Advances in complex function theory: Proceedings of seminars held at Maryland University 1973 74

Lecture Notes in Mathematics Edited by A. Dold and B. Eckmann Subseries: Departmentof Mathematics, University of Maryland,College Park Adviser: L. Greenberg

505 Advances in Complex Function Theory Proceedings of Seminars Held at Maryland University, 1973/74

Edited by W. E. Kirwan and L. Zalcman |11

|

I

I

Springer-Verlag Berlin.Heidelberg. New York 1976

Editors William E. Kirwan Lawrence Zalcman Department of Mathematics University of Maryland College Park Maryland 20742/USA

Library of Congress Cataloging in Publication Data

Main entry under title: Advances in complex function theory. (Lecture notes in mathematics ; 505) An outgro~-th of a year long program of seminars, lectures and discussions presented at the University of Maryland, 1973-74, sponsored by the Dept. of Mathematics. 1. Functions of complex variables--Addresses, essays, lectures. I. Kirwan~ William E., 1938II. Zalcman, Lawrence. III. Maryland. University. Dept. of ~athematics. IV. Series: Lecture notes in mathematics (Berlin) ; 505. QA3.L28 no. 505 [QA331] 510'.8s [515'.9] 75-45187

AMS Subject Classifications (1970): 30A24, 30A32, 30A34, 30A36, 30A38, 30A58, 30A60, 3 0 A 6 6

ISBN 3-540-0?548-8 Springer-Verlag Berlin 9 Heidelberg 9 New York ISBN 0-38?-0?548-8 Springer-Verlag New York 9 Heidelberg 9 Berlin This work is subject to copyright. All rights are reserved, whether the whole or part of the material is concerned, specifically those of translation, reprinting, re-use of illustrations, broadcasting, reproduction by photocopying machine or similar means, and storage in data banks. Under s 54 of the German Copyright Law where copies are made for other than private use, a fee is payable to the publisher, the amount of the fee to be determined by agreement with the publisher. 9 by Springer-Verlag Berlin 9 Heidelberg 1976 Printed in Germany Printing and binding: Beltz, Offsetdruck, 6944 Hemsbach

PREFACE The past decade has been a period of r e m a r k a b l e a c t i v i t y for complex f u n c t i o n theory. long standing,

An unusual n u m b e r of open problems, many of

have been settled.

At the same time,

new t e c h n i q u e s of

e x c e p t i o n a l power continue to be developed.

These methods have

already yielded a great deal; their promise,

if anything,

their past success.

exceeds

An optimist will see in these d e v e l o p m e n t s

cations of a r e n a s c e n c e of f u n c t i o n theory,

the a c h i e v e m e n t s

indi-

of which

may u l t i m a t e l y rival the great triumphs of the past. It was thus e s p e c i a l l y a p p r o p r i a t e

for the U n i v e r s i t y of M a r y l a n d

M a t h e m a t i c s D e p a r t m e n t to designate the a c a d e m i c year 1973-74 a Special Year in Complex Function Theory. complex analysts

The present volume

special year.

well over thirty

from the United States and abroad p a r t i c i p a t e d

y e a r - l o n g p r o g r a m of seminars, sions.

Altogether,

lecture courses,

in a

and informal discus-

is an o u t g r o w t h of, and a m e m o r i a l to, this

Partly a (very incomplete)

record of m a t e r i a l p r e s e n t e d

at seminars during the year and partly an a n t h o l o g y of results a c t u a r y o b t a i n e d during that period,

it ranges over a r e l a t i v e l y broad expanse

of classical and m o d e r n function theory: harmonic

functions,

conformal mapping, disparate topics

Fuchsian groups,

c o e f f i c i e n t problems,

automorphic

functions,

functions of several variables.

is a certain emphasis,

sub-

quasi-

Uniting these

in point of view or in method,

on problems having concrete geometric content.

This seems only natur-

al, for geometric function theory has been the source not only of some of the most difficult and important problems of the general theory, but also of many of its most beautiful and seminal results. Future volumes,

to be published

in the Springer Lecture Note

Series and the U n i v e r s i t y of M a r y l a n d D e p a r t m e n t of M a t h e m a t i c s

Lec-

ture Note Series, will be devoted to m a t e r i a l p r e s e n t e d in various lecture courses during the special year.

For help in editing the

IV

present volume, we are grateful to Bernard Shiffman and, especially, Leon Greenberg.

Thanks also are due Pat Berg and Paula Verdun for an

excellent job of typing, and the Mathematics Department of the University of Maryland for having made available the resources required for the preparation of the manuscript. We hope the reader will find in these papers ample evidence of the continued vigor of the insights of the classical masters and their successors.

Complex analysis

is indeed alive and well.

W,E, KIRWAN LAWRENCE ZALCMAN

TABLE OF CONTENTS

C.A.

BERENSTEIN An E s t i m a t e for the N u m b e r of Zeros of A n a l y t i c F u n c t i o n s in n - D i m e n s i o n a l Cones . . . . . . . . . . .

1

P E T E R L. D U R E N A s y m p t o t i c B e h a v i o r of C o e f f i c i e n t s of U n i v a l e n t Functions . . . . . . . . . . . . . . . . . . . . . . . W.K.

17

HAYMAN On the D o m a i n s W h e r e a H a r m o n i c or S u b h a r m o n i c F u n c t i o n is P o s i t i v e . . . . . . . . . . . . . . . . .

24

ALBERT MARDEN Isomorphisms

B e t w e e n F u c h s i a n Groups

. . . . . . . . .

56

ALBERT PFLUGER On a C o e f f i c i e n t

P r o b l e m for S c h l i c h t F u n c t i o n s

....

79

CH. P O M M E R E N K E On I n c l u s i o n R e l a t i o n s for Spaces of A u t o m o r p h i c Forms . . . . . . . . . . . . . . . . . . . . . . . . .

92

EDGAR REICH Q u a s i c o n f o r m a l M a p p i n g s of the Disk w i t h G i v e n B o u n d a r y Values . . . . . . . . . . . . . . . . . M.

i01

S C H I F F E R and G. S C H O B E R A Distortion

T h e o r e m for Q u a s i c o n f o r m a l M a p p i n g s

138

URI SREBRO Quasiregular Mappings T.J.

. . . . . . . . . . . . . . . . .

148

SUFFRIDGE Starlike

Functions

as Limits

of P o l y n o m i a l s

......

184

PARTICI PANTS IN THE SPECIAL YEAR

Lars V. Ahlfors

(Harvard University)

Albert Baernstein II

(Washington University,

Carlos A. Berenstein

(University of Maryland)

Douglis M, Campbell

(Brigham Young University)

David Drasin

(Purdue University)

Peter L. Duren

(University of Michigan)

F.W. Gehring

(University of Michigan)

Leon Greenberg

(University of Maryland)

Walter Hayman

(Imperial College, London)

Mauriee Heins

(University of Maryland)

J.A. Hummel

(University of Maryland)

James A. Jenkins

(Washington University,

W.E. Kirwan

(University of Maryland)

Jan Krzyz

(Maria Curie-Sklodowska University, Lublin)

011i Lehto

(University of Helsinki)

Albert E. Livingston

(University of Delaware)

Thomas H. MacGregor

(SUNY, Albany)

Albert Marden

(University of Minnesota)

Petru Mocanu

(University of Cluj)

Raimo N~kki

(University of Helsinki)

Bruoe Palka

(Brown University)

John A. Pfaltzgraff

(University of North Carolina)

Albert Pfluger

(ETH, Zurich)

George Piranian

(University of Michigan)

Christian Pommerenke

(Technische Universitat Berlin)

Edgar Reich

(University of Minnesota)

M.M, Schiffer

(Stanford University)

Glenn Schober

(Indiana University)

Uri Srebro

(Technion, Haifa)

St. Louis)

St. Louis)

Viii

Kurt Strebel

(University

of Zurich)

Ted J. Suffridge

(University

of Kentucky)

Jussi V~is~l~

(University

of Helsinki)

Lawrence

(University

of Maryland)

(University

of Maryland)

Mishael

Zalcman Zedek

AN ESTIMATEFOR THE NUMBEROF ZEROESOF ANALYTIC FUNCTIONS IN n-DIMENSIONALCONES CARLOS A, BERENSTEIN*

!.

