inequality yields the range of values for the first sum on the left.
With his method, Milin also obtained [1964b] a number of new necessary and sufficient conditions for univalence of functions in a multiply connected domain, Singh [1962] found necessary and sufficient conditions for univalence and boundedness of a function that is regular in a multiply connected domain. Ozawa [1952] obtained necessary and sufficient conditions for a function that
tained [1964b] this second inequality by a different procedure and he showed not only
is regular in a given multiply connected domain to map that domain onto a Riemann
that it is sharp but that it defines the range of values for !fez), zl for fixed z € B
surface with area not exceeding
and function fez) ranging over the class of univalent functions in question, and
IT.
A convenient method for obtaining extremal properties of univalent functions
he found all boundary functions of the problem. He also obtained a number of more
is the use of the classical minimal property of the Dirichlet integral
general sharp inequalities. We give a theorem that is the analogue of Theorem 3
JB J(c/J';
of Chapter IV,
§ 2 for
Let B denote a finitely connected domain in the z-plane including the point at infinity with boundary consisting of closed analytic Jordan curves. Let '1 (B) denote the class of all functions F (z) that are univalent in B and regular in that =
00
=
+ c/J~2) dx dy. A systematic exposition of the possible use of this method
in the theory of univalent functions was given by Nehari [1953], who used this
a multiply connected domain.
domain except for a simple pole at z
(c/J, c/J) B
and that satisfy the condition F'(",,)= l.
method to obtain a large number of both familiar and new extremal problems. We present the following theorem of Nehari: Let Band B 1 denote two domains in the z-plane with boundaries C and C 1 respectively, Suppose that these boundaries consist of a finite number of closed
F or arbitrary given t € B, where t ~ 00, and () € [0, 77], consider a function j e(z, t) that satisfies the condition j e(t, t) = 0 in the class '1 (B) and maps the
analytic Jordan curves. Suppose that Be B
domain B onto· the plane with cuts along arcs of logarithmic spirals of inclina
larities all in the domain B. Let p (z) and PI (z) denote functions with the following prop
tion () to the rays issuing from the origin. Finally, we define
erties: the sums p (z) + S(Z) and Pl(z) + S(Z) are single-valued and harmonic
I
1 !2 (z,
t)
1
P (z t) = -- log 1t. ( t) , ' 2 Jo Z,
R (z, t) = 2 z, tEB.
For arbitrary complex numbers x
I ..! I
N
k. v=1
Zk
k
eS \
~
(z) on ds ~
~1 ds, d S (z) an \
1
-wi
This theorem enables us to show that the quantity N
xkYvlog F(Zk)-F(t y )
..!
xkYvR (Zk, tv)
k. y=1
N
~ J! k.~1
where the differentiation is in the direction of the outer normal to the domains B and B l'
in the domain B, the inequality
-tv
Let S(z) denote a function that is
mains; p (z) and p/z) are equal to zero on C and C 1 respectively. Then
and Yk (for k = 1,2, •.• ,N, where N;::: 1)
k
,
respectively in the domains Band B 1 and continuous in the closure of these do
(Z-t)2 log j1t!2 (z, t) jo (z,.u
Then we have the following theorem. and for arbitrary values of zk and t
1
univalent and harmonic in the closure of B 1 except for a finite number of singu
n
I
rn { where the (
N
XkXP(Zk, z.)
k.~/kJiP(tk'
t.)
(1)
~
fl,
~
n rtll CiyK
(~fl'~) -
y=1
are arbitrary fixed points in
fl,
~
ev!i (~I',~y)},
y=1
B and the
constants, decreases with broadening of the domain B.
a
~
are arbitrary complex
This result had been obtained earlier by Bergman and Schiffer [195I] by the variational procedure.
638
SUPPLEMENT
§3. FUNCTIONS ANALYTIC IN MUL TIPL Y CONNECTED DOMAINS ,
Nehari [1953] used this theorem to obtain inequality (2) of §2, subsection 1° of this supplement.
denote the set of all points (X, Y) == (1[;(zI)I, 1[~(z2)1) in the plane XOY in the class 9JHa , a ; zI' z2; B1' 8 2), Let F )z, a) denote the function described in I 2 Theorem 2, Chapter XI, §3 corresponding to the domain B v ' where v'= 1, 2.
Beginning with Kelvin's principle, Kubo [1954, 1955, 1955a] proved for uni valent mappings of multiply connected domains theorems analogous to certain
We have the following theorem:
theorems of Nehari [1953].
The set &(a , a ; zI' z2; B1' B 2) is the closed domain 2 I
Nehari's method [1953], in conjunction with the subordination principle (see
O:E;; Xy ~ I QI - Q2IgF~ (Zl'
below, subsection 3°) made it possible to obtain (Alenicyn [1956, 1958]) other extremal properties of functions defined on a multiply connected domain. We pre
with the boundary point
sent two theorems which generalize to conformal mappings (both univalent and
[or [unctions defined by equations o[ the form
nonunivalent) of a multiply connected domain the familiar theorem of Lavrent'ev Let B denote a finitely connected domain in the z-plane with no isolated boundary points and including the point z = 0. Let a and a denote distinct
I[v (z) I
1
[or arbitrary p>
[v (z) into the boundary component of the domain D v that is retained as the domains D v' for v = 1, 2, are broadened to a system of disjoint domains. Let =
[(z, 0, C) denote a function that maps the domain B univalently onto the unit
C) =
-
-all - a1
It)
=
= -1P Ell FII (Z,
ZII
)
1.
over (see Alenicyn [1966]) the results of Bergman and Schiffer [1951] mentioned above to the case of several functions that are univalent and have no common values in a multiply connected domain. Specifically, we have the theorem:
Let B denote a bounded and finitely connected domain in the z-plane bounded by closed analytic Jordan curves. Let [k(z), [or k
I 1: {!
with equality holding only in the case of functions defined by the equations
I, (z) I, (z)
'=
I122 (z) (z) -- all a1
)
Z, Zl,
= 1,'"
, n denote [unctions
denote arbitrary constants [or /l = 1," . , Nand k = 1, ... , n. Then, N n
Q21 2 1f' (0, 0, C1)f' (0, 0, Cg) \'
II (z) - a 1 = pf (z, 0, C1), 11 (z) -a.
1
common values in that domain. Let (kIJ- denote arbitrary points in 8 and let akIJ
0, where v'= 1, 2. Then
If; (O)f~ (0) I:E;; I Ql
It)
F (
that are regular and univalent in the domain 8 and suppose that they have no
disk with concentric circular cuts such that C v is mapped into the unit circle and [(0, 0,
and
pEl
Starting with an examination of suitable DiricWet integrals, one can carry
2
denote a sequence of functions that map B univa
lendy and conformally onto disjoint domains D v' Define a v = [)O) for v'= 1,2. Suppose that the boundary component C v of the domain B is mapped by the func tion w
°
Z2) F~ (Zg, ZIl)
deleted. Equality holds in the second inequality only
00
11 (z) - a1 = I 1 (Z ) - a2
(Chapter IV, §4) on univalent conformal mappings of a disk onto two disjoint domains.
finite numbers. Let
639
p..
>=1
~
2
2-f (z, 0, CIl),
IJ.kp.lJ.k. [Uk
k=I
+ 1 (~kp.' ~kv)] 0
V 1: IJ.kl'-~k.Ko (Ckl'-' ~k»'
I] (~jr,) Ii. (Ckl') } I IJ.jl'lJ.". (f· (C· )-h (~k»lI ~ ~
+-; l~j
p
(~"p.' ~kv)
1
11'
I'
n
1'- •• =1 k=l
where p is an arbitrary positive number.
where
Let Bland B 2 denote any two finitely connected domains in the z-plane that have no isolated boundary points and do not include the point z and a
2
=
00.
Let a
given numbers in the domains 8
1
and B 2 respectively. Let [I(z) and [/z) de
note functions that are meromorphic in Bland B 2 respectively and that satisfy the conditions [v (z v)
B 2' Let
=
m(aI'
a v for v
'=
1, 2 and [1 «() -f- [2 (r) for arbitrary ( €
81
z l' z 2; B l' 8 2) denote the class of all pairs of func tions [/z) and [/z) with these properties. Let &(a , a 2; zI' z2; 8 1 , B 2) I
and T €
l
Uk(z,
denote any two given distinct finite numbers. Let z 1 and z 2 denote
a ; 2
and Ko(z,
Z)
1[
II. (z) I;' (C) Q=-; (fk(z)-lk(~»2 -
1] ,
(Z-C)2
z, CEB,
and lo(z, () are the Bergman kernels o[ the domain B with respect
to the class L~ (B). For n [1951].
=
1, the theorem yields the inequality found by Bergman and Schiffer
640
SUPPLEMENT "
As a result, one obtains analogous inequalities for the class ~ (B) of func
Q(z, a, b)
§4
of
of Chapter V defined on B. Then, for arbitrary points z.
tions [(z) that are regular and univalent in the domain B and satisfy the condi
z.
~on fez 1 )[(z 2) .;, 1 for arbitrary z 1 and z 2 in the domain B and for the class
1, 2,'" , m), we have the inequality
~(l) (B) of functions
1
~(B),
1:
then
~(l/B)'
)9
+ 1C1o (z, z)
~ 1tKo(z, z),
Z
EB.
m
Ik~O •• fJ m "" ..::..l V,
,,'=1
, (z, J \1k;r.!""/k /v
(for j
I
1V
, )
Z ·v'
I
J
~~ ~ n
and
lv
0, 1,' •. , n and v
=
=
m
''[j/ij"pj(z/v, Zj'" (1j),
1=0 " .'=1
,-, (ZJ" ' z'j"1 a) '1j.'1j"Pj J
where In (fj(z)-!k(C))(Ok-Oj) (fj (z) - Ok) (fk (C) - OJ) ,
If' (z) /9 12)2 . 6!' z +1tlo(z, z) ~1tKo(z, z)- (1-I/(z) 1 I
-
(2)
All these inequalities remain valid if we replace in each of them the kernels
o
n
j= 0
'I
and y.
1V
then
1 {}
K
1
n
1 {!, z } ± ( (z) (z) If \ 1I_' 19
If fez) €
J v'
in B. and arbitrary numbers y.
fez) that are univalent and bounded in the domain B:
1[(z)1 < 1. In particular, we have If fez) €
641
§3. FUNCTIONS ANALYTIC IN MULTIPLY CONNECTED DOMAINS
and 1 with the kernels K and 1 with respect to the class 0
L2 (B), but, in
general, such a substitution yields weaker inequalities. Inequality (2) with kernels K and 1 had been obtained earlier by Singh [1962]. Furthermore, necessary and sufficient conditions have been obtained for func
l
j#-k i'#O k#-O' ,
,
1
In/o(C)-!i(Z) J'--I-O k=O' 10 (C) - OJ' -r-' ,
rnkJ (t:, z) = T
In 10 (z) - Ik (C) J' = lo(z)-ak '
Irn
°k
--I- 0' '-r- ,
(fk(Z)-!k(C))lk(ak) (fk (z) _ Ok) (fk (C) _ Ok)
+
.
qk
lln[(!o(Q-!o(Z»!~(C1o»)+qo(t:,
(~, z, C1k), J = k -::j::. 0;
z, (10),
j=k=O,
tions regular in a given multiply connected domain to be univalent in that domain
and those branches of the function ¢k' «(, z) are chosen for which ¢k' «( , a . )
and to have no common values, In the case of a single function, these conditions
and where [~(ao) means limz-a.o (l/(z - ao) [o(z)).
1
1
1
yield the familiar necessary and sufficient conditions found by Bergman and Schif
Mitjuk [1961, 1961a, 1961b, 1964, 1965, 1965a, b, d) took numerous results
=0
fer [1951] for univalence of a function in a multiply connected domain. In the case
known for functions that are analytic in finitely connected domains and carried
of several functions, they constitute a new result even for a disk. Also, neces
them over to functions that are analytic in infinitely connected domains. In doing
sary and sufficient conditions have been found for a function to belong to the
so, he sharpened certain res ults known already for a circle (various covering
class
Ii (B).
theorems, theorems on the product of conformal radii of disjoint domains, etc.).