INTRODUCTION The r e l a t i o n b e t w e e n the order of g r o w t h of an entire function

in

Cn

and the area of its zero-variety,

and more g e n e r a l l y

N e v a n l i n n a theory in several complex variables, studied in the recent past by Chern, among others

(see,

e.g.,

has been e x t e n s i v e l y

Griffiths,

Lelong,

[12] for references).

Stoll,

The t e c h n i q u e s used

by these authors are e s s e n t i a l l y similar to the d i f f e r e n t i a l - g e o m e t r i c m e t h o d employed by N e v a n l i n n a and A h l f o r s

in the case of a single

variable. M a n y problems

in analysis require a similar e x t e n s i o n

to several variables) angular r e g i o n s of

C I.

(from one

of results known for f u n c t i o n s d e f i n e d in For reasons that will become a p p a r e n t below,

it is not p o s s i b l e to r e d u c e the problem to the o n e - v a r i a b l e case; nevertheless,

using a p o t e n t i a l - t h e o r y a p p r o a c h one can still o b t a i n

the r e q u i r e d estimates

(Theorem 2 of

w

below).

I w i s h to thank P r o f e s s o r M. Schiffer for the very helpful comments he m a d e in our conversations.

2.

PRELIMINARIES Let us recall

d e r i v a t i v e in _~i ( [ _ B) 4~

~n

some standard n o t a t i o n can be w r i t t e n as

(cf.

[7]).

d = B + ~ ,

we o b t a i n dd c : i 2~

~[.

* This research was supported in part by NSF Grant GP-38882.

The exterior

and w i t h

de =

In

~m

we indicate

so it makes

B(0,r)

~

~n =

= B r = {llzll

A2n

E C n,

=< r } ,

B(a,r)

the Laplace

operator,

to functions

Ag =

of n-complex

x_~, j=l ax. ] variables

~2n.

z = (Zl,...,Zn)

generally, r

A : Am

sense to apply

by identifying If

by

Sr

llzll2 = IZl 12 + .-. + IZn 12,

=

:llzll

(z

= r}

= {z : llz-all ~ r}.

for

we w-rite

0 < r < ~.

We can define

two

More

(l,l)-forms

by n

r : ddClfzll 2 -_ ~-~ i j=E1 dzjAdzj

= dd c

Then

Cn = r ^ "'" ^ r

logll zll 2 ,

(n times)

more generally

the restriction

linear variety

is the euclidean

other hand,

~n-I

under unitary

transformations

(i)

~2n-I

O.

is the volume

of

is a measure

z~

Ck

form of

~n ,

to any k-dimensional

area form of the variety. of "projective"

and complex

: dc l~

area:

dilations,

and

(complex) On the

it is invariant and

^ ~n-l'

is the area form in the unit sphere

S 1 = {Ilzll = I},

~

~2n-i

i.

1 If i.e.

f

is an analytic A2n ioglf(z) I

the analytic

(2)

defines

variety

~(r)

a positive

measure,

= 0}.

of (2) defines

we can define : ~

loglf(z) I

is subharmonic, whose

support

is

Moreover,

^ Cn-i = ~(A loglf(z)I)r

that the l.h.s,

As usual,

then

V = {z : f(z)

ddC l~

it follows V.

function,

the countin~

ddC l~

^r

the euclidean function

area form in

by

0 < r < ~.

r

More usually, and

if

D r = D n B r,

D

is a cone

then

in

~n

(having vertex at the origin)

(3)

sD(r) : ID dd c log If(z)I 2^ Cn-l" r

Similarly,

we have the projective

9(r)

: ~

area of

V,

defined

dd c log IfCz)12^ ~n-l" r

If we assume formula

further

that

in Nevaniinna

f(0)

~ 0,

we have the following

theory

(4)

~(r) : ~(r)

r2n-2

Sketch of the proof. one sees easily

9(r)

that

: I

Clearly

~n-i

dCn_l

ddCl~

: I

I.

2.

Furthermore,

by Stokes theorem,

dCl~

^~n-I

Sr d eloglf(z)I 2 A ~Cn-i

= ~ 1

r This

For

fails when

of additional

n = I,

a(r)

a(r).

r

simple relation

due to the appearance Remark

: 0.

hence,

= [

Sr Remark

= d~n_l

= IIzll-2n+2r

Br

~D

crucial

boundary

= 9(r)

~

is replaced

by

terms.

= number of zeroes of

f

in

Br 9 The next important ducing

it to the one variable

(5)

~(r)

where the operator inner

formula

integral

allows

us to compute

It is Crofton's

case.

~(r)

by re-

formula

[ii]

I dd c loglf(X~)l 2, = [ J ~s 1~2n_l(~ ) IXJ~r dd c

acts on the complex variable

just counts

the number of zeros of

g(k)

l,

so the in

: f(kz)

{fxl A r}. Let us recall (p > 0) that

and finite

that a function

f

is said to be of order

type if there exist constants

A, B

> 0

P such

If(z)l For such functions,

it is known

r§

and therefore

lim r~

Similarly,

Crofton's

degree

then

complex

half-plane g(0)

# 0.

{~ :

I~

-

THEOREM

Denote

= lim 9(r) r§ rp

shows that if

two theorems Let

g

by

< =

f

ePB.

is a polynomial

from the theory of functions

be an analytic of order

9g(r)

If

[9, p.185]

Sg(8)

Cp

=< epB

of

p

function

and finite

the number

defined

type,

of zeroes

of

of one

in the

such that g

in the disk

r/21 ~ r / 2 } .

lim r+~

where

that

< m.

{Re ~ ~ 0},

I.

function

o(r) r p+2n-2

formula

9(r)

variable.

[9, p.44])

(5)

by (4) and

We now recall

(cf.

~i IlkI~ r d d C l o g l f ( k z ) 1 2

(6)

m,

=< A exp {BllzllP}.

p > 1

then there exists an increasing

such that ~g(r) rp

< =

(i +llp) p ~ 1 2 J-~2

2~(p-l)

cosPe dSg(e)

is a positive constant independent of

< C B = P

g

and

B

is the

constant involved in the definition of finite type. Remark theorem

3.

By using conformal

for functions

larg I I ~ e/2.

of order

This possibility

since by a theorem of Liouville

mappings,

p > T/e,

we can obtain a similar

defined

does not exist

in the angle

in

the only conformal

C n, ' n > 2, maps are the M~bius

transformations. The generalization

of theorem

of this paper and appears

in w

I to cones

in

Cn

is the objective

Suppose open cone

f

D

is h o l o m o r p h i c of order

in

C n.

p

and finite type,

We define the indicator f u n c t i o n of

I I h (z) = lim lim log,f(ry), y+z r+~ rp yED

(7)

f

in an by

z # 0.

This f u n c t i o n is ( p l u r i ) - s u b h a r m o n i c and h o m o g e n e o u s of degree p.

For

n = i,

h

is even continuous. We say

all

z

the outer

f

lim

is not n e c e s s a r y and the f u n c t i o n

is of c o m p l e t e l y r e g u l a r growth i__qn D

if for almost

we have

E D n SI ,

* loglf(rz)1 h (z) = lim

(8)

r§

r p

Then we have the f o l l o w i n g

T H E O R E M II.

p > 0

Let

[9, p.182]

and completely

g

be an analytic

regular growth

there exists an increasing

lim r+ ~

The m e a n i n g of of formula

3.

9

g

g

in {k E ~l : Re k > 0}.

function

(r) (9)

function of order Then

such that

Sg(e)

712 _

rP

1

[

2~p ~-~/2

cosP8 dSg(8)

is the same as in T h e o r e m I.

(9) to several variables

<

The g e n e r a l i z a t i o n

is due to G r u m a n

[7].

FUNCTIONS OF COMPLETELY REGULAR GROWTH We a s s u m e the number of v a r i a b l e s

define

N

is

to be the cone g e n e r a t e d by

(10)

N

and as before

Nr = N O B

: {tz : z ( N,

n > 2. N,

If

N r Sl,

we

i.e.

t > 0},

r"

Using the m e t h o d of L. Gruman,

we prove the f o l l o w i n g result.

PROPOSITION

i.

C SI,

f

and

K c N,

Let

p

a f u n c t i o n analytic in

f(z)

uniformly

in

K .

an open set

such that for every compact

= p(z) + O(IIzll-I)

Then

2-2n lim ~K (r) ~ < ~. r~ log r

(12)

Proof.

f

N

N

we have

(ii)

Pm

be a non-zero polynomial,

If

z E SI,

is a homogeneous

polynomial

are of completely

pm(Z)

# 0

t > 0,

regular

p(tz)

: tmpm(Z)

of degree

growth

m.