By extending to multiply connected domains the method of his earlier work on
We present some of his results.
in integral form and a number of new inequalities of a general nature for univalent
Suppose that the function w = fez) = Wo + ap(z - zo)P + ap+/z - zO)P+l+ ... , where a p .;, 0, is regular in a domain B of the z-plane that has an interior radius r
functions that map finitely connected domains onto disjoint domains. The follow
about a point Zo belonging to B (see, for example, Hayman [1958]). Suppose that
disjoint domains ([1961], see
§ 1),
Lebedev obtained a generalized area theorem
Zo is one root of the equation fez)
ing result is a particular case of one of these inequalities: Let B., / = 0, 1", . , n, be finitely connected domains in the z-plane with no 1
isolated boundary points. Let a. denote an arbitrary point in B .. For each /, let
f.1 (z)
1
1
denote a function defined on B.. Suppose that these functions perform 1
function of the domain B. Let R denote the interior radius about the point
B;
plane) or of the domain
a.=[.(a) for /';'0 and fo(ao)=lXJ. Let p(z, (, a) and q.(z, (, a) denote
line passing through w O' Then,
111
1
1
1
1
1
W
the image B f of the domain B (here and in what follows considered in the w
univalent conformal mappings and that their ranges are disjoint domains. Define functions in the domain B. corresponding to the functions P (z, a, b) and
= woo Let zk' for k = 1, 2,"" denote the
other roots and let p k denote their multiplicities. Let g 8(z, () denote the Green's
obtained from
o
Bf by symmetrization about a ray or
of
____
. --.--...,.....::;;=-.--=-----=-=--==---;;=--=.-=-=--=..--==-----,----.-_ _ --=--=.=-;-.;;..-,-..:.==::=..:.0-;;'_==_-=-=.==--=-=...
~--,-.--
642
-=._._.._ ..
.~.
- __._.
..,....__._-=
="-',,"',-""-.
SUPPLEMENT
§3. FUNCTIONS ANALYTIC IN MULTIPLY CONNECTED DOMAINS
643
I
extremal properties have been established by using the capacity of such a domain
R;:::: I ap I rP exp ~ PkgB (ZO, Zk)·
as its characteristic [1965bl.
k~l
This result [1964a] generalizes to an arbitrary domain and strengthens the familiar symmetrization principJe of Hayman [1958] for a disk.
lem of boundary distortions for multiply connected domains.
Mitjuk obtained [1965a] the following inequality for the area S (B f) of the image B f of the domain B under the function w = f(z) defined above:
S (Sf);:::: 1t [ I ap I r P exp ~ PkgB (zo, Zk) k~l
New results have been obtained by a number of authors as regards the prob
r
Thus, by combining the extremal-metric method with the method of symmetriza tion, Jenkins [1956] obtained a number of theorems on boundary distortions for uni valent mappings of a multiply connected domain and, in particular, he generalized to multiply connected domains the familiar theorem of Loewner [1923] for a disk.
For Lavrent'ev's theorem on univalent mappings of a disk onto two disjoint
Dealing with this class of questions is the article by Tamrazov [1965a], in
domains, which we have repeatedly referred to, Mitjuk obtained [1965a] the follow
which he established extremal properties of univalent conformal mappings of mul
ing generalization and sharpening.
tiply connected domains with specified boundary components.
Let w
f.J (z J,) = w.J
=
f.J (z)
denote functions that are regular in a domain B. We write
B. Suppose that these functions map B onto B f for j = 1,2. J j Suppose that one or the other of the following two conditions is satisfied: for z. €
t
and B f
2
have disjoint domains symmetric about certain
lines or rays passing through the points w
t
and w . 2
sent some new results dealing with an extension of the subordination principle to multiply connected domains. Here, in the case of a disk, we shall stop only for those results that are a direct sharpening of the corresponding theorems in
Then,
Chapter VIII, §8.
IJ (Pll (Zl)Jr:9) (Z2) I 1
in the unit disk and a number of theorems on subordinate functions were given. In recent years, the theory of subordination has been developed further. We now pre
a) the domains B fj and B f2 are disjoint, b) the domains B f
3°. The principle of subordination for multi ply connected domains. In Chapter VIII, §8, the concept of subordination was defined for functions that are regular
~
I 'W1 -
19
'W :I
r P1 9 1 r P9
exp
[_
'\1 P"gB (Zt>
J.
z;') - ~ p'kgn (Z2' z'k)] ,
k~.t
where p. is the multiplicity of the root z. of the equation
k~t
f.J (z)
- w. = 0, for
As a supplement to Theorem 3 of §8 of Chapter VIII regarding the majoriza tion of the first two coefficients of a subordinate function, Xia Dao-xing (see the article by Xia Dao-xing and Zhang Kai-ming [1958]) obtained the following result, that cannot be improved:
Suppose that two functions f(z) = I.~=o akz k and F(z) = I.~=o Akz k are regu 1, 2,'" , are respectively the roots of the equa la~in the disk Izi < 1 and suppose that f(z)-<. F(z) in Izi < 1. Then, la31::; tions f/z) - w t = 0 and f/z) - w 2 = 0 distinct from Zt and z2 and with multi y2 max<\ At l, IA2 1, IA31L plicities p~ and p~ and where r j is the interior radius of the domain B about Chapter VIII, §8 includes two theorems of Biernacki (Theorems 6 and 7), and the point Z., for j = 1, 2. J the conjecture was made that the number ~ in Theorem 6 can be replaced by These inequalities are special cases of more general results obtained by (3 -15)/2 and the number 0.12··· in Theorems 7 by 3 The validity of Mitjuk from the "principle of admissible transformations" [1965d], which include, J
j
=
1, 2, and z~ and z~, for k
J
J
=
vB.
in particular, symmetrization about a straight line, circular symmetrization,
these conjectures of Goluzin was proved by Xia Dao-xing [1957, 1957a], who ob
Szego-Markus symmetrization (see § 2 of this supplement), and the operation of
tained the following theorems:
broadening a domain until it is simply connected by adjoining to it all components
Suppose that two functions f(z) and F (z) such that f(o)
Izi < 1.
of its complement with respect to the plane except for one component (suitably
lar in
chosen). We point out also that, for functions that are regular in a multiply con
argf'(O)
nected domain whose boundaries contain analytic Jordan curves, a number of
=
Suppose that F(z) is univalent in the disk
=
F (0)
Izi < 1.
=
0 are regu
Suppose that
argF'(O) and f(z)-<. F(z) in that disk. Suppose that f(z) ~ F(z). Then
644
SUPPLEMENT
§3. FUNCTIONS ANALYTIC IN MULTIPLY CONNECTED DOMAINS
645
I
1) the inequality If(z)1 < IF(z)1 in the annulus 0 < Izi < (3 - '1/5 )/2, but,
for an arbitrary point
Zo such that Izol
=
(3 -
i"5)/2,
fying all the conditions listed for which If(zo)!
I['(z)! < jF'(z)1 such that !z 0 I > 3 -
2) the inequality
arbitrary point z 0
=
there exist functions satis
IF(zo)!;
in the annulus 0
< Izl ~
3-
is, but, for an
'1/ 8, there exist functions satisfying the
Thus, Biernacki's problems posed in Theorems 6 and 7 have obtained a
=
and satisfy the conditions f(o)
1. Suppose that F (z) is univalent in
1
=
F (0)
0,
=
If(z)! :::; F (z) in Izi < 1. Find the greatest radius R of a disk Izi < R in which Lewandowski [1961] showed that such a number
R exists and that 0.21 < R ~
0.29' • '. Under the additional assumption that F (z) is starlike in
!z I <
1, the
number R is a root of the equation R 3 + R2 + 3R - 1 = 0 (Lewandowski [1%la]). On the other hand, if we replace the assumption that F(z) is starlike with the assumption that fez) is univalent, R is a root of the equation 10g[(I+R)/(I-R)]+ =
17/2 (Bielecki and Lewandowski [1%2]).
In Chapter XI,
§ 3,
v
that maps the domain B univalently onto the unit disk with con
centric circular cuts. If, as we assume, the supplementary condition ['(a, a, C »0 v
~
f€Jl(l)(B,a)
I['(a)! is satisfied only by a function fez, a, C ), v
for which F'(a, a)
v
=
max v =l,- ", m ['(a, a,
Obviously, the function
F(z,
C).
1" .. , m,
""
We denote this function by F(z, a).
a) in general is not unique.
If the domain B is the disk Izi < 1, then F(z, a) =
=
F'(a, a)
=
1/(1 - lal
2
=
F(z, a)
=
(z-a)/(J-az)
).
Let fez) denote a function that is meromorphic in a domain B. We shall say that fez) is subordinate in B about a point z 1 E B to a function g (r) that is meromorphic in the disk
I rl
< 1 if, for every z in B, we have fez) = g(w(z)),
where w (z) is a function in the class at(l/B, z /
We indicate this relation by
j(z) -< g('t), zEB.
(1)
ZI
If in addition w (z) E at(l)(B, z 1)' we shall say that the function fez) is uni
Theorem 2 illustrated the role of the function F (z, a) in
theorem can be regarded as one of the simplest applications of the following ex tension to multiply connected domains (Alenicyn [1961]) of the subordination prin ciple in the case of a circle. Let B denote a finitely connected domain in the z-plane with no isolated boundary points and not including the point z
=
00.
Let ~(1) (B) denote the class
of all functions fez) that are regular and satisfy the inequality If(z)ll< 1 in B. Let ~(l) (B, a) den~te the subcl~ss of functions in ~(l)(B) such that f(a)
=
0,
where a E B. Let ~(l)(B) and ~(l/B, a) denote the subclasses of univalent functions in ~(l/B) and ~(l/B, a) respectively. Thus, F(z, a) is that one of the functions in the cl~s ~(l)(8, a) for which F'(a, a)
=
maxf€~(l)(B,a) I['(a)[.
Consider the function F (z, a), which plays an analogous role with respect to the
~(0(8,
C)
valently subordinate to the function g (r) and we shall indicate this by
the solution of certain extremal problems in the class of bounded functions. This
class
at(l)(B, a,
I['(a)! is solved by a function fez, a, C ) in the class
and, consequently, F'(a, a)
zl < 1 and that
the function fez) does not assume values that the function F(z) assumes in it.
2 arctanR
~
fGJl(o(B,a,C v )
""
inverse of Biernacki's problems. Specifically, let fez) and F (z) denote functions ['(0) ~ 0, and F'(O)
""
max
Lewandowski [1961, 1961al posed a problem that in a certain sense is the
Iz I < 1
a) that map the C veach into an external boundary
is satisfied, this function is unique. It follows that the problem of finding
definitive solution.
that are regular in the disk
~(o(B,
component of the image of the domain B. We know that the problem of finding max
I['(zo)l> 1F'(zo)j.
conditions listed, for which
all functions f(z) in
a). Specifically, suppose that the""boundary components of an m
connected domain Bare C 1 ,"', C m and that lR( 1 ) (B, a, C v ) is the subclass of
f(z)':< g('t), zEB. zi
If a function g (r) is univalent in the disk
(2)
•
Irl
< I, the relation (1) means
simply that the function f(z) does not assume in the domain B those values that the function g(r) does not assume in the disk Irl < 1 and that f(zl)
=
g(O). The
relation (2) means in addition that fez) is univalent in B.
If the domain B is the unit disk Izl < I and z 1 respectively the familiar relations
=
0, then (1) and (2) are
j(z)-
If fez) -< ZI g(r) in a domain B and if the function fez) is regular at a point z l' then I['(z 1)1 :::: Ig '(0) I F'(z l' z 1)' with equality holding (provided g' (0) 0)
t
646
§ 3.