+ o(tm-l),

Clearly both

in the sense that if

and

z ( N

and

lira loglf(rz) I _ lira log [p(rz) ] = m. r~ log r r+~ iog r

Take any such

z E N

and pick

{w ( S 1 : llw-zll < s} c N. almost all obtains

s > 0

I

formula

e,

Let

we have

the Jensen

(14)

0 < s < I,

such that

f(sz)

# 0,

so from Crofton's

in n-variables

loglf(s(z+e~))[m2n_l(~)

loglf(sz) I

s

:

Gsz(t)

= I

i~

ddC l~

dt asz(t) ~

'

r

B(sz,t) The right hand side of (14) satisfies s

i~ kl(e)

dt ~

is a positive

~

D' =

D = {w ( S 1 : llw-zll < s/2}.

S1

where

p

then

(13)

where

where

~ kl(e)

constant.

~sz(3/4 s2n-2

es)

For

formula

one

In other words, (i5)

for any

kl(e)

r > 1

asz(

1

es)

I

< ~2n_l(~) -- S1 Since

r > i,

(16)

ds s 2n-I Ir If(zs+es~)I log 1 if(sz) I

we can find an integer (l+e/4) m < r <

we have

m ~ i

ds -s

such that

(l+e/4) m+l.

Define (17)

a q : (i + El4) q

From the definition of (18)

D

q : 0,-..,m.

it follows that for

Da \ Da r B(sz, 3~ s) q q-i 4

aq_ 1 < s < a q

q : l,...,m.

Hence (aq - aq_ I) ID

a\Da q

ddC l~

12 ^ r

q-i a

: ( a q - aq_ I) la q d~D (s) q-i

< aq2n-I

ds

q ~sz ( -- s) 2n-i q-i s

Therefore a a2n-i ( ~ - l ) q-i aq_ 1

and we obtain

a a l-2n (~cs) ds fa q s2-2ndq D~(s) <= a2n-I q q L s sz 4 q-i q-i

a

a

4 +s~ laq e s 2-2n d~ q-i

(19)

By adding the inequalities

(s) < laq sl-2n Osz (~ es) ds. q-i

in (19) for

q = l,.'.,m

and (16) we obtain with a new constant

(20)

k2(~)

~

< =

a0

and using

k2(e) > 0

~2n_l(~)

log

S1

if(sz) I

From (ii) it follows that for

(15)

s

w E D'=\D'\1

loglf(w) I < m logllwll + 0(i) loglf(sz) I : m log s + 0(i). Therefore

the integral on the right hand side of (20) can be

bounded by (constant) parts,

log r;

k3,

k4

~

k 3 log r + k4,

are positive constants

depending

obtain a similar inequality for any compact finite covering of Remark 4. defined by

K

by sets

D

f

in

K\K R for

R

sufficiently

of an algebraic

algebraic and therefore

9(r)

that for any f(z+k~)

K c N,

and

z.

f

of Proposition

large.

(k E C)

variety

V

of the variety

It follows from a V

in

Cn

lies

variety then it is itself

is bounded.

Additional

assumptions

1 should enable one to eliminate

from the conclusion; z,~ E C n,

II~ll = i,

We

by choosing a

lies within an c-neighborhood

within an e-neighborhood

log r

e

as above.

theorem of Rudin [i0] that if an analytic

the function

on

From (ii) it follows that the analytic variety

Vp = {z : p(z) = 0}

factor

by

and we finally obtain

~D(r) --~ r where

the left hand side can be integrated

for example~

in a disk of radius

1

the

one might assume

the number of zeroes of

g(l)

on

=

is bounded independently

of

z

and

~.

An example above remark

(21)

of a function

is the exponential f(z)

Here the

aj

are distinct

f

=

[ a j(z) j=l

are non-zero and

is of completely

regular

(22)

with the property

mentioned

in the

polynomial,

exp

s > 2.

polynomials,n

~'3 = (aj,l,-.',aj,n)

~ cn

=

[ Zkaj, k. (cf. [13].) In this case, k=l type of order 1 with indicator function

h (z) = max Re. J

J Then by Crofton's

(23)

formula

lim o(r) r-~ ~

As a corollary

= I

of Proposition

with very few zeroes for

f;

N (k) = {z ( S I : h*(z)

fk(z)

4.

GENERAL CASE

Taking

= e

-

and

into account

problem of estimating function

f(z)

in a cone

in

D

is more general T,

cones

in

we obtain

it is enough = Re

N~k)

the number

below

t e.g.

1

formulas

(cf.

(2) and

the existence

function

of regions

to take > max Re} j~k '~J [2]).

(3),

we can reduce

of zeroes of a non-zero

to estimating

Riesz mass of the subharmonic

By

h*(z)m2n-l(Z)" SI

the

analytic

qD (r) = fD u.

we shall restrict

Au, the so-called r Though the method used

ourselves

to circular

~m. x = (Xl,.-.,Xm),

if

D fl SI

Ixl

we denote

respectively

has a smooth boundary with bounded curvature.

a point

in

~m

i0

and its euclidean norm.

To keep the notation u n i f o r m we will assume

m ~ 3,

m : 2

though the case

polar coordinates (24)

is easier to deal with.

(r,81,...,em_ I)

0 < r : Ixl,

where the remaining

x

: x/r,

8's

We introduce

by 81 : arc cos x I

are defined

(0 ~ e I ~ ~)

in the usual manner.

Then the

Laplacian can be written as A m = A : r l-m ~-~ 8 tr . m-i ~-~) 8. + r - 2 ~ ,

(25)

where

~

is an operator

ly the Laplace-Beltrami

involving only the a n g u l a r variables, operator on the sphere

~

name-

= {x E ~ m : ix I = i}.

There is only one case where we need an explicit description of Assume the harmonic 81 ,

and that

function

v(r)

v

6.

depends only on the coordinates

= v(r,8 I) = rPf(81) ,

p > 0.

r,

Then

Av = vrr + m-lr Vr + r-2 Ve 8 + ( m - 2 ) r - 2 ( c o t

81)v e

i !

= 0 l

or

(26)

f"(8 I) + (m-2)cot

By the change of variable

(27)

(l-~2)g"(~)

81 f'(8 I) + p(p+m-2)f(8 I) : 0.

~ : cos e l,

f(e I) : g(~),

we have

- (m-l)~g'(~)

+ p(p+m-2)g(~)

: 0.

The solutions of (27) that are regular for functions,

given explicitly

(28)

m-2 2 (~) = F(p+m-2) g(~) = Cp F(p+l)F(m-2)

where t = 0.

F(a;8;y;t)

exactly

F. Klein

[p+l]

p.276]

g(1)

F(p+m-2) = F(p+l)F(m-2)

(see [8, p.286])

zeroes in (-i,i]

if

are the Gegenbauer

by

F(p+m-2;-p;~;

denotes the hypergeometric

Furthermore,

theorem of

[i, vol.3,

~ = i

~ 0

~_i~)

function regular for and it follows from a

that the function p

,

not integral,

and

g

has p

zeroes

ii

if

p

is a p o s i t i v e A circular

integer

open

cone

(where

K(e),

=

[s]

integral

0 < a < ~,

part

is d e f i n e d

of

s).

b y the

condition

(29)

K(~)

Let

S(a)

tions

f

= K(a) and

= {x

O S I = {x

~ 0 : 0 ~ e 1 < e}.

: r = i,

eigenvalues

u

0 ! e I < ~}.

of

6

in

in

S(e),

S(e)

Then

are

the

eigenfunc-

defined

b y the

condition

(30)

6f + ~f

6

Since ues

= 0

is an e l l i p t i c

operator,

0 < Pl < ~2 < "''"

(31)

we o b t a i n

on

~S(e).

a sequence

of eigenval-

T h e n we c a n w r i t e

p = p(p+m-2),

Corresponding

f : 0

eigenfunctions

c a n be

p > 0.

found

which

are

e I)

where

the

functions

only

of

m-2 of

81,

namely

terized

= Cp 2 (cos

f0(81)

O'S a r e c h a r a c -

by the c o n d i t i o n m-2

(32)

For

C 2 (cos e) P instance,

dimension

for

0.

we o b t a i n

Pl = l,

independent

of

the

m.