SUPPLEMENT
only for fez)
g(fF(zl' zl»' where
=
"-
If I =
1. This assertion remains valid if we
FUNCTIONS AN AL YTIC IN MUL TIPL Y CONNECTED , DOMAlNS
II
0:0 _
H( Z,. a)-- e i~
"-
replace -< with -< and F with F.
Z-Ol
_
I-all
n=1
(1 _ q~I1.3..-) (I _ 01
647
q211 01 )
Z _ I . , (1-q2n atZ )(1_q211 alZ) \
By means of this subordination principle, we can find theorems of a general nature that enable us to obtain new distortion theorems for functions that are ana
where a is defined by the condition H (1; a \) = 1.
lytic in a multiply connected domain and, in particular, to carryover to multiply connected domains certain familiar theorems for a disk. For example, we obtain the following two theorems in this way: Let PTt denote the class of all functions fez) that are regular in the disk
Izi < 1
For arbitrary given a € B q and Zo € B q H(z, 1; ao(zo)' a)
of the covering taken into account) does not exceed
Tt.
Then, if a function fez) ~ F'(z, z)
anYj>oint in B to one of the functions in the class P Tt' then
1['(z)1
(~ F'(z, z)) for z €
z \ only for functions
B with equality holding at the point z
=
with equality holding (when Zo 1= a) only for fez)
o is a simple uncovered point of the function fez) in the domain B if the function log [fez) - w o] is regular in that domain. Then: If the function fez) is regular in a domain B and satisfies the conditions =
=
1, where a € B, and if If(z)!
MF'(a, a) (x> 0) in the disk
< M (MF'(a, a) > 0, then
Iwl < Me -x
When the domain B is the unit disk, this theorem yields Landau's familiar theorem [1929].
fH(z, 1; ao(zo)' a), where
In the case of the annulus B q ,we have (Alenicyn [1958]) the following rela
§4. We shall now elaborate on this question, using the remaining results of that paper and Alenicyn's paper [19611. Let B q denote a circular annulus q
n
H(z; a),
~=-argH'(a,
1;-
~, a).
Starring with Robinson's inequality (3), which is obviously a generalization of the first of the inequalities in the Schwarz lemma (see Chapter VII,
§ 1)
to the
case of a circular annulus, we can obtain theorems of a general nature that en able us to carry over to the case of a circular annulus certain distortion theorems and its simplest consequences. For brevity, we write H (z, 1; a (z), a) = H (zL o Let 9Jl* denote a class of functions g (z) that are meromorphic in the disk
Izi
< 1 and let 9Jl a (B) denote the class of all functions fez) that are meromorphic
in the domain B and that are subordinate in that domain about a point a in it to
Certain results from Robinson's paper [1943] were presented in Chapter XI,
< Izi < 1,
one of the functions in the class 9Jl*. We indicate this relationship between these two classes by writing '!!J!a(B) -< 9Jl*" Let us suppose that (w) is a real function
(lzl)
Let M
a 1 € Bq
Under these conditions, we have the theorem:
there exists a unique function H (z; a 1 ) that is regular in the closure of
the annulus B q , that is univalent in B q itself, and that satisfies the conditions
1, H(l;at)=l. H(at;at)=O,IH(z;at)I,Ii! 1_ - =Iatl, q . IH(z; at)l\ z /_1By using the corresponding results of Robinson [1943], we find, after some easy
m*. Izl
defined for all values of w assumed in the unit disk by functions in the class
where q> O. We know from Robinson's paper [1943] that, for arbitrary given
transformations,
=
(3)
with which we are familiar for a circle. We present one of these general theorems
but it is not always true
that this function does not have simple uncovered points in a larger disk.
,
zo,
I f I = l-
F (z, a)= ei@z-tH(z; -
W
0, ['(a)
arg
1!(zo)I~IH(zo, I; adzo), a)1
Suppose that a function f(z) is regular in a domain B. We shall say that a
=
(q/\al) e i
in the annulus B q and an arbitrary function f(z) in the class ~(1)(B q' a), we have
!(z)=!(zt)+eF(z, Zt) (!(zt)+eF'(z, Zt», lel=I.
f(a)
=
tionship between the functions exhibited by AhHors and Robinson:
of the form
x -1 sinh x
we set ao(zo)
z-\ H(z; ao(zo»H(z; a). Then, for an arbitrary given point Zo
and map it onto surfaces the area of each of which (with the multiplicity
that is regular in a domain B is subordinate (univalently subordinate) in B about
point
=
,
1
(M/1z!)) denote a decreasing (increasing) function of
Izi
for
If one of the inequalities
M t (I
Z
l)~cD(g(z» ~M2 ([ z I),
I Z 1< 1,
holds in the class '!!J! , then the corresponding inlquality
*
(4)
~~
--
-
~.
648
_
..
~
;"";;;;"""~"~.-".~
Mt(IH(z)I)~(j(z))~M2(IH(z)I),zEB q ,
m*,
Here, equality holds at a point z
gfH(z)fH(z, 1; a O(zl)' a)), where
for fez)
=
1Zo I < 1
the function g
1fl
=
problem of finding =
1 and for a point
z 1 (~a) only Zo
with
(z) is extremal for the corresponding inequality (4).
the constant e
that annulus about a point a in it to one of the functions in the class S,
1-11~\ 12'
"_ ",_____
;;;.;:;;.;",,~~
___,_
._....._.. "'
Z
sUPf€iR(l)(Bl'.r cf(z)w(z)dz!
649
either is identically equal to
(where a is a real constant) or maps B onto the m-fold covered Nk
r
.
2
=1
The problem considered in this theorem generalizes, in particular, the problem
For an arbitrary function fez) that is regular in the annulus B q and subordinate in
I H (z) I
ia
J
From this theorem it follows, in particular, that
_I
,,"c,"
unit disk, where m ~ n. If w(z) = ~k=1 ~j=1 Il kj /(z - z)J, where a EB and the k
+ ~~ N k'
Ilk. are constants, then n S m -
zo
Ij(z) I ~ (1
";'~~~~
§3. FUNCTIONS ANALYTIC IN MUL TIPL Y CONNECTED DOMAINS
SUPPLEMENT
holds in the clas s ~I)B q) <
;~,
E B q,
(5)
of finding sUPf€~(l)(B) 1~:=1 Akf(ak)j, (where a k E B and the A k are arbitrary complex constants) and the problem of finding sUPf€ i R(l)(Bl 1['(a)l, The first of these problems was investigated by Garabedian [1949]; the second was solved by AhHors. With regard to Havinson's generalization [1961] of the inequality in the
with equality holding at a point z
=
z
1 (~a)
only for
j(z) = eH(z, 1; ao(zt), a)/(I_eiargH(zl) H(z, 1; ao(zt), a))2,
Schwarz lemma to arbitrary domains, the question is as follows: let C denote a
jel=1.
closed bounded set in the z-plane, let B denote whichever of the domains Com plementary to C includes the point z
One can show that inequality (5) yields the range of values of the functional
f (z)
for fixed z E B q and the function
f
as it ranges over the entire clas s of
functions considered here. AhHors's theorem (Chapter Xl, §3, Theorem 1) can be regarded, on the one tiply connected domain onto a multiply covered disk and, on the other hand, as a generalization of the inequality
1['(0)1
S 1 in the Schwarz lemma for a disk to
00,
and let y denote an arbitrary Contour
encircling C. The problem is to investigate the properties of the extremal func tions in the problem of finding logue, for the case
hand, as an example of a problem in which an extremal function maps a given mul
=
a~ e
z
ia =
00.
a=
00,
sUP €iR(1)(B) f
(1/277) Ifyf(z) dzl, which is the ana
of the problem of finding sUPf€~(l)(B) 1f'(a)1 when
Havinson showed that the extremal function f*(z) is unique up to a factor
(where a is a real constant), that all its zeros except a zero at the point 00
lie in the convex envelope of the boundary set C, and that this function
aSSumes in B all values in the unit disk except possibly a set of values of ana
finitely connected domains. Havinson showed in a series of articles []953, 1953a,
lytic capacity zero (see op. dt.). Because of the complexity of the formulation of
1955, 1961] that this geometric property of an extremal function holds in a very
the other properties of the extremal function f*(z), we cannot give them here.
broad class of problems and he generalized the inequality mentioned to arbitrary domains. The methods used in solving these more general problems and, for that matter, the very statement of the problems would take us beyond the framework of the -geometric theory of functions, and we present only a few of the simpler results of Havinson. Let
B
denote an n-connected domain in the z-plane. Suppose that its bound
ary C consists of closed analytic curves. Let
E 1 denote a class of functions
that are regular in the domain B and that can be represented in that domain by means of a Cauchy integral in terms of their boundary values. Then we have the theorem:
Suppose that w (z) is an analytic single-valued function on each of the bound ary curves of the domain B and that it does not coincide on any of these curves with any function in the class E 1 in B. Then the extremal function in the
_
;, ---
-
- ,-- -
-,~.
,~
r
-- -
-'
-_.
-_."
- _.. _-------,,-_ .._--------_._,--"'---------_._-_._.,--_.,_._----,._---- '~~.-~~~~~~~~~~
"
~~ \
:i
I',,~
BIBLIOGRAPHY FOR THE SUPPLEMENT H. Abe
1958/59 On univalent functions in an annulus, Math. Japon. 5, 25-28. MR 21 #4248. N. I. Ahiezer and M. G. Krein
1938
Some questions in the theory of moments, Nauen. Tehn. Izdat. Ukrain. Harkov; English trans., Trans!. Math. Monographs, vol. 2, Amer. Math. Soc., Providence, R. I., 1962. MR 29 #5073. L. V. Ahlfors
1930
Untersuchungen zur Theorie der konformen Abbildung und der ganzen Funktionen, Acta Soc. Sci. Fenn. A. 1, no.9, 1-40.
L. V. Ahlfors and A. Beurling
1946
Invariants conformes et problemes extremaux, C. R. Dixieme Congres Math. Scandinaves, 341-351, Jul. Gjellerups Forlag, Copenhagen, 1947. MR 9, 23. I. A. Aleksandrov 1958 The range of values of the functional 1= wnf'(w)m /f(w)n If(w)1 1 in the class S, Tomsk. Gos. Univ. Ueen. Zap., no. 32, 41-57. (Russian) 1958a Conditions for convexity of the image region under mappings by func tions regular and univalent in the unit circle, Izv. Vyss. Ucebn. Zaved. Matematika no. 6(7), 3-6. (Russian) MR 23 #A3834. 1959 On the star-shaped character of the mappings of a domain by functions that are regular and univalent in the circle, Izv. Vyss. Uce~n. Zaved. Matematika no.4(11), 9-15. (Russian) MR 26 #2601 1960 Domains of definition of some functionals on the class of func tions that are univalent and regular in a circle, in Series on Modern Problems in the Theory of a Complex Variable, Fizmat giz, Moscow, 39-45. (Russian) MR 22 #5744.
1963
f,
f ')
Boundary values of the functional J = J (f, f', on the class of holomorphic functions univalent in a circle, Sibirsk. Mat. 4, ]7-31 (Russian) MR 26 #3892.
Z.
1963a Variational formulas for univalent functions in doubly connected do mains, Sibirsk. Mat. Z. 4, 961-976. (Russian) MR 27 #5902.
651
652
BIBLIOGRAPHY FOR THE SUPPLEMENT
BIBLIOGRAPHY FOR THE SUPPLEMENT
653
1
1963b Variation of non-univalent analytic functions, Trudy Tomsk. Gos. Univ. Ser. Meh.-Mat. 163, 155-159. (Russian) 1963c Extremal properties of the class S(w o)' Trudy Tomsk. Gos. Univ. Ser. Meh.-Mat. 169, 24-58. (Russian) MR 29 #3627. 1964 Geometric properties of schlicht univalent functions, Trudy.Tomsk. Gos. Univ. Ser. Meh.-Mat. 175, 29-38. (Russian) MR 32 #2571. 1965 The range of systems of functionals, Trudy Tomsk. Gos. Univ. Ser. Meh.-Mat. 182, 59-70. (Russian) MR 33 #7514. I. A. Aleksandrov and V. V. Cernikov 1963 Extremal properties of star-like mappings, Sibirsk. Mat. Z. 4, 241 267. (Russian) MR 26 #6387. I. A. Aleksandrov and V. Ja. Gutljanskil 1965 Extremal problems for classes of analytic functions having a structural formula, Dokl. Akad. Nauk SSSR 165, 983-986 = Soviet Math. Dokl. 6 (1965), 1531-1535. MR 33 #5900. 1966 Extremal properties of close-to-convex functions, Sibirsk. Mat. Z. 7, 3-22. (Russian) MR 33 #4270.
1965
Conformal mappings of multiply connected domains onto surfaces of several sheets with rectilinear cuts, Izv. Akad. Nauk SSSR Ser. Mat. 29,887-902. (Russian) MR 32 #2587. 1966 On univalent functions without common values in a multiply connected domain, Dokl. Akad. Nauk SSSR 167, 9-11 = Soviet Math. Dokl. 7 (1966), 305-307. MR 33 #7515. V. A. Andreeva, N. A. Lebedev and A. V. Stovbun 1961 On the range of certain systems of functionals in various classes of regular functions, Vestnik Leningrad. Univ. 16, no.7, 8-22. (Russian) MR 24 #A243. G. K. Antonjuk 1958 On the covering of areas for functions regular in an annulus, Vestnik Leningrad. Univ. 13, no. 1, 45-65. (Russian) MR 20 #2434. I. Ja. Asnevic and G. V. Ulina
1955
On ranges of values of analytic functions represented by a Stieltjes integral, Vestnik Leningrad. Univ. 10, no. 11, 31-42. (Russian) MR 17, 599. I. E. Bazilevic I. A. Aleksandrov and V. I. Popov 1951 On distortion theorems and coefficients of univalent functions, Mat. Sb. 1965 Solution of a problem of T. E. Bazilevic and G. V. Korickit on star 28 (70), 147-164. (Russian) MR 12, 600. shaped arcs of level curves, Sibirsk. Mat. Z. 6, 16-37. (Russian) 1951a On distortion theorems in the theory of univalent functions, Mat. Sb. MR 30 #3202. 28 (70), 283-292. (Russian) MR 13, 640. Ju. E. Alenicyn 1957 Regions of the initial coefficients of bounded univalent functions of 1956 On univalent functions in multiply connected domains, Mat. Sb. 39 (81), p-fold symmetry, Mat. Sb. 43 (85), 409-428. (Russian) MR 20 #3991. 315-336. (Russian) MR 18, 292. 1959 On an estimate of the mean modulus in a class of bounded univalent 1956a A contribution to the theory of univalent and Bieberbach-Eilenber~ functions, Mat. Sb., 48 (90), 93-104. (Russian) MR 22 #1679. functions, Dokl. Akad. Nauk SSSR 109, 247-249. (Russian) MR 18, 293. 1961 On an estimate of the mean modulus and coefficients of univalent 1958 On functions without common values and the outer boundary of the do functions, in Series on Modern Problems in the Theory of Functions of main of values of a function, Mat. Sb. 46(88), 373-388. (Russian) a Complex Variable, Fizmatgiz, Moscow, pp. 7-41. (Russian) MR 20 #6532. 1961a An asymptotic property of the derivatives of a class of functions that 1961 An extension of the principle of subordination to multiply connected are regular in a disk, in Series on Modern Problems in the Theory ~of regions, Trudy Mat. Inst. Steklov. 60, 5-21; English trans!., Amer. Functions of a Complex Variable, Fizmatgiz, Moscow, pp. 216-219. Math. Soc. Trans!., (2) 43 (1964), 281-297. MR 25 #1281. (Russian) 1962 On the ranges of variation of systems of coefficients of functions rep 1964 Generalization of an integral formula for a subclass of univalent func resentable as a sum of Stieltjes integrals, Vestnik Leningrad. Univ. tions, Mat. Sb. 64(106),628-630. (Russian) MR 29 #3625. 17, no. 7, 25-41. (Russian) MR 25 #2179. 1965 On the spread of the coefficients in the expansions of univalent func 1964 Conformal mappings of a multiply connected domain onto many-sheeted tions, Mat. Sb. 68(110),549-560; English trans!., Amer. Math. Soc. canonical surfaces, Izv. Akad. Nauk SSSR Ser. Mat. 28, 607-644. Trans!. (2) 71 (1968), 168-180. (Russian) MR 29 #1315.
654
BlI3LIOGRAPHY FOR THE SUPPLEMENT
I. E. Bazilevic and G. V. Korickii
1953
MR 14, 632.
Some properties o[ level curves on univalent con[ormal mappings, Mat. Sb. 58(00), 249-280. (Russian) MR 26 #2600. I. E. Bazi1evic and N. A. Lebedev 1962
1966 On the dispersion of the coe[[icients o[ mean p-valent functions, Mat. Sb. 71 (13), 227-235; English trans!., Amet. Math. Soc. Trans!. (2) 72 (968), 107-117. MR 33 #7516. S. Bergman
1959
Domaine de variation des coef[icients A 2 et A 3 des fonctions uni valentes et bornees, Bull. Soc. Sci. Lettres l.odz, 10, no. 4. MR 23 #A3841. Z. Charzynski and M. Schiffer 1960
J. C1unie 1959
1951
J.
The kernel [unction and con[ormal mapping, Math. Surveys, no. 5, Amer. Math. Soc., Providence, R. 1. MR 12, 402. S. Bergman and M. Schiffer Kemel functions and conformal mapping, Compositio Math. 8, 205-249. MR 12, 602. A. Bielecki and Z. Lewandowski 1962
Sur certaines majorantes des [onctions holomorphes dans Ie cercle unite, Colloq. Math. 9, 299-303. MR 27 #2625. M. Biernacki 1936
Sur les [onctions univalentes, Mathematica (Cluj) 12, 49-64.
1956
Sur les coef[icients tayloriens des [onctions univalentes, Bull. Acad. Polon. Sci. Cl. III, 4, 5-8. MR 17,957.
S. Bochner
Uber orthogonale Systeme analytischer Funktionen, Math. Z. 14, 180-207.
E. Bombieri 1963 Sui problema di Bieberbach per Ie [unzioni univalenti, Atti Accad. Naz. Lincei Rend. C!. Sci. Fis. Mat. Natur. (8) 35, 469-471. MR 29 #3622. C. Caratheodory
A new proof o[ the Bieberbach conjecture for the fourth coefficient, Arch. Rational Mech. Anal. 5, 187-193. MR 22 #5746.
1960a A geometric proo[ of the Bieberbach conjecture [or the fourth coeffi cient, Scripta Math. 25, 173-181. MR 22 #9615.
1950
1922
655
Z. Charzynski and W. Janowski
On some properties o[ univalent con[ormal mappings, Mat. Sb. 32 (74), 209-218. (Russian)
BIBLIOGRAPHY FOR THE SUPPLEMENT 1
On meromorphic schlicht [unctions, J. London Math. Soc. 34, 215-216. MR 21 #5737. 1959a On schlicht [unctions, Ann. of Math. (2) 69, 511-519. MR 21 #6438. C1unie and C. Pommerenke 1966
On the coef[icients o[ close-to-convex univalent functions, J. London Math. Soc. 41, 161-165. MR 32 #7734. L. E. Dunducenko 1956 Certain extremal properties o[ analytic [unctions given in a circle and in a circular ring, Ukrain. Mat. Z. 8, 377-395. (Russian) MR 19, 25. P. L. Duren 1963
Distortion in certain conformal mappings o[ an annulus, Michigan Math. J. 10, 431-441. MR 28 #199. P. L. Duren and M. Schiffer 1962
A variational method for functions schlicht in an annulus, Arch. Ra tional Mech. Anal. 9, 260-272. MR 25 #179. 1962a The theory of the second variation in extremum problems for univalent [unctions, J. Analyse Math. 10, 193-252. MR 27 #284.
Ja. S. Fel'dman 1963 On certain extremal domains connected with univalent functions, Vestnik Leningrad. Univ. 18, no. 2, 67-85. (Russian) MR 26 #6391. I. M. Ga1perin 1911 Uber den Variabilitiitsbereich der Fourier'schen Konstanten von posi 1965 Some estimates for functions bounded in the unit circle, Uspehi Mat. tiven harmonise hen Funktionen, Rend. Circ. Mat. Palermo 32, 193-217. Nauk 20, no. 1 (21), 197-202. (Russian) MR 30 #2150. T. Carleman P. R. Garabedian, G. G. Ross and M. M. Schiffer 1922/23 Uber die Approximation analytischer Funktionen durch lineare Aggre 1965 On the Bieberbach conjecture for even n, J. Math. Mech. 14,975 gate von vorgegebenden Potenzen, Arkiv Mat. Astronomi Fysik, 17. 989. MR 32 #207. V. V. Cernikov P. R. Garabedian and H. A. L. Royden 1962 Extremal properties on certain classes of analytic functions with real 1954 The one-quarter theorem for mean univalent functions, Ann. of Math. coefficients, Dissertation, Tomsk State University. (Russian) (2) 59, 316-324. MR 15,613.
656
BIBLIOGRAPHY FOR THE SUPPLEMENT
BIBLIOGRAPHY FOR THE SUPPLEMENT ,
657
1949
Gong Sheng (Kung Sheng, Kung Sun) 1955 Contributions to the theory of schlicht functions. II, The coefficient problem, Sci. Sinica 4, 359-373.
1954
A. W. Goodman 1950 On the Schwarz-Christoffel transformation and p-valent functions, Trans. Amer. Math. Soc. 68, 204-223. MR 11, 508. 1958 Variation formulas for multivalent functions, Trans. Amer. Math. Soc. 89, 129-148. MR 20 #3293. 1958a Variation of the branch points for an analytic function, Trans. Amer. Math. Soc. 89, 277-284. MR 20 #5873a. 1958b On the variation formulas for univalent functions, Trans. Amer. Math. Soc. 89, 285-294. MR 20 #5873b. 1958c On the critical points of a multivalent function, Trans. Amer. Math. Soc. 89, 295-309· MR 20 #5874.