It f o l l o w s we can

~ = ~/2

=

find

f r o m the a b o v e

a harmonic

that

function

for a n y

v

in

p >

K(~)

0,

p ~ pl~P2 ,-.-

and a p o s i t i v e

P constant

K

P

(33)

the

properties

Iv0(x) I ~ Kpr p,

In fact~ pn ~ a n d find

with

we c a n t a k e e'

harmonic

v

= -r p

to be a c o n s t a n t

P

sufficiently functions

Vp(X)

close v

P

to

e

(actually

for

multiple

(0 < e'

< e)

depending

also

x

E ~K(e).

of

f . For P

we c a n on

p =

similarly

e')

in

12

K(~')

such that the conditions

replaced

by

3K(a)

3K(e').

Let us denote by with pole at ~n'

above are satisfied with

a.

normalized

G(x)

Let

= g(x,a)

~n

the Green's

be the above

by the condition

[

function of

K(~)

eigenfunction with eigenvalue I~n 12 ~m-i

= i.

Then follow-

J S(a) ing Bouligand

[3],

Lelong-Ferrand

(34)

G(x,a)

= ca

~

has proved that for

t pn ~On. ~n(a~)~n(X~)

n:l where

On = -Pn - m + 2,

instance

[4] or [6]).

we can conclude

positive constants

(35)

/(m-2) ~ + 4~ n

and

ca

is the area of

S(~)

From known estimates of these

that if kl,

r > lal = t

a ~

(i,0,...,0),

k2

such that

F.]_ ~ G(x) dist(x*,~S(a)) -I r

r ~ 2,

(see for

~'s

and

~'s

then there exist

Pl+m-2 ~ k2

and we have also

Ol-i (36)

S-~G Sr (x)

= olr

t

Pl

~l(X~)~l(a *)

c /(m-2)

where

r § =.

with pole at

For a

R ~ 2,

Ol-i + o(r

)

+4U 1

the Green's function

GR(X)

= GR(X,a)

of the region

(37)

KR(a)

: {x E KCa)

: r < R}

can be found to be

(38)

R m-2

GR(X)

: G(x) - (~)

R2

G(-yx). r

Hence

(39)

it follows from (36) that there exists a constant O I-I ~G R(x) I 0 _<_ - ~

ir = R

< kS R

k3 > O

13

Finally, subsets obtain

of

we

need

KR(a).

from

a lower

Take

x

bound

such

on

that

GR

on

sufficiently

2 ~ r = eR,

large

0 < E < i;

we

(35)

GR(X)

~ dist(x ~,~S(~))R

-Pl-m+2( kl

Pl+m_2

Pl)

k26

9

c The

expression

fore,

there

in p a r e n t h e s e s exists

c0 > 0

increases such

that

to

+~

when

for a n y

r,

c § R

0+ "

There-

satisfying

2 ~ r ~ EoR,

(40)

We c a n now

THEOREM

i.

harmonic

constant

Let

near

constants

prove

u 0

B,

and

satisfies

p.

Then

M = M(u,8)

I KR.(6)

Proof.

and

Since

therefore

we c a n

(i,0~0,-'',0) function R>

such

find

P

function

u(x)

for any

such

aside

u Z -%

that

w = u - Bv

~ BrP+c

for

- C,

v

P

R ~

p~

which

some

we

i8

positive

can f i n d

a

2

: max(p,pl).t

for

the m o m e n t

the

set

{u

exceptional

cases

has m e a s u r e

zero

as c l o s e

Applying

as d e f i n e d

the

= -~}

a 6 K(e)

~ -~.

K(~),

for

0 < 8 < ~,

that

a point u(a)

in

as we w a n t

Green's

in (33),

formula

we h a v e

I

where

n

GR(x,a)Au

denotes

p = pI

+ w(a)

KR(~) the

we have

:

I

for

w -SGR ~ ~m-i ~KR(~)

inner

normal.

to take

Clearly,

P* = P I + e,

w ~ 0 e > O.

on

to

to the

2

(41)

$for

result.

Au < MR p*+m- 2 ' =

Let us l e a v e

P = Pl,P2,'''.

pr,i n c i p a l

be a 8 u b h a r m o n i c

C,

M,

our

3K(e);

14

therefore,

setting

I

SR(e) = {x : x* E S(e),

~GR <

w -~-n- = A(I+Kp )Rp

I

~KR(~)

These inequalities ~i"

(42)

Using I

3GR

SR(~ )

<

(40),

M~R

Ixl = R},

< MIR ~n =

P+~

we have

[ ~m-i JSR(~)

P-Pl

follow from (33),

(39),

and the definition of

we conclude that

Au < M3RP+m-2 Ke0R(8 ) =

+ M4R of+m-2 lw(a)I + I

Au. K2(a )

From here the conclusion of the theorem clearly follows, the fact that exceptional

u

is harmonic near zero implies

since

[ Au < ~. J K2(~)

The

cases are treated similarly by means of the construction

in (33) above. Remark 5. when

Clearly,

P > Pl"

If

for references) term

lw(a) l

tion

u ~ -~.

P > Pl'

P ~ PI'

the Phragmgn-LindelSf

shows that

in inequality

Remark 6. for

the interesting case of theorem 1 occurs

w ~ 0

everywhere

in

theorem

(cf. [5,6]

K(e);

the dominant

(42) just gives the integrability condi-

The constant

M3

in (42) is proportional

to

B.

Thus,

theorem I assets that

R-P-m+2 I

Au

< C(6)B.

KR(6) This estimate can be improved in Theorem I

slightly to a bound analogous to that

(w

Because of its importance, analytic functions.

we restate Theorem i in terms of

15

THEOREM 2. Suppose Cn

such that

type. If

as(R)

f

f(0) # 0

is an analytic function in the cone and

f

has order

p > pl(~)

denotes the area of the variety

K(~)

and finite

V N KR(8)

(8 < ~)j

we have

~BCR) lim R+~ ~ (B

of

< C(8)B

is the constant appearing in the definition of the type of

f)

REFERENCES i.

Bateman Manuscript Project, McGraw Hill, 1953J

2.

C.A. Berenstein and M. Dostal,

3.

G. Bouligand,

4.

R. Courant and D. Hilbert, Methods of Mathematical vol. I,II, Interscience Publishers, 1962.

5.

B. Dahlberg, Mean values of subharmonic functions, Arkiv f~r Matematik ii (1973), 293-309.

6.

M. Ess~n and J.L. Lewis, The genera~lized Ahlfors-Hein8 theorem in certain d-dimensional cones, Math. Stand. 33 (1973), 113-124.

7.

L. Gruman,

8.

E.W. Hobson, The Theory of Spherical and Ellipsoidal Harmonics, Cambridge Univ. Press, 1931.

9.

B. Ja. Levin, Distribution of Zeroes of Entire Functions, of Mathematical Monographs, vol. 5, AMS, 1964.

10.

W. Rudin, A geometric criterion for algebraic varieties, Journal of Math. and Mech. 17 (1968), 671-683.

ii.

B. Schiffman,

to appear.

Sur le8 fonction8 de Green et de Neumann du cylindre, Bull. Soc. Math. de France 42 (1914), 168-242. Physics,

Entire functions of several variables and their asymptotic growth, Arkiv f~r Matematik 9 (1971), 141-163.

Transl.

Applications of geometric measure theory to value distribution theory for meromorphic maps, in Value Distribution Theory, part A, M. Dekker,

1973.

12.

W. Stoll, Value Distribution Theory,

13.

R. Tijdeman,

part B, M. Dekker,

1973.

On the distribution of the values of certain functions. Ph.D. thesis, Universiteit van Amsterdam, (1969).

ASYMPTOTICBEHAVIOROF COEFFICIENTSOF UNIVALENTFUNCTIONS PETER L, DUREN

The a u t h o r Tauberian

remainder

coefficients related

[3] r e c e n t l y theorems

of u n i v a l e n t

results,

pointed

out a c o n n e c t i o n

and the a s y m p t o t i c

functions.

including

estimation

The present

an i m p r o v e m e n t

between of the

note d e s c r i b e s

on a t h e o r e m

some

of

Bazilevich. As usual,

a nal y t i c Hayman

S

is the class

f(z)

= z + a2z2

and u n i v a l e n t

[4,5]

asserts

(i)

lim

with

equality

only

k(z)

+ a3 z3 + ...

in the unit

that

!anl n

n~

of all f u n c t i o n s

for each

< i.