P. R. Garabedian and M. Schiffer Identities in the theory of conformal mapping, Trans. Amer. Math. Soc. 65, 187-238. MR 10, 522. 1955 A coefficient inequality for schlicht functions, Ann. of Math. (2) 61, 116-136. 19S5a A proof of the Bieberbach conjecture for the fourth coefficient, J. Ra tional Mech. Anal. 4, 427-465. MR 17,24. S. A. Gel/fer
The variation of multivalent functions, Dokl. Akad. Nauk SSSR 98, 885-888; English transl., Amer. Math. Soc. Transl. (2) 26 (1963), 1-4. MR 16, 459; MR 27 #1581. 1956 The method of variations in the theory of p-valent functions, Uspehi Mat. Nauk 11, no. 5 (71), 60-66; English transl., Amer. Math. Soc. Trans!. (2) 18 (1961), 37-43. MR 18,648; MR 23 #A1789. A. W. Goodman and E. Reich 1958 On the maximum of the conformal radius of the fundamental region of a 1955 On regions omitted by univalent functions. II, Canad. J. Math. 7,83-88. given group, Mat. Sb. 44 (86), 213-224. (Russian) MR 20 #1782. MR 16, 579. 1960 On the maximum conformal radius of a fundamental domain of a group of fractional-linear transformations, Mat. Sb. 52 (94), 629-640. A. Grad MR 22 #11112. 1950 The region of values of the derivative of a schlicht function, in the 1962 An extension of the Goluzin-Schiffer variational method to multiply appendix to the book by Schaeffer and Spencer (1950]. MR 11, 508. connected regions, Dokl. Akad. Nauk SSSR 142, 503-506 = Soviet T. H. Gronwall Math. Dokl. 3 (1962), 70-73. MR 24 #A3286. 1914/15 Some remarks on conformal representation, Ann. of Math. 16, 72-76. 1964 Typically real functions, Mat. Sb. 64 (106), 171-184. (Russian) H. Grotzsch MR 29 #1317. 1928 Uber einige Extremalprobleme der konformer Abbildung, Ber. Verh. 1966 Variational method in the theory of functions of bounded type, Sachs. Akad. Wiss. Leipzig, 80, 367-376. Mat. Sb. 69 (111), 422-433; English transl., Amer. Math. Soc. 1928a (her einige Extremalprobleme der konformer Abbildung. II, Ber. Verh. Transl. (2) 72 (1968), 51-64. MR 33 #1464. Sachs. Akad. Wiss., Leipzig, 80, 497-502. S. A. Gel'fer and L. V. Kresnjakova 1955/56 Zum Haufungsprinzip der analytischen Funktionen, Wiss. Z. Martin 1965 A variational method in the theory of analytic functions with bounded Luther Univ., Halle-Wittenberg Math.-Natur. Reihe 5, 1095-1100. mean modulus, Mat. Sb. 67(109),570-585. (Russian) MR 33 #288. MR 18, 726. N. V. Genina (Semuhina) H. Grunsky 1962 On an extension of a variational method of G. M. Goluzin to doubly 1932 Neue Abschatzungen zur konformen Abbildung ein- und mehrfach connected domains, Tomsk Gos. Univ. Ueen. Zap. 44,226-240. (Russian) E. G. Goluzina 1962 On typically real functions with fixed second coefficient, Vestnik Leningrad. Univ. 17, no. 7,62-70. (Russian) MR 25 #2180. 1965
On ranges of values of certain systems of functionals in the class of typically real functions, Vestnik Leningrad. Univ. 20, no.7, 45-62. MR 32 #4277.
zusammenhangender Berichte, Schr. Math. Seminars Inst. Angew. Math. Univ. Berlin 1, 93-140. 1939 Koeffizientenbedingungen fur §chlicht abbildende meromorphe Funk tionen, Math Z. 45, H. 1. 29-61. S. Ja. Havinson 1953 On extremal properties of functions mapping a region on a multi-sheeted circle, Dokl. Akad. Nauk SSSR 88, 957-959. (Russian) MR 14, 967.
_1111.
7~
1111
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11I1111~IIIIIIIIIIIII'I~llIIlI'I!\I.!IIII~""~~'Jllk~_""ffi,@?i"'~;"'l'f:''I<_'1i_'''''''~j''''''',_,_"'''_A''''''''''''"""""
BIBLIOGRAPHY FOR THE SUPPLEMENT
658
1953a On some nonlinear extremal problems for bounded analytic functions, Dokl. Akad. Nauk SSSR 92, 243-245. (Russian) MR 15, 515. 1955 Extremal problems for certain classes of analytic functions in finitely connected regions, Mat. Sb. 36 (78), 445-478~ English trans!., Amer. Math. Soc. Trans!. (2) 5 (1957), 1-33. MR 17, 247; MR 18, 728. 1961 The analytic capacity of sets, joint nontriviality of various classes of analytic functions and the Schwarz lemma in arbitrary domains, Mat. Sb. 54(96), 3-50; English trans!., Amer. Math. Soc. Trans!. (2) 43 (1964), 215-266. MR 25, #182. W. K. Hayman
1951
Symmetrization in the theory of functions, Tech. Rep. no. 11, Navy Contract N6-ori-l06, Task Order 5 (No. R-043-992), O. N. R., Washington. MR 12, 401. 1951a Some applications of the transfinite diameter to the theory of functions, J. Analyse Math. 1, 155-179. MR 13, 545. 1955 The asymptotic behaviour of p-valent functions, Proc. London Math. Soc. (3) 5, 257-284; Russian transl., Matematika 2(958), no. 1, 55-80. MR 17, 142.
BIBLIOGRAPHY FOR THE SUPPLEME!'jT
659
J. A. Jenkins 1949 Some problems in conformal mapping, Trans. Amer. Math. Soc. 67, 327-350. MR 11, 341. 1951 On a theorem of Spencer, J. London Math. Soc. 26, 313-316. MR 13, 338. 1953 On values omitted by univalent functions, Amer. J. Math. 75, 406-408. MR 14, 967. 1953a Symmetrization results for some confom.al invariants, Amer. J. Math. 75, 510-522. MR 15, 115. 1954 On a problem of Gronwall, Ann. of Math. (2) 59, 490-504. MR 15,786.
1954a On the local structure of the trajectories of a quadratic differential, Proc. Amer. Math. Soc. 5, 357-362. MR 15, 947. 1955 On circumferentially mean p-valent functions, Trans. Amer. Math. Soc. 79, 423-428. MR 17, 143. 1956 Some theorems on boundary distortion, Trans. Amer. Math. Soc. 81, 477-500. MR 17,956. 1957 Some new canonical mappings for multiply connected domains, Ann. of Math. (2) 65, 179-196. MR 18, 568. 1958 Multivalent functions, Cambridge Univ. Press., Cambridge; Russian 1957a On a conjecture of Spencer, Ann. of Math. (2) 65, 405-410. trans!., IL, Moscow, 1960. MR 21 #7302. MR 19, 25. 1958a Bounds for the large coefficients of univalent functions, Ann. Acad. 1958 Univalent functions and conformal mapping, Springer-Verlag, Berlin, Sci. Fenn. Ser. A, I. Math. no. 250/13. MR 20 #3292. 1958; Russian trans!., IL, Moscow, 1962. MR 20 #3288. 1963 On successive coefficients of univalent functions, J. London Math. 1960 On univalent functions with real coefficients, Ann. of Math. (2) 71, Soc. 38, 228-243. MR 26 #6382. 1-15. MR 22 #5741. 1964 Paper read at seminar on theory of functions of a complex variable, 1960a An extension of the general coefficient theorem, Trans. Amer. Math. Moscow University, March, 1963; Russian trans!., Matematika 8 (1964), Soc. 95, 387-407. MR 22 #8126b. no. I, 142-150. 1960b On certain coefficients of univalent functions. II, Trans. Amer. Math. 1965 Coefficient problems for univalent functions and related function Soc. 96, 534-545. MR 23 #A309. classes, J. London Math. Soc. 40, 385-406. MR 31 #3592· 1960c On the global structure of the trajectories of a positive quadratic dif G. Ja. Hahlija ferential, Illinois J. Math. 4, 405-412. MR 23 #AI794. 1958 On a covering theorem for functions that are regular in doubly connec 1963 On some span theorems, Illinois J. Math. 7, 104-117. MR 27 #278. ted domains, Trudy Kutaissk. Gos. Ped. Inst. 18, 251-258. (Russian) 1963a An addendum to the general coefficient theorem, Trans. Amer. Math. E. Hille Soc. 107, 125-128. MR 26 #5154. 1949 Remarks on a paper by Zeev Nehari, Bull. Amer. Math. Soc. 55, 1965 On Bieberbach-Eilenberg functions. III, Trans. Amer. Math. Soc. 119, 552-553. MR 10, 697. 195-215. MR 31 #5969. J. A. Hummel J. A. Jenkins and D. C. Spencer 1957 The coefficient regions of starlike functions, Pacific J. Math. 7, 1951 Hyperelliptic trajectories, Ann. of Marh. (2) 53, 4-35. MR 12, 400. 1381-1389. MR 20 #1780.
,. :. 660
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BIBLIOGRAPHY FOR THE SUPPLEME,NT
BIBLIOGRAPHY FOR THE SUPPLEMENT
W. Kaplan 1952
Close-to-convex schlicht functions, Michigan Math. J. 1, 169-185. MR 14, 966. A. Kobori and H. Abe
1959 Une remarque sur un theoreme de M. Hayman, Japan. J. Math. 29, 32-34. MR 23 #A3832. L. I. Kolbina 1952 Some extremal problems in conformal mapping, Dokl. Akad. Nauk SSSR 84, 865-868. (Russian) MR 14, 35. 1952a On the theory of univalent functions, Dokl. Akad. Nauk SSSR 84, 1127-1130. (Russian) MR 14, 35. 1955 Conformal mapping of the unit circle onto mutually nonoverlapping domains, Vestnik Leningrad. Univ. 10, no.5, 37-43. (Russian) MR 17, 26. Y. Komatu
661
1962 On regular functions with bounded mean modulus, Dokl. Akad. Nauk SSSR 147, 290-293"" Soviet Math. Dokl. 3(962), 1590-1594.
MR 27 #1603. 1963 Some estimates for regular functions with bounded mean modulus, Izv. Vyss. UCebn. Zaved. Matematika, no. 1 (32), 94-97. (Russian) MR 26 #6422. J. Krzyz 1964 Some remarks on close-to-convex functions, Bull. Acad. Polon. Sci. Ser. Sci. Math. Astronom. Phys. 12, 25-28. MR 28 #5174. J. Krzyz and Z. Lewandowski 1963 On the integral of univalent functions, Bull. Acad. Polon. Sci. Ser. Sci. Math. Astronom. Phys.ll, 447-448. MR 27 #3791. T. Kubo 1954
Symmetrization and univalent functions in an annulus, J. Math. Soc. Japan 6, 55-67. MR 15,948. 1943 Untersuchungen iiber konforme Abbildung von zweifach zusammen 1954a Kelvin principle and some inequalities in the theory of functions. I, hangender Gebieten, Proc. Phys.-Math. Soc. Japan 0) 25, 1-42. Mem. ColI. Sci. Univ. Kyoto Ser. A. Math. 28, 299-311. MR 16, 122. MR 7, 514. 1955 Kelvin principle and some inequalities in the theory of functions. II, 1957 On the coefficients of typically-real Laurent series, Kodai Math. Sem. Mem. ColI. Sci. Univ. Kyoto SeI. A. Math. 29, 17-26. MR 16,914. Rep. 9, 42-48. MR 19, 404. 1955a Kelvin principle and some inequalities in the theory of functions. III, 1958 On analytic functions with positive real part in a circle, Kodai Math. Mem. ColI. Sci. Univ. Kyoto Ser. A. Math. 29, 119-129. MR 20 #3977. Sem. Rep. 10, 64-83. MR 20 #3998. 1958 Hyperbolic transfinite diameter and some theorems on analytic func 1958a On analytic functions with positive real part in an annulus, Kodai tions in an annulus, J. Math. Soc. Japan 10, 348-364. MR 21 #3551. Math. Sem. Rep. 10, 84-100. MR 20 #3999. P. P. Kufarev Y. Komatu and M. Ozawa 1943 On one-parameter families of analytic functions, Mat. Sb. 13 (55), 1951 Conformal mapping of multiply connected domains. I, Kodai Math. 87-118. (Russian) MR 7, 201. Sem. Rep., 81-95. MR 13, 734. 1946 On integrals of a very simple differential equation with movable polar 1952 Conformal mapping of multiply connected domains. II, Kodai Math. singularity in the right-hand member, Tomsk. Gos. Univ. Veen. Zap. Sem. Rep., 39-44. MR 14, 461. no. 1, 35-48. (Russian) G. V. Koricki! 1947 A theorem on the solutions of a certain differential equation, Tomsk. 1955 On curvature of level lines and of their orthogonal trajectories in Gos. Univ. Ueen. Zap. no. 5, 20-21. (Russian) conformal mappings, Mat. Sb. 37 (79), 103-116. (Russian) MR 17,26. 1947a On a special family of one-sheeted domains, Tomsk. Gos. Univ. Ueen. 1957 Curvature of level lines in univalent conformal ma ppings, Dokl. Akad. Zap. no. 5, 22-36. (Russian) Nauk SSSR 115, 653-654. (Russian) MR 19, 845. 1947b On a method of numerical determination of the parameters in the 1960 On the curvature of level curves of univalent conformal mappings, Schwarz-Christoffel integral, Dokl. Akad. Nauk SSSR 57, 535-537. Uspehi Mat. Nauk 15, no. 5, (95), 179-182. (Russian) MR 23 #AI793· (Russian) MR 9, 277. 1947c A remark on integrals of Lowner's equation, Dokl. Akad. Nauk SSSR L. V. Kresnjakova 57, 655-656. (Russian) MR 9, 421. 1961 Analytic functions with bounded mean modulus, Izv. Vyss. Ueebn.
Zaved. Matematika, no. 1 (20), 98-103. (Russian)
MR 24 #A3298.
._
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662
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1948
On a system of differential equations, Tomsk. Gos. Univ. Ueen. Zap. 8, 61-72. (Russian) MR 11, 21
1950
On conformal mapping of complementary regions, Dokl. Akad. Nauk SSSR 73, 881-884. (Russian) MR 12, 401 1951 Remark on extremal problems in the theory of univalent functions, Tomsk. Gos. Univ. Ueen. Zap. no. 14, 3-7. (Russian) 1954 On a property of extremal regions of the problem of coefficients, Dokl. Akad. Nauk SSSR 97, 391-393. (Russian) MR 16, 122. 1955 Remark on the problem of coefficients, Tomsk. Gos. Univ. Ueen. Zap. 25, 15-18. (Russian) MR 19, 404. 1956 On a method of parametric representations and the variational method of Goluzin, Proc. Third All-Union Math. Congress. Vol. 1, Moscow, pp. 85-86. (Russian) 1956a Methods and results in the theory of univalent functions, Proc. Third All-Union Math. Congress. Vol. 2, Moscow, pp. 29-31. (Russian) 1956b On a certain method of investigation of extremum problems in the theory of univalent functions, Dokl. Akad. Nauk SSSR 107, 633-635. (Russian) MR 17, 1069. 1958 Certain methods and results in the theory of univalent functions, Proc. Third All-Union Math. Congress. Vol. 3, Moscow, pp. 189-198. (Russian)
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BIBLIOGRAPHY FOR THE SUPPLEMEJ>lT
BIBLIOGRAPHY FOR THE SUPPLEMENT
1947d On the theory of univalent functions, Dokl. Akad. Nauk SSSR 57, 751-754. (Russian) MR 9, 507.
""!'lt~(1,~b.,VW~,_"iii"_",,$ .......... ~M"'M"~";
663
M. P. Kuvaev and P. P. Kufarev
1955
An equation of Lowner's type for multiply connected regions, Tomsk. Gos. Univ. Ueen. Zap. 25, 19-34. (Russian) MR 19, 401.
G. V. Kuz'mina
1962 1965
Some covering theorems for univalent functions, Dok1. Akad. Nauk SSSR 142, 29-31 = Soviet Math. Dok1. 3 (1962), 21-23. MR 24 #A1383. Covering theorems for functions which are regular and univalent in the circle, Dokl. Akad. Nauk SSSR 160, 25-28 = Soviet Math Dokl. 6, (1965), 21-25. MR 30 #3204.
E. Landau
1929
Cher die Blochsche Konstante und zwei verwandte Weltkonstanten, Mith. Z. 30, H. 4, 608-634. M. A. Lavrent' ev
1934 On the theory of conformal mappings, Trudy Fiz.-Mat. Inst. Akad. Nauk SSSR 5, 195-246. (Russian) , v M. A. Lavrent ev and B. V. Sabat
1958
Methods of the theory of functions of a complex variable, GITTL, Moscow, 1951; 2nd rev. ed., 1958; German trans 1., Mathematik fur Naturwiss. und Technik, Band 13, VEB Deutscher Verlag, Berlin, 1966. MR 14, 457; MR 21 #116. N. A. Lebedev
1951
The method of variations in conformal mapping, Dok1. Akad. Nauk SSSR 76, 25-27. (Russian) MR 12, 491. 1951a Certain e'timates and problems on an extremum in the theory of con P. P. Kufarev and A. Fales formal mapping, Dissertation, Leningrad State University. (Russian) 1951 On an extremal problem for complementary regions, Dokl. Akad. Nauk 1955 Majorizing region for the expression 1= In(z-\f'(z)l-Ajf(z)A) in the SSSR 81, 995-998. (Russian) MR 14, 262. class S, Vestnik Leningrad. Univ. 10, no. 8, 29-41; English trans1., P. P. Kufarev and N. V. Semuhina Amer. Math. Soc. Transl. (2) 22 (1962), 43-57. MR 17,248. 1956 On the extension of G. M. Goluzin's variational method to doubly con 1955a Certain estimates for functions regular and univalent in the unit disc, nected regions, Dokl. Akad. Nauk SSSR 107, 505-507. (Russian) Vestnik Leningrad. Univ. 10, no. 11, 3-21; English trans1., Amer. Math. Soc. Trans1. (2) 22 (1962),59-80. MR 17, 599. MR 17, 1193.
1963
On G. M. Goluzin's variational formula, Trudy Tomsk. Gos. Univ. Ser. Meh.-Mat. 163, 58-62. (Russian)
E.
M. P. Kuvaev 1959 Generalizations of an equation of the L6wner type for automorphic functions, Trudy Tomsk. Gos. Univ. Ser. Meh.-Mat. 144, 27-30. (Russian) 1959a A new derivation of Lowner's equation for doubly connected regions, Trudy Tomsk. Gos. Univ. Ser. Meh.-Mat. 144, 45-55. (Russian)
1955b On the theory of conformal mappings of a circle onto nonoverlapping regions, Dok1. Akad. Nauk SSSR 103, 553-555. (Russian) MR 17, 250. 1955c On a parametric representation of functions regular and univalent in a ring, Dokl. Akad. Nauk SSSR 103, 767-768. (Russian) MR 17, 356. 1955d On domains of values of functionals defined on classes of analytic functions, Dissertation, Leningrad State University, (Russian)
664
BIBLIOGRAPHY FOR THE SUPPLEMENT
BIBLIOGRAPHY FOR THE SUPPLEMF;NT
665
1957
On the domain of values of a certain functional in a problem of non Ju. D. Maksimov overlapping domains, Dokl. Akad. Nauk SSSR 115, 1070-1073. 1955 Extremal problems in certain classes of analytic functions, Dokl. Akad. (Russian) MR 19,951. Nauk SSSR 100, 1041-1044. (Russian) MR 16, 810. 1961 An application of the area principle to non-overlapping domains, Trudy 1955a On locally f-convex and locally f-starlike multivalent functions, Dokl. Mat. Inst. Steklov. 60, 211-231. (Russian) MR 24 #A1384. Akad. Nauk SSSR 103, 965-967. (Russian) MR 17, 357. 1966 Application of the area principle to the problems on non-overlapping 1961 Extension of the structural formula for convex univalent functions to a finitely connected regions, Dokl. Akad. Nauk SSSR 167, 26-29 = multiply connected circular region, Dokl. Akad. Nauk SSSR 136, 284 Soviet Math. Dokl. 7(966), 323-327. MR 33 #5865. 287 = Soviet Math. Dokl. 2 (1961), 55-58. MR 22 #812l. N. A. Lebedev and I. M. Milin M. Marcus 1951 On the coefficients of certain classes of analytic functions, Mat. Sb. 1964 Transformations of domains in the plane and applications in the theory 28(70), 359-400. (Russian) MR 13, 640. of functions, Pacific J. Math. 14, 613-626. MR 29 #2382. 1965 An inequality, Vestnik Leningrad. Univ. 20, no. 19, 157-158. H. Meschkowski (Russian) MR 32 #4248. 1952 . Einige Extremalprobleme aus der Theorie der konformen Abbildung, Z. Lewandowski Ann. Acad. Sci. Fenn. Ser. AI. 00.117, 1-12. MR 14, 367. 1958 Sur l'identite de certaines classes de fonctions univalentes. I, Ann. 1953 Verzerrungssatze fur mehrfach zusammenhangende Bereiche, Compo Univ. Mariae Curie-Sklodowska, Sect. A. 12, 131-146. MR 24 #A217. sitio Math. 11, 44-59. MR 15, 116. 1960 Sur l'identite de certaines classes de fonctions univalentes. II, Ann. 1954 Verallgemeinerung der Poissonschen Integralformel auf mehrfach Univ. Mariae Curie-Sklodowska, Sect A. 14, 19-46. MR 28 #200. zusammenhiingende Bereiche, Ann. Acad. Sci. Fenn. Ser. AI. 166. 1961 Sur les majorantes des fonctions holomorphes dans le cercle Izl < 1, MR 15, 695. Ann. Univ. Mariae Curie-Sklodowska, Sect. A, 15, 5-11. MR 26 #314. 1962 Hilbertsche Riiume mit Kernfunktion, Die Grundlehren der Math Wis 1961a Starlike majorants and subordination, Ann. Univ. Mariae Curie senschaften, Bd. 113, Springer-Verlag, Berlin. MR 25 #4326. Sklodowska, Sect. A. 15, 79-84. MR 25 #2186. I. M. Milin Li En-pir 1964 The area method in the theory of univalent functions, Dokl. Akad. 1953 On the theory of univalent functions on a circular ring, Dokl. Akad. Nauk SSSR 154, 264-267 = Soviet Math. Dokl. 5 (964), 78-8l. Nauk SSSR 92, 475-477. (Russian) MR 15, 516. MR 28 #1283. 1953a On typically real functions on a circular ring, Dokl. Akad. Nauk SSSR 1964a Closed orthonormal systems of analytic functions in domains of finite 92,699-702. MR 15, 516. connectivity, Dokl. Akad. Nauk SSSR 157, 1043-1046 = Soviet Math. Dokl. 5 (964), 1078-1082. MR 34 #4469. 1953b Certain questions in the theory of univalent and typically real func tions on a circular annulus, Dissertation, Leningrad State University. 1964b The area method in the theory of univalent functions, Dissertation, (Russian) Leningrad State University. (Russian) K. Loewner 1965 A bound for the coefficients of schlicht functions, Dokl. Akad. Nauk 1923 Untersuchungen aber schlichte konforme Abbildung des Einheits SSSR 160, 769-771 = Soviet Math. Dokl. 6(965), 196-198. MR 30 #3206. kreises. I, Math. Ann. 89, 103-12l. J a. S. Mirosnieenko V. G. Lozovik 1963
On a class of functions which are univalent in the unit circle, Izv. Vyss. Zaved. Matematika, no.2 (33), 63-69. (Russian) MR 26 #5150. 1963a Functionals defined on certain classes of analytic functions, Ukrain. Mat. Z. 15, 95-99. (Russian) MR 27 #1575.
1951
On a problem .in the theory of univalent functions, Ueen. Zap. Donetsk. Ped. Inst. 1,63-75. (Russian)
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666
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BIBLIOGRAPHY FOR THE SUPPLEMENT
1. P. Mitjuk
Univalent conformal mappings of multiply connected domains, Dopovidi Akad. Nauk Ukratn. RSR, 158-160. (Ukrainian) MR 23 #A3252. 1961a Some theorems on univalent conformal mappings of multiply connected regions, Dopovidi Akad. Nauk Ukrai'n. RSR, 420-423. (Ukrainian) MR 24 #A2030. 1961b A generalization of some theorems on univalent conformal maps of doubly connected domains, Dopovidi Akad. Nauk Ukrai'n. RSR, 1115 1118. (Ukrainian) MR 24 #A2660. 1964 A generalized reduced module and some of its applications, lzv. Vyss. Ucebn. Zaved. Matematika, no.2 (39), 110-119. (Russian) MR 29, #2376. 1964a The symmetrization vrinciple for multiply connected domains, Dokl. Akad. Nauk SSSR 157, 268-270 = Soviet Math. Dok1. 5 (1964), 928 930. MR 29 #2375. 1965 The inner radius of a domain and certain of its properties, Ukrain. Mat. Z. 17, no. 1,117-122. (Russian) MR 32 #7730.