A t h e o r e m of

f E S,

= e < i, =

for a r o t a t i o n Z

-

IzI

disk

of the Koebe

function

z + 2z 2 + 3z 3 + ''.

(i - z) 2

The proof c o n s i s t s simple

argument

growth,

that

in the

(2)

of two parts. each

It is first

f E $

has a d i r e c t i o n

complicated considerably recently

is

O

step

of m a x i m a l

difficult

discovered out

= ~,

for e v e r y o t h e r direction.

is to deduce

more

As pointed

e i80

sense that

lira (i - r)2if(r~8~ r+l

and the limit

shown by a r e l a t i v e l y

(I) from

for

~ > 0

a sim~er approach in [3],

(2).

The second and more

Hayman's

than for in the case

the d e d u c t i o n

of

argument

~ = 0.

Milin

~ > 0.

(I) f r o m

(2)

is

is [8,9]

18

essentially

a Tauberian

step.

f(z) g(z)

:

To be s p e c i f i c ,

(i - z) 2

~

:

let ~

z

fCz)

n

:

b

n=O

z

n

Then n Sn =

[ k=0

bk

=

an+ 1 - a n

n ~ n T1 1 k=0

Sk

=

an+ ~ n+l

and

_

On

After

a suitable

(8)

rotation

tim

of

IgCr)l

f,

we m a y a s s u m e

that

= ~ ;

r+l in o t h e r of the

words~

series

Consider the u n i t number

8

= 0.

while

n o w the

class

disk onto e

as

in

normalized

by

(8).

application

Thus

0 [ bn ,

the

(2).

(1)

is a s t a t e m e n t

is a s t a t e m e n t

S (~)

exterior Assume

Then

of the

(2)

about

about

of f u n c t i o n s

of an a n a l y t i c

that

~ > 0

a square-root

the A b e l

the C e s ~ r o

f E S

which

arc a n d h a v e

and

that

Schwarz

reflection

principle

show

~z

{i + Cl(l

- z) + c2(i

means. map

Hayman

f 6 S (~)

transformation

means

is

and an that

f

has

the

form

f(z)

(4)

-

in a n e i g h b o r h o o d with

Ill

= e.

has o b t a i n e d

the

-

of

the p o i n t

For

- z) 2 + "''}

z) 2

(i

each

z = i,

f E S~(e)

where

with

l

m > 0,

is a c o m p l e x

number

Bazilevich

[1,2,9]

estimate

lanl < ~n + B1 l~i~g n + B 2 ,

where We

the c o n s t a n t s

shall

strengthen

B1 this

and

B2

result

depend by

only

eliminating

on the

a,

c I,

and

logarithmic

c 2. term.

ig

The proof is simpler than that of Bazilevich,

since it depends only

upon an e l e m e n t a r y T a u b e r i a n r e m a i n d e r theorem,

w h i c h we state first

as a lemma.

LEMMA.

Let

the f u n c t i o n

g(z)

:

[

bkzk

k=l

be a n a l y t i c

in

the

sums

partial

then

o

]z I < 1 and

and

on

at

the

the p o i n t

Ces~ro

means

z : i.

Let

sn

denote

of

b k.

If

sn

the

: g(1) + O(i/n).

n

This lemma is c o n t a i n e d in a paper of K o r e v a a r follows from a deeper result proof of the lemma

h(z)

The f u n c t i o n hypothesis,

:

~

h

[6],

(Theorem I.i, p.47).

(as suggested by Korevaar)

we now proceed to give it.

w h e r e it

Since the direct

is short and elegant,

Let

Is n - g(1)] z n .

:

1 - z

n=O

is also analytic

in

the c o e f f i c i e n t s of

of a c l a s s i c a l t h e o r e m of Fatou m o d i f i c a t i o n s of the proof of 76),

= 0(1),

Izl < 1

h

and at

are bounded.

(proved,

z : i.

By

H e n c e by a variant

for example,

by obvious

M. Riesz as p r e s e n t e d in [7],

pp.

the partial sums n

[ k=O are also bounded.

T H E O R E M I. e

ie o

For

of maximal

Is k - g(1)]

In other words,

each

f E S (~)

a n : g(1) + O(i/n).

with

~ > 0

growth, -ie O

an e

where

Ikl

= ~ .

:

1)

~ + 0(~

,

and with

direction

73-

20

COROLLARY.

For

each

lanl Proof e o : 0,

f ~ S (~)

~

a n + C,

of Theorem.

and c o n s i d e r g(z)

-

(i

with

-

n = 2,3,--..

Assume again

Z

z) 2

~ > O,

without

the

loss

of g e n e r a l i t y

that

function

f(z)

=

bn zn

Izl < 1

n=O

Since

f

has

analytic

the f o r m

there

f

: ~{i

+ Cl(l

is u n i v a l e n t

are b o u n d e d ~

z : i,

it f o l l o w s

that

g

is

and

g(z)

Since

(4) n e a r

as

and

shown

in

- z) + C2(i

~ > 0~ [3].

- z~ 2 + "'" } 9

the p a r t i a l

Thus

by the

sums

sn

of

~he

bk

lemma,

a _

n

~n-1

as the

theorem

It

should

which

that

appeal

THEOREM

f

is b o u n d e d

that

this

that

true Here

if

(and hence complex

f E S,

has a pole ~

Under

full

of

f

is m o r e

We w i s h

can obtain direct

to t h a n k

and

its

the a d d i t i o n a l

each neighborhood one

u s e of

of

z = i,

an e v e n

and m a k e s

W.K.

Hayman

no for

of a r g u m e n t .

of finite

constants

a role.

outside

the a r g u m e n t

did not m a k e

the u n i v a l e n c e

f E S (a),

and 8uppose

D c : {z : JzJ < i, onto a region

proof

Only

played

theorems.

type

Let

the

f E S*(e).

to T a u b e r i a n

2.

n

z = 1

result.

suggesting

O(l_)

near

is c e r t a i n l y

stronger

+

be o b s e r v e d

structure

hypothesis

~

asserts.

the a s s u m p t i o n local

_

n

area. o f order

and

~,

that for each

Jz-

at most

f

maps

> ~}

ii

Suppose

e > 0,

f

is m e r o m o r p h i c

two there).

at

z : 1

Then for some

21

(5)

a n = nk

If

COROLLARY.

Proof

f ~ S (~)

of T h e o r e m .

f(z)

where

k

is the

~

Koebe

is a n a l y t i c

choose

+ Us

has

the

form

+ ~(z) ,

function,

~

and

are

constants,

z

at

z : i.

= ~(z)

the c o n s t a n t

and univalent

f

(5) holds with

1 - z '

r and

then

~ > O,

By h y p o t h e s i s ,

_

and

with

= kk(z)

s

n = 2,3,-..

+ ~ + o(i/~),

in some

r

:

Let

- vz,

v

so that

disk

r

Iz - 11 ~

# 0. e,

E > 0.

Then

r

is a n a l y t i c

Let

7. c n z n , n=O

so t h a t

(6)

an

In v i e w of integral

:

kn

+

(6) we n e e d

over

~

only

the u n i t

prove

imply

(7),

D r

n

'

show

that

r

has

=

2~3,

finite

....

Dirichlet

n=l

that

c

n

: o(i/vrn),

we use the hypothesis

I

Thus

n

disk:

I~I
This will

+c

as the

theorem

asserts.

To

that

Ilf'(z)l 2 dx dy <

E

is t h e

s u m of f o u r

univalent

functions,

each having

finite

22

Dirichlet

integral over

De,

and so

I Il~'(z)12

dx dy < ~.

D

c

But by the o r i g i n a l choice of

c,

%

also has finite D i r i c h l e t

integral over the part of the unit disk inside the circle e.

Hence the full integral

complete.

in (7) is finite,

Iz - i I :

and the proof is

REFERENCES i.

I.E. Bazilevich,

2.

I.E. Bazilevich,

On the dispersion of the coefficients of univalent functions, Mat. Sb. 68 (1965), 549-560 (in Russian). On a univalence criterion for regular functions and the dispersion of their coefficients, Mat. Sb. 74 (1967),

133-146

3.

(in Russian).