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BIBLIOGRAPHY FOR THE SUPPLEMENT,
1953
1961
667
Some inequalities in the theory of functions, Trans. Amer. Math. Soc. 75, 256-286. MR 15, 115.
H. Nishimiya 1957 On a coefficient problem for analytic functions typically-real in an annulus, KOdai Math. Sem. Rep. 9, 59-67. MR 20 #101. 1959 On coefficient-regions of Laurent series with positive real part, Kodai Math. Sem. Rep. 11, 25-39. MR 21 #7310. O. S. Nosenko
1963
On the range of Stieltjes functionals with restrictions of equality type, Dopovidi Akad. Nauk Ukrai'n. RSR, 1563-1567. (Ukrainian) MR 29#5996.
M. Ozawa
1952 1965
On functions of bounded Dirichlet integral, Kodai Math. Sem. Rep. 4, 95-98. MR 14, 967. On the sixth coefficient of univalent function, Kodai Math. Sem. Rep. 17, 1-9. MR 31 #2394.
E. Peschl 1965a The principle of symmetrization for multiely connected regions and 1936 Zur Theorie der schlicht en Funktionen, J. Reine Angew. Math. 176, certain of its applications, Ukrain. Mat. Z. 17, no. 4, 46-54. 61-96. (Russian) MR 33 #7510. V. V. Pokorny! 1965b Certain properties of functions regular in a multiply connected region, 1951 On some sufficient conditions for univalence, Dokl. Akad. Nauk SSSR Dokl. Akad. Nauk SSSR 164, 495-498 = Soviet Math. Dokl. 6 (1965), 79, 743-746. (Russian) MR 13, 222. 1252-1255. MR 32 #5862. C. Pommerenke 1965c The symmetrization principle for an annulus and certain of its appli 1961/62 Uber die Mittelwerte und Koeffizienten multivalenter Funktionen, cations, Sibirsk Mat. Z. 6, 1282-1291. (Russian) MR 35 #4396. Math. Ann. 145, 285-296. MR 24 #A3282. 1965d Certain extremal problems in the geometric theory of functions, Disser 1962 Uber einige Klassen meromorpher schlichter Funktionen, Math. Z. 78, tation, Kiev State University. (Russian) 263-284. MR 28 #1285. P. T. Mocanu 1962a On starlike and convex functions, J. London Math. Soc. 37, 209-224. 1957 Une generalisation du theoreme de la contraction dans la classe S de MR 25 #1279· fonctions univalentes, Acad. R. P. RomIne. Fi1. Cluj. Stud. Cere. Mat. 1963 On starlike and close-to-convex functions, Proc. London Math. Soc. 8, 303-312. (Romanian) MR 21 #5735a. (3) 13, 290-304. MR 26 #2597. 1958 Sur une generalisation du theoreme de contraction dans la classe des 1964 Linear-invariante Familien analytischer Funktionen. 1, Math. Ann. fonctions univalentes, Acad. R. P. RomIne. Fi1. Cluj. Stud. Cere. Mat. 155, 108-154. MR 29 #6007. 9, 149-159. (Romanian) MR 21 #5735b. 1964a Lacunary power series and univalent functions, Michigan Math. J. 11, M. A. Monastyrskil 219-223. MR 29 #4883. 1959 On an application of the method of positive functionals for C-functions, V.1. Popov Ueen. Zap. Sahtinsk. Gos. Ped. lost. 2, no. 6, 109-117. (Russian) 1965 The range of a certain system of functionals on the class S, Trudy Z. Nehari Tomsk. Gos. Univ. Ser. Meh.-Mat. 182, 107-132. (Russian) 1949 The Schwarz ian derivative and schlicht functions, Bull. Amer. Math. MR 33 #7521. Soc. 55, 545-551. MR 10,696.
•
~
~"
668
.
~ __,.
2
!
2
7"--
-1
fu~_=
..
.
j=~!';g~rg1l~_~~_
BIBLIOGRAPHY FOR THE SUPPLEMENT
H. Prawitz
1927/28 Uber Mittelwerte analytischer Funktionen, Arkiv Mat. Asrronom. Fysik 20A, no. 6, 1-12.· B. N. Rahmanov
1953
On the theory of univalent functions, Dokl. Akad. Nauk SSSR 91, 729 732. (Russian) MR 15, 413.
1936 1962
M. O. Reade
Estimates for the transfinite diameter of a continuum, Math. Z. 85, 91-106. MR 30 #4921. E. Reich and S. E. Warschawski 1960 On canonical conformal maps of regions of arbitrary connectivity, Pacific J. Math. 10, 965-989. MR 22 #8120. M. P. Remizova 1959 On regions of values of analytic functions reyresented by the sum and product of Stieltjes integrals, Ukrain. Mat. Z. 11, 175-182. (Russian) MR 22 #771. E.
Ren~el
1933
Uber einige Schlitztheoreme der konformen Abbildung, Schr. Math. Sem· und Inst. Angew. Math. Univ. Berlin 1, no. 4, 141-162.
=--=-
"'=-"""""-"---"J""","""'""-=_=__ '_~="'_.
669
Sur certains systemes singuliers d'equations integrales, Ann. Sci. Ecole Norm. Sup. 28, 33-62. M. S. Robertson 1935
1964
--~
F. Riesz 1911
On the theory of univalent functions, Dokl. Akad. Nauk SSSR 78, 209 211. (Russian) MR 12, 816.
1960 On the range of values of a certain functional defined on certain classes of bounded univalent functions, Tomsk Gos. Univ. Ueen. Zap. no. 36, 33-50. (Russian) 1962 The range of the functional 1= In(w\,b'(wp-'A¢'(OP'/¢(w)A) in the class 5 1[1¢(w)lJ, Izv. Vyss. Ueebn. Zaved. Matematika, no.2 (27), 119-129. (Russian) MR 25 #4086. 1962a The range of the functional [= In(w'\'¢'(w)1-'A¢'(0)l'/¢(w)A\¢(w)I K ) in the class 51' Izv. Vyss. Ucebn. Zaved. Matematika, no.4 (29), 134-142. (Russian) MR 25 #4087. 1963 The range of values of the functional 1= [(few), few), f'(w), f'(w), ['(0)) in the class S1[lf(w)lJ, Trudy Tomsk. Gos. Univ. Ser. Meh.-Mat. 169, 59-68. (Russian) MR 29 #1324. E. Reich and M. Schiffer
BIBLIOGRAPHY FOR THE SUPPLEMENT
1951
1955 On close-to-convex univalent functions, Michigan Math. J. 3, 59-62. MR 17, 25. M. I. Red'kov
._'=.,,",";jl>o~~~_".:'==--:::;:o;'.~~,<;
On the coefficients of a typi<:ally-real function, Bull. Amer. Math. Soc. 41, 565-572. Analytic functions star-like in one direction, Amer. J. Math. 58,465-472. Variational methods for functions with positive real part, Trans. Amer. Math. Soc. 102, 82-93. MR 24 #A3288.
1963
Extremal problems for analytic functions with positive real part and applications, Trans. Amer. Math. Soc. 106, 236-253. MR 26 #325. R. M. Robinson 1943
Analytic functions in circular rings, Duke Math. MR 4, 241. W. Rogosinski 1932
J.
10, 341-354.
Uber positive harmonische Entwicklungen und typisch reelle Potenz reihen, Math. Z. 35, 93-121.
S. Sato
1955 Two theorems on bounded functions, Sugaku 7, 99-101. 1955a On values not assumed by bounded functions, Kenkju hokoku. Sidzen Kacaku, Liberal Arts J. Natur. Sci. 6, 1-6. (Russian) A. C. Schaeffer and D. C. Spencer 1950
Coefficient regions of schlicht functions, Amer. Math. Soc. Colloq. Publ., vol. 35, Amer. Math. Soc., Providence, R. 1. MR 12, 326.
M. Schiffer 1938 A method of variation within the family of simple functions, Proc. London Math. Soc. (2) 44, 432-449. 1938a On the coefficients of simple functions, Proc. London Math. Soc. (2) 44, 450-452. 1943 Variation of the Green function and theory of p-valued functions, Amer. J. Math. 65, 341-360. MR 4, 215. 1953 Some recent developments in the theory of conformal mapping, appen dix to Russian transl. of R. Courant, Dirichlet's principle, conformal mapping and minimal surfaces, Interscience, New York and London, 1950. (Russian) M. Schiffer and O. Tammi
1965
The fourth coefficient of a bounded real univalent function, Ann. Acad. Sci. Fenn. Ser. A I, No. 354. MR 35 #4394.
670
BIBLIOGRAPHY FOR THE SUPPLEMENT
BIBLIOGRAPHY FOR THE SUPPLEMENT
1965
Theorems on the covering of lines under a conformal mapping, Mat. Sb. 66 (08), 502-524. (Russian) MR 31 #2392. 1965a Certain extremal problems of the theory of univalent conformal map pings, Mat. Sb. 67 (109), 329-337. (Russian) MR 33 #7523. 1965b A conformally metric theory of doubly-connected regions, and a gener alized Blaschke product, Dokl. Akad. Nauk SSSR 161, 308-311 == Soviet Math. Dokl. 6 (965), 432-435. MR 31 #327. 1965c Univalent functions and conformally metric theory of multiply con nected domains, Dissertation, Kiev. (Russian)
I965a On the fourth coefficient of bounded univalent functions, Trans. Amer. Math. Soc. 119,67-78. MR 32 #2579. V. Singh 1962 Grunsky inequalities and coefficients of bounded schlicht functions, Ann. Acad. Sci. Fenn. Sec. A I, no. 310. MR 26 #1444. G. G. Slionskil 1958 On the extremal problems for differentiable functionals in the theory of univalent functions, Vestnik Leningrad. Univ. 13, no. 13, 64-83. (Russian) MR 20 #4652. 1959 On the theory of bounded schlicht functions, Vestnik Leningrad. Univ. 14, no. 13, 42-51. (Russian) MR 22 #2704. V. I. Smirnov 1932 Sur les formules de Cauchy et de Green et quelques problemes qui s'y rattachent, Izv. Akad. Nauk SSSR Ser. Mat. 7, 337-372.
O. Teichmiiller
1938
Untersuchungen aber konforme und quasikonforme Abbildung, Deutsche Math. 3, 621-678. G. C. Tumarkin and S. Ja. Havinson 1960
D. C. Spencer
On mean one-valent functions, Ann. of Math. (2) 42, 614-633. MR 3,78. O. V. Stepanova 1963 Certain properties of level curves for univalent conformal mappings, Mat. Sb. 61 (103), 350-361. (Russian) MR 27 #1579. 1965 A property of level lines in univalent conformal mappings, Dokl. Akad. Nauk SSSR 163, 1330 = Soviet Math. Dokl. 6 (965), 1124-1125. MR 32 #2577. G. Szego 1955 On a certain kind of symmetrization and its applications, Ann. Mat. Pura Appl. (4) 40, 113-119. MR 17,1074.
1941
P. M. Tamrazov 1962 Relative boundary distortion under a schlicht conformal mapping of a doubly connected domain, Dopovidi Akad. Nauk Ukrain. RSR, 338 340. (Ukrainian) MR 26 #3883. 1962a On the theory of schlicht conformal mappings of doubly connected domains, Dopovidi Akad. Nauk Ukrafn. RSR, 563-566. (Ukrainian) MR 26 #3884. 1962b On univalent conformal mapping of doubly connected domains, Dopovidi Akad. Nauk Ukrafn. RSR, 1142-1145. (Ukrainian) MR 27 #276. 1963 Some estimates in the theory of univalent conformal mappings of doubly connected domains, Dopovidi Akad. Nauk Ukrai'n. RSR, 1160 1163. (Ukrainian) MR 29 #6008.