P.L. Duren, Estimation of coefficients of univalent functions by a Tauberian remainder theorem, J. London Math. Soc. 8

(1974), 279-282. 4.

W.K. Hayman, The asymptotic behaviour of p-valent functions, Proc. London Math. Soe. 5 (1955), 257-284.

5.

W.K. Hayman, Multivalent 1958.

6.

J. Korevaar, Another numerical Tauberian theorem for power series, Nederl. Akad. Wetensch. Proc. Set. A. 57 -Indagationes Math. 16 (1954), 4 6 - 5 6 .

7.

E. Landau, Darstellung und BeKr~ndung einiKer neuerer ErKebnisse der Funktionentheorie, Zweite Auflage, J. Springer, Berlin, 1929.

8.

I.M. Milin, Haymants regularity theorem for the coefficients of univalent functions, Dokl. Akad. Nauk. SSSR 192 (1970), 738-741 (in Russian).

9.

I.M. Milin, Univalent Functions and Orthonormal "Nauka", Moscow, 1971 (in Russian).

Functions,

Cambridge University

Systems,

Press,

Izdat.

ON THE DOMAINSWHEREA HARMONICOR SUBHARMONICFUNCTIONIS POSITIVE W.K,

i.

PLANE RESULTS, % R e c e n t l y

If

v

both

is h a r m o n i c connected,

a positive

result

Kuran raised

in t h e p l a n e

is it the

HAYMAN

and

the

case that

previously

[i0].

the

following

sets

v

v > 0,

is l i n e a r ?

A complete

question.

v < 0 He h a d

answer

are obtained

is c o n t a i n e d

in

THEOREM

k

Let

i.

v

be the number

nite

of

harmonic

of components

if and only

the degree

be a real

if

v,

v

of the set

is a polynomial.

we have

(i 9 i)

the sharp

If

k : 2,

n = 1

v > 0

in

of

plane

harmonic

given

in

THEOREM

D

D

point

Then

v # 0.

Then

k

In thi8 case,

if

Let

is fin

is

inequalities

so that

Let us say t h a t a f u n e t i o n

all

in the plane.

< n < k. ~i k --

COROLLARY.

if

function

2.

entire only

and

from

Let

v--+0

inside

functions

D

function8

D.

v(z) as

has

is linear 9

a domain

approaches

It is n a t u r a l

can have

be a plane

3 cases

the

domain

f = u + iv

the f o l l o w i n g

z

v

such

same

and

any

to ask tract.

let

that

F(D)

v

has

as a tract

D

finite

boundary

in w h a t w a y A full

two

answer

be the class D

is

of

as a tract.

are p o s si b l e

tThe results of this section represent Joint work with Brannan, Fuchs and Kuran. Full proofs will be published in [3].

25

(i)

F(D)

is empty;

(ii)

F(D)

is not empty but contains no n o n - v a n i s h i n g

entire function.

fo

is any m e m b e r of

all other members are given by

F(D)

a > 0

(iii)

In this case if

and

b

is real;

contains a n o n - v a n i s h i n g

F(D)

In this case

fo"

where

f = af 0 + b,

consists of all functions

F(D)

where

af 0 + b/f 0 + c,

entire f u n c t i o n

a ~ 0,

ab > 0

b [ 0,

f =

and

c

is

real.

Before

considering

Example

i.

ly c o n n e c t e d not

Take

but

sufficient

Example Zn---+~

v

is n o t

Let n

proof

zn

and

of T h e o r e m

v = 1 - e x cos y .

to c o n s i d e r

2.

with

the

the

an

of

<

some

v > 0

examples.

is e v i d e n t -

in T h e o r e m

1 it is

v.

of r e a l

be p o s i t i v e an

set

Thus

tracts

be a s e q u e n c e

let

The

a polynomial. just

1 we give

numbers

numbers

such

such

that

that

~.

n=l l+IZnl Set

f(z)

=

u + iv

an z-- z

:

n-

n:l f(z)

Then

y.

Since

tal, tions

is m e r o m o r p h i c f(z)

it is c l e a r with

functions

in the p l a n e

c a n be r a t i o n a l that

Theorem

poles.

Also,

f(z),

Theorem

since

v

of a r b i t r a r y

1 does there

2 does

and

not

not

has degree

extend

is a v e r y extend

the

same

either.

as

or t r a n s c e n d e n -

to h a r m o n i c wide

sign

choice

funcfor

the

26

i.i.

Proof of T h e o r e m

i.

The proof of T h e o r e m

L EMM A

1 is based

Under the hypotheses of Theorem I, let

i.

tracts of

v.

Then the finite boundary

s e c t i o n a l l y a n a l y t i c Jordan curves finite, U = 0

on two s u b s i d i a r y

which go from infinity on

D

~ = 1

consists of

to

p,

to infinity in the

from c o n t i n u i t y

sidering

the b e h a v i o u r

that

has

J o r d a n arcs curves.

r ,

of

be one of the

z

p

where

p

plane.

is

Also,

F.

It is evident

F

r

D

results.

The

of

sectionally going

from

latter

F

that

case

on

F.

near a b r a n c h point

analytic ~

u = 0

to

character.

~

and

is h o w e v e r

of

Thus

(possibly)

excluded

Also, by conv,

F

we see

consists

of closed

of

Jordan

by the m a x i m u m

princi-

ple. It remains finite.

to show that

It is clear

domain

D

where

the total number

that each

v < 0.

F

Further,

Since the total n u m b e r of tracts

of

p

of curves

separates

D

all these

D9

-v

F

is

from at least one are distinct.

is finite,

p

must

be

finite.

L EMM A

ber.

2.

With the n o t a t i o n of Lemma I let

Then the equation

where

f

f(z)

has at most

= w0

is an entire f u n c t i o n having

We map the unit

disk

I~I < 1

(1.2)

z

onto

:

w0

v

D

be any c o m p l e x nump

roots in

D,

as imaginary part.

by a c o n f o r m a l

mapping

t(~),

and set

F(~)

Then

U

is p o s i t i v e

=

f{t(~)}

and h a r m o n i c

in

=

U + iV.

I~I < 1

and

V

vanishes

27

continuously sible by

as

~

exception

of

(1.2).

It t h e n

approaches

any p o i n t

p

~i

points

follows

for f u n c t i o n s

of p o s i t i v e

6.13,

that

p.

179)

f r o m the imaginary

F(~)

=

ic ~=i

where

the

Thus

c

are r e a l

F(~)

evidently tion

f(z)

Lemma

2.

has

= w0

It f o l l o w s only

have

it f o l l o w s tial

from

from

Lemma

=

the

is f i n i t e Set

and

f = u + iv

number

and

of c o m p o n e n t s

show

each

the

total

(i.i).

let

F0

of the

e.g.,

0, the

I~I z

[6],

~ = 1

to theorem

Theorem

< 1.

the

to

equation

v

equation

in the p l a n e f(z)

the

This

that

= w0

equa-

proves

and f(z)

for a n y

cannot

p. F(~)

Thus

= t(~).

have

-v = w0

w 0.

Now

an e s s e n -

is a p o l y n o m i a l . i, it r e m a i n s number

To see this be the

correspond

set

complement

of

k

to s h o w that

of t r a c t s

we a r g u e

v = 0. F0

of

if

Ivl

as follows.

Let

in the

k

be the

closed

plane.

that

k

s

is the n u m b e r

counted

with

To p r o v e F0

and

in

f(z)

of T h e o r e m

then

(1.3)

where

so

the p o s -

+ ic 0,

c~ ~

that

with

representation

the h y p o t h e s i s

of r o o t s

and

satisfies

and

= i

which

(see,

~+~ -9 ~-~

roots

theorem

proof

is a p o l y n o m i a l

part

of tracts, that

number

at

p

I~I

classical

~v1

2, and

number

~p

function

roots

Picard's

singularity

f(z)

p

at m o s t

a finite

To c o m p l e t e

We

has

a finite

has at m o s t

constants

is a r a t i o n a l at m o s t

to

on

considered

(1.3)

n + s + i,

of f i n i t e

correct we

=

branch

points

of

f

on

F

0'

multiplicity.

study

as a n e t w o r k .

the

disection

As v e r t i c e s

of the we t a k e

closed the

plane

finite

by branch

28

points

Vg,

9 = 1

multiplicity of order Let

to

s ,

q

of

together

f

on

r0,

which

with

~,

which

we

acts

supposed

as a b r a n c h

to have point

n - i. e

be t h e

total

number

of edges,

to a v e r t e x ,

possibly

the

same

2e

arcs

r0

coming

analytic

branch

point

arcs.