671
Qualitative properties of solutions of certain types of extremal prob lems in Series on Modern Problems of the Theory of Functions of a COqIplex Variable, Fizmatgiz, Moscow, pp.77-95. (Russian) MR 22 #11136.
G. V. Ulina
1960
On the domains of values of certain systems of functionals in classes of univalent functions, Vestnik Leningrad. Univ. 15, no. 1, 34-54. (Russian) MR 22 #5745.
Xia Dao-xing (Hsia Tao-hsing)
1956 1957
On the functions univalent in a circular ring, Acta Math. Sinica 6, 598-618. (Chinese) Goluzin's number (3 - V5)f2 is the radius of superiority in subor dination, Sci. Record 1, 219-222. MR 20 #6530.
1957a On the radius of superiority in subordination, Sci. Record 1, 329-333. MR 20 #6531. 1958 Some covering properties of convex domains in the theory of conformal mapping, Sci. Sinica 7, 816-828. MR 21#2046b. Xia Dao-xing (Hsia Tao-hsing) and Zhang Kai-ming (Chang K'ai-ming)
1958
v.
Some inequalities in the theory of subordination, Acta Math. Sinica 8, 408-412 = Chinese Math. Acta 9 (967), 116-120. MR 21 #7306. A. Zmorovic 1952 1953
On certain variational problems of the theory of univalent functions, Ukrain. Mat. Z. 4, 276-298. (Russian) MR 15, 301. On some classes of analytic functions univalent in a circular ring, Mat. Sb. 32 (74), 633-652. (Russian) MR 14, 1075.
-
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672
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BIBLIOGRAPHY FOR THE SUPPLEMENT
1954 1956
On some special classes of analytic functions univalent in a circle,
Uspehi Mat. Nauk 9, no. 4 (62), 175-182. (Russian) MR 16, 459.
On certain classes of analytic functions in a circular ring, Mat. Sb.
40 (82), 225-238. (Russian) MR 18, 648.
SUBJECT INDEX
1958 On a generalisation of Schwarz's integral formula on n-connected cir cular domains, Dopovidi Akad. NaukUkrai"n. RSR, 489-492. (Ukrainian) MR 20 #5277_ Accessible boundary points 35 Analytic curve 9
1959 Theory of special classes of univalent functions_ I, Uspebi Mar. Nauk Anharmonic ratio 330
14, no.3 (87), 137-143. (Russian) MR 22 #3819. Approximation in mean 448-449 1959a Theory of special classes of univalent functions. II, Uspebi Mat. Nauk Asymptotic value 351
Automorphic function 64, 258
14, no.4 (88), 169-172. (Russian) MR 22 #3820. 1965 On a class of extremal problems associated with regular functions Bieberbach conjecture 605-606, 608-612
generalized 589
with positive real part in the circle Izl < 1, Ukrain. Mat. Z. 17, no.4, Blaschke function 398
12-21; English transI., Amer. Math. Soc. Trans!. (2) 80 (1968), 215 Bloch constant 363 226. MR 33 #5901.
Classes: C 563, 575; E g 528; E p 438;
"-
Hp 402,498; h p 385; L 563; L 563; "
LP 388; N 393; R 563; R 563; S 110,
563; SM 563; Sa 152; Sp 488; SO)
563; SR 563; S(k) 563; S* 530, 563;
S 563; T 563; T r 541; I 110, 563;
I m 563; I p 488; IO 563; I(k) 563;
I*
Boundary continua 7
Bounds for area of
an annulus 605
the image of a disk 595
the star of the image of a disk 180
Bounds for bounded functions 514-526
Bounds for coefficients of p-valent
functions 490, 492, 493
Bounds for coefficients of univalent
functions 187, 197, 199-204, 605-618
Bounds for curvature of level curves
563;
I
563;
lR
Closedness condition Conformal mapping of nected domain onto Conformal mapping of
563
451
a multiply con
a disk 254-255
Riemann surfaces
461-467
Conformal radii of disjoint domains 156,
583, 584, 587
Conformal radius 29, 313
Convergence of a sequence of domains
to a kernel 54, 228-229
Convergence of sequences of analytic
functions 11-14
Convergence of sequences of harmonic
functions 19-22
Convex domain 166, 201
Convexity of level curves, conditions
for 601-602
Correspondence of boundaries under
conformal mapping 31-46,262-265.
599-603
Bounds for mean modulus of a function
595 Bounds for the growth of adjacent coefficients of
close-to-convex functions 617
p-valent functions in mean 615
univalent functions 193, 195, 614, 615 Brouwer theorem 252
417-428
Covering theorems 73-75, 89, 494
of arcs of a circle by image of disk
and annulus 595, 597, 605
of segments 174, 177., 597, 604
Cut 9
Capacity of a set 310 Caratheodory convetgence theorem 55 Caratheodory-Fejer problem 497 Caratheodory·Fejer theorems 500, 502 Cauchy integral formula 408, 435-441
Cauchy integral theorem 408 Cauchy singular integral 431
v v Cebysev constant 294, 295
Cebysev polynomial 294, 295
Circular domain 234, 235
Circular polygon 78
Dirichlet problem 267
Entire function of finite order 352
Equivalent points 260
Family of polynomials complete in a domain 449
Function of bounded variation 385
673
674
SUBJECT INDEX
SUBJECT INDEX
Gauss's differential equation 84 General theorem of Jenkins coefficients
573 Gluing theorems 454, 456 Green's formula 273 Green's function 269,310,311 Gronwald problem 590 Grotzsch principle 171 Grotzsch theorem, generalized 603 Hadamard three-circle theorem 344 Harmonic function 19, 266 Harmonic measure 315, 342 Harnack's theorem 21 Heine-Borel covering theorem 6 Helly theorem 385 Hilbert theorem 213-214 Holder inequality 388 Hyperbolic disk 332 Hyperbolic distance 331 Hyperbolic metric 332, 336, 361 Hyperbolic transformation 259 Hyperconvergence 356 Inner measure 420 Integral formula, Cauchy 408, 435-441 Integral representation of classes of func tions regular in a disk (annulus) 529,
574,619-623 Integral theorem, Cauchy 408 Jenkins coefficients, general theorem of
573 Jensen formula 323 Jensen-Schwarz formula 323 Jordan contenr 300, 302 Jordan theorem 8 Schoen flies supplement to 36 Kellogg theorem 426 Kernel of sequence of domains 54, 228 Koebe covering theorem 49 generalized and sharpened 597, 598,
602 Koebe lemma 31 Landau consrant 363 Landau theorem 338 Laplace differential equation 19 Laurent system of functions 634 Lavrent'ev formula 156 Lavrent'ev theorems, generalized 638,
639, 642
Lemmas on mean moduli 183, 184, 186 Limiting value of Cauchy integral 431 Limits of convexity 166, 598 Limits of generalized starlikeness 167
170, 599 Limits of starlikeness 167, 599 Lindelof lemma 33 Lindelof principle 30, 339 Lipschitz condition 413, 426 Loewner differential equation 91 Majorant, univalent 368 Mapping, conformal univalent 23 continuous 239 quasiconformal 456 topological 239 Mapping a disk onto mutually disjoinr domains 157, 583-586 Mapping a doubly connected domain 205
210, 593-594 onto a circular annulus 208 Mapping a multiply connected domain Onto a disk with cutS along arcs of loga rithmic spirals 242-243 a finite surfa ce 629, 630 a multiply covered disk 648-649 a plane with cutS along arcs of loga rithmic spirals 220, 243 a plane with rectilinear cuts 242 a plane with rectilinear parallel cutS
213-214, 277 an annulus with cuts along arcs of logarithmic spirals 243 mutually disjoint domains 638-641 Mapping an n-connected domain onto an n-sheeted disk 278, 468 Mapping the exterior of the unit disk onto a plane with cut along arc of ellipse (hyperbola) 135 Mapping the interior of a circular polygon onto the unit disk 78~81 Mapping the interior of a regular n-sided circular polygon onto the unit disk 83-87 Mapping the interior of a rectilinear poly gon onto the unit disk 76-78 Mapping the interior of a regular n-sided polygon onto the unit disk 77 -78 Maximum principle generalized 267 Mean p-valent function 587, 588 Measurable set 420 Meromorphic function 23, 321
Method of continuity 234, 239-243 Method of contour integration 225, 283,
284, 567 Method of extremal meuics 572 Method of integral representations 574
576 Method of parametric representations
89-99, 110-122, 568, 579, 591-592 Method of quadratic differentials 572-574 Method of symmetrization 574 Metric, continuous 360 Metric, regular 360 Metric, support 361 Minimal property of Dirichlet integral 637 Minimization of area 30, 227, 631 Minimization of the maximum modulus 29 Minkowski inequality 389 Modulus of doubly connected domain 209 Modulus of n-connected domain 244 Modular function 64 Modular grid 63 Modular Riemann surface 64 n-sheeted disk 277 Natural boundary 64 Necessary and sufficienr condition for univalence of a function 609 Nontangential path 382, 428 Normal family of analytic functions 67 Normality test 68, 70 Order of enthe function 352 Orthogonal polynomials 451 Outer area 631 Outer measure 420 p-valent function 487 Parabolic transformation 259 Parametric representation of univalent functions 89-99 Picard's theorem 72 Poincare's theorem 462 Poisson formula 20, 268 Poisson integral 380 Poisson-Stieltjes integral 384 Prime ends 40 Principle, hyperbolic metric 336 Principle of areas 47, 490, 565-567, 587 Principle of compactness 19 Principle of condensation 14-18 Principle of extension 342-343 Principle of length and area 572
<:
675
Principle of subordination
for annulus 648 for disk 368-369, 643-644 for multiply connected domains 645
.
647 Principle of symmetrization 588, 603 generalized for multiply connected domains 642 Privalov's uniqueness theorem 428 Problem of coefficients 605, 617 Quasiconformal mapping 456 Range of a functional on a given class
117 Range of values of functionals and sys tems of functionals on classes of univalent functions
117, 133, 136, 139,580-583 representable by means of Stieltjes integrals 624-628 Riemann surface, constructive definition
461-462 Riemann surfaces of elliptic, hyperbolic and parabolic types 462 Riemann theorem 25-29 Robin's constant 310 Scalar product of regular functions 226 Schoenflies supplement to Jordan theorem
36 Schottky function 287 Schottky theorem 337-338 Schur's theorems 498, 505 Schwarz derivative 636 Schwarz formula 92, 322 Schwarz invariant 79, 580 Schwarz lemma 25, 329, 332; generalized and sharpened 329-330, 333, 361, 373,
648-649 Schwarz-Christoffel formula 77 Spread of coefficients of univalent functions 614 Star of a domain 177 Starlike domain 167, 202 Stadikeness generalized 167, 599 Starlikeness of level curve 599-602 Subordinate function 368, 369 Summable function 431 Symmetrization method of Marcus 603 Symmetrization principle 588, 603; generalized 642
-------_.
676
------
--''-~-;~~_.-
----==~=
SUBJECT INDEX
Theorems of area 47; of p-valem func rions 490
generalized 565, 566, 578, 586
generalized for multiply connected
domains 567, 633, 634, 635
Theorems of disrorrion 51, 134; of chords
577
generalized 587-590
Theorems of rotation 51, 110
Transfinire diamerer 294
Typically-real funcdon 540
Uniform boundedness in the interior of a
domain 15
Uniform convergence in the interior of a
domain 11
Uniqueness theorem of Privalov 428
Univalent function 23
Universal covering surface of n-connected
domain 261
Variation of univalem functions 100-108
Variational formulas in classes E g 527,
528; Sand! 106, 107; Sa 108, 109
Variational method 99-108, 128-165, 526,
569, 572, 592
Variational-geomeuic method of Lavrent 'ev
569
Variational-parametric method of Kufarev
571
Vitali's theorem 18
Weierstrass's theorem 12
I
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