Summing

of

of order

s

of

over all the

one

(namely

out

F0

each ~).

are

points

are

thus

vertices.

2s+2

of

from a vertex

There

of all t h e

there

branch

going

At

just a

such analytic

F0

including

~,

we

see t h a t

2e

q [ ~=I

:

(2s~+2)

+ 2n

:

2s + 2n + 2q,

i.e.,

e

On t h e

other hand

it f o l l o w s

k

which

is

finite,

We note

k

also

must since

in t h e

f

plane

from

is

ples

f = zn ,

show that

(i.i).

is

degree

n- 1

+

q.

Euler's formula

:

is sharp.

n

and

completes

f : zn + i

(i.i)

n

that

s + n + 1

that

since

q

and

so

e

is

be f i n i t e . has

This

+

incidentally

n + 1

which

s

e - q + 1

(1.3).

Finally points

:

:

<

so

k

total

number

0 < s < n-l.

<

s

of branch

Thus

2n,

the p r o o f

for which The

the

of T h e o r e m

s : n - I,

corollary

0

follows

i.

The e x a m -

respectively, immediately

from

(i.i).

1.2.

Proof

of Theorem

The proof

2.

of Theorem

2 is l o n g a n d w e m u s t

refer

for

it to o u r

29

forthcoming

paper

[3].

However

I should

like to m e n t i o n

one key

step.

LEMMA and

3. Vl,

If

fl = Ul + ivl'

v2

have a common

f2 = u2 + iv2

tract

D,

are entire functions

then

Vl,

v2

have the same

sign in the whole plane.

The p r o o f of L e m m a analytically

related

series development

3 is b a s e d on the

for

fl

in t e r m s of

obtained

from points

of b o t h

fl

f2

2.

Rs

examples

Most

satisfactorily

3. Set

t i v e i n t e g e r and

D

are

in the p o w e r

can be a p p r o x i m a t e d

or by t a k i n g

of the r e s u l t s

and

v2

is no a n a l o g u e

v2(x,y,z)

are u n b o u n d e d

= z + Re(x+iy) n,

= z + ex siny.

But

1 and

We give

where

Then

2 do

some more

is a p o l y n o m i a l

is t r a n s c e n d e n t a l .

Thus in h i g h e r n < k

since n o n - c o n s t a n t

above

and b e l o w ,

in

is a posiand

Iv21

of a r b i t r a r y dimensions

there

(I.i).

harmonic

the set

n

IVll

vI

of the i n e q u a l i t y

We n o t e that

functions

v ~ 0

v

in

Rm

m u s t have at least

two c o m p o n e n t s .

Example a2 + b2 = 1

4.

Consider

va

positive

numbers

a, b

such that

and set

va

Then

by

the c o n j u g a t e s

of T h e o r e m s

to h i g h e r d i m e n s i o n s .

Vl(X,y,z)

e x a c t l y two tracts.

degree

in

f2

[3].

Example

have

z

f2

and

in such e l e m e n t s .

IN SPACs

not e x t e n d

fl

in such a w a y that all e l e m e n t s

elements

and

fact that

is h a r m o n i c

=

cos x sinh (ay) sinh (bz)

in

R3

but all the f u n c t i o n s

va

h a v e the

30

same sign e v e r y w h e r e for

0 < a < i.

Thus there does not appear to

be a s a t i s f a c t o r y a n a l o g u e of T h e o r e m 2.

However, we note the fol-

lowing result of Kuran [ii].

T H E O R E M 3.

If

P

monic in

Rm

where

is a n o n - n e g a t i v e constant.

c

and

is a harmonic p o l y n o m i a l in vP > 0

Rm

outside a compact set,

and then

v

is har-

v = cP,

These results raise two interesting open problems.

Q u e s t i o n i. growth,

Does Theorem

3 extend to harmonic functions

P

of slow

e.g., such that

max

[xr~r Q u e s t i o n 2.

log

IP(x)l

=

o(r)?

If two harmonic f u n c t i o n s

v I, v 2

in

Rm

have a

common tract, do they have the same sign everywhere?

T h e o r e m 2 shows That if

m : 2

the a n s w e r to both these questions

is yes. The only p r o b l e m That we have not yet e o n s i d e r e d is the inequal1 ~ k ~ n

ity

in (i.I).

This inequality does in faet extend even to

subha~monic functions and not m e r e l y h a r m o n i c p o l y n o m i a l s in

R m.

The r e m a i n d e r of these lectures will be concerned with this extension. We recall some definitions: Let

u(x)

be defined in a domain

to be subharmonie

(s.h.) in

D

D

in

R m.

Then

u(x)

if The following c o n d i t i o n s are sat-

isfied.

(i)

(ii)

-~

< u(x)

u(x)

<

is said

+|

is upper s e m i c o n t i n u o u s

(u.s.c.)

in

D;

31

(iii)

u(x)

for

some

r,

where

to

(m-l)-dimensional

x If

< I(x,r)

and

u(x)

I(x,r)

radius

is s.h.

and h e n c e

denotes

the

surface

for all

average area

arbitrarily

of

u

on the

with

sphere

small

respect

of c e n t r e

r.

in

Rm

let

B(r)

=

max

u(x).

Ixl =r We d e f i n e

the

l

order

:

~ r§

As a s p e c i a l

l

and

io~ + B(r) log r

case

we m a y

is a h a r m o n i c

polynomial

(i),

(gig)

k

(ii)

and

in T h e o r e m

number hand

of d i s t i n c t

~

consider

=

satisfied the

of

function

n.

u(x)

and

I = Z = n.

number

of t r a c t s

of the

set

of

(i.i)

plausible

u

is 8.h.

by

u = Ivl,

Evidently

components make

~

io~ + B(r) log r

rlim ~

the

of d e g r e e

are

order

u > 0. the

u

where

is s.h.

Also, of

since

the n u m b e r

u,

This,

following

v

i.e., and

the

result

the left

of

[8].

THEOREM

If

4.

the lower order If r n sin and

lower

'

1 is p r e c i s e l y

inequality

Heins

the

~

k

is e v e n

he,

where

u =

examples

now

when

extend

if

pDints

u

k 8

is sharp. u : r 89

is odd,

since

is h a r m o n i c

(gig)

= x + iy,

For

by

8 + 2~. except

is e v i d e n t l y

tracts,

where

k > i,

v : I m ( x + iy) n =

polynomials

8 + i sin e)

k.

k

1 ~ ~ ~ k.

inequality

to odd

even

and has

satisfies

the

we r e p l a c e and

R2

the h a r m o n i c

Iv I s h o w that

everywhere, at t h e s e

u

z = r(cos

dently o n e - v a l u e d changed

of

in

However,

isin(}kS) u

at p o i n t s

these is evi-

I

Also,

satisfied,

1 n = [ k

and

I

is un-

is c o n t i n u o u s where

so t h a t

u = 0~ u

is still

s.h. However,

Heins'

theorem

lies m u c h

deeper

than

the

corresponding

32

result for harmonic polynomials. Denjoy

conjecture

proved by Ahlfors

totic values of an entire

k

Thus there exist paths f(z)--+a

Since the values ferent

9

a9

in the open plane

as

z-~

F

are disjoint paths going from

zV

lies in

from

0

to

z

We complete

in common.

D

for

from a classical each domain

9 = 1

to

k,

a~ ~ ag+ I.

from dif-

F

~

in

Izl ~ R,

by straight from

0

so that

y9

Yk+l

that

~

having only

arranged

and

= Yl"

f(z)

line segments

to Yv

where

Y~+I

in antibound a

Also it follows

must be unbounded

in

since f(z)--~a9

and

Julia

F

Thus we may assume that the to

where

values

F .

We may assume the

theorem of

D ,

~.

f(z)

such that

distinct the paths

y9

clockwise order around the origin, domain

along

z9

and obtain paths

these endpoints

finite asymptotic

F

will not intersect near

Izl = R.

To see this, suppose that

distinct

are supposed

the famous

on the number of distinct asymp-

function.

is an entire function with a .

In fact, it contains

Thus

as

z --~

If(z) I < M, yg,

along

Yv"

say on the union of the paths while

If(z)l

in each domain

> M

somewhere

D v-

Consider now u(z) u

: u(z)

the plane and yg,

E : u(z) E

> 0

while

u(z)

> 0

somewhere

in each

has at least one component

has at least

k

If(z)l1

{----~I

: Yk+l

Then

paths

l~

components

and

u(z)

is clearly u(z) D .

= 0

s.h.

in

on the

Thus the set

in each domain has at least

D9 k

and so

33

tracts.

4,

u(z)

has

and since the lower order of

u(z)

coincides

definition

of

if

Thus,

by T h e o r e m

of the lower order

k ~ 2.

in

k

we deduce

Ahlfors'

1 P ~ 7

also yields

and

which have at least

Rm

The r e s t r i c t i o n func t i o n

in

any o r d e r

if

theorem

k = i.

m

both greater than one,

let

Rm 1

k > 2

k

to

What is

is e s s e n t i a l

has at least

subject

tracts.

since

one tract and

0 < I < ~.

s

any u n b o u n d e d

such a f u n c t i o n

In view of T h e o r e m

s.h. m a y have

4 we have

1 = ~ k.

s

It turns m > 2 when

out that

into

One reason

k

congruent

divides

into

k

AN UPPER BOUND

moni c

Heins' m e t h o d

though the results k = 2.

plane

3.

w i t h ~he c l a s s i c a l

be the lower bound of the Sower order8 of s.h. functions

s

Rm

1 ~ k,

at least

the f o l l o w i n g

Given integers

PROBLEM.

f(z),

In fact the a r g u m e n t

We can now formulate

lower order

polynomials

examples

sharp w h e n

or

in all cases

shall

see,

sible

in certain

THEOREM

5.

good bounds

k = 2

m = 2

Let

for and

for h a r m o n i c

for general

except

fact that we cannot

divide

s.h.

which is even.

polynomials

some harThese

are c e r t a i n l y

They may even be although,

as we are pos-

cases.

m, n

be integers,

k

is defined as follows.

k

of degree

m ~ 2,

v

where

u,

2~/k.

of tracts.

functions, slight i m p r o v e m e n t s

harmonic p o l y n o m i a l tracts,

opening

to c o n s t r u c t

large n u m b e r

s k

in the way that the

of a n g u l a r

We proceed a relatively

to the case sharp,

cones

sections

s

quite well

are no longer

is the

circular

congruent

having

sharp

can o b t a i n

for this

right

FOR

[3] yield

~

extends

n

in

R m,

n ~ i.

There exists a

such that Suppose

that

Ivl

has

34

n + m - 2

(3.1)

where

p, q

are integers

(3.2)

and

k

We n o t e

that

:

k : 2n

(3.3)

:

(m-l)p

+ q,

0 ~ q < m-l.

Then

2pm-l-q(p+l) q

if

m : 2.

k = 2n,

If

if

m > 2,

we h a v e

n < m

and 2 { x - i / ( 4 x ) } m-I

(3.4)

<

k

<

2x m-l,

if

n > m~

where

n+m-2

(3.5)

x

F r o m this

COROLLARY.

we can

deduce

We have for

Z(k,m)

~(k,m)

where

[y]

4.

polynomial

k > 2,

m > 2

m

rl~ k] + 1 Llog 2J

<

(m-l)( } k) I/m-I

Theorem

polynomials

polynomials

y : P~l)(t) (-1,1)

5.

which

by G. Szeg8

For any positive

in the interval (3.6)

following

-

to p r o v e

of u l t r a s p h e r i c a l

LEMMA

m-i

if

integer

of degree

k > 2m

y. We

shall

need

we

shall

quote

certain

properties

f r o m the

b o o k in

[13].

k kj

and satisfying

( l - t 2 ) y '' - (2k+l)ty'

k < 2m

'

i8 the integral part of

We p r o c e e d

orthogonal

<

the

-

and

~ > O, k

having

there exists a distinct

the differential

+ k(k+2k)y

:

0.

~ero8

equation

35

pkk)(t)c

Also k

is an even or odd f u n c t i o n of

t

c o r r e s p o n d i n g as

is even or odd.

The polynomials nomials.

P~l)(t)

The property

and the equation statement

are the so-called

of the zeros

(3.6) is given

is formula

(4.7.4),

ultraspherical

is discussed

poly-

on p. 117 of [14]

on p. 80, formula (4.7.5). The last

p. 80.

We deduce LEMMA

5.

u = U(Xl,X2,''',x m)

Let

nomial of degree

xI

in

m+l x~) 89 R = ([v=l

and set

v v

~

=

be a homogeneous

x m.

to

Let

t = Xm+i/R.

be a positive

integer

Then if

~k~(1) ~ ~k (t)u(x),

is a homogeneous

k

harmonic poly-

1

1 ~ + ~ (m-l]

=

harmonic p o l y n o m i a l of degree

s + k

in

xI

to

Xm+ 1 9 Suppose

first that

~i k

degree

in

k

is even.

t 2 = x ~+I/R2 Rkp~l)(t )

is clearly Similarly

a homogeneous if

k

Rk_(~ r k )(t) where

~

homogeneous

is a polynomial polynomial

It remains x = Xm+ I,

:

=

is a polynomial

of

Thus

89 cv x2vm+IRk-2v ~=0 of degree

k

in

xI

to

Xm+ I.

k = 2p + i, RtR2P~(t 2) of degree

of degree

to show that

p2 = [mv:l xg, 2

P~I)

by Lemma 4.

polynomial

is odd,

Then

v

k

=

Xm+iR2Pr

p,

so again

in

xI

is harmonic.

t : x/R, Q

=

Rkp(x/R)

to

Rkp~l)(t)

is a

Xm+ I.

To see this,

we write

36

where

P = P~k).

(3.7)

~-~ P(t)

and for

9 : !

~x This

P(t)

leads

We note :

to

that for

p2 P'(t) ' ~-~

~Q ~x

kRk-2xp + Rk-302p '

:

m xx9 R3 P'(t),

:

x : Xm+ I

3Q

kRk_2x

:

~x

p

-

Rk_3xx 9

P'

to

(3.8)

82Q 3x 2

:

(kR k-2 + k ( k - 2 ) R k - 4 x 2 ) p + (2k-3)xo2Rk-Sp ' + p4Rk-6p".

(3.9)

~2Q Bx2

=

(kRk_2 + k(k_2)Rk_4x~)p

_ x{R2 + (2k_3)x~}Rk_Sp ' 2 2~k-6~,, + x x~

~ 9

Then V2(Qu)

=

QV2u + uV2Q

=

uV2Q + 2

+ 2

~ ~=i

8u 3xv

9 8__q__ 8x9 3u

m

~

(kRk-2p - R k - 3 x p ' ) x u ax

~=i :

by Euler's

theorem,

ic and d e p e n d s V2Q + 2s

on

k-2P

=

u{V2Q + 2 Z ( k R k - 2 p

since xI

u

to

_ Rk-3xp,)}

is h o m o g e n e o u s

xm

only.

From

of degree (3.8)

and

s

harmon-

(3.9) we have

_ Rk-3xp,)

p ( k ( m + l ) R k-2

+ k ( k - 2 ) R k-2 + 2~kR k-2)

+ x p , ( ( 2 k - 3 ) p 2 R k-5 _ mR k-3

_ ( 2 k _ 3 ) p 2 R k-5

_ 2s k-3)

+ p,,(p4 + x 2 p 2 ) R k - 6

: in v i e w of

Rk-2{(l-t2)p"

(3.6).

We now p r o c e e d

Thus

- (m+2Z)tP'

v = Qu

to p r o v e

+ k(k+m+2s

is harmonic.

Theorem

5.

Let

This n

= proves

0 Lemma

be a p o s i t i v e

5.

37

integer.

Let

k2

to

km

be non-negative

integers

k 2 + k 3 + --- + k m

such that

n.

We define

u2(x) This function that

Up(X)

=

is harmonic

Rg(xl+ix2)k2.

and

has been defined

UP is a harmonic kp+ 1 = 0

lu2(x) I

polynomial

we define

v

m = p.

:

of degree

k 2 + --- + kp

Up+ 1 = Up.

tp : Xp+i/Rp+ I, The set

where

p = 2.

any values of

xI

of

Vp+ 1

tp

fixed,

where tp

Up+ l(tp)

to

has

Xp

-I

largest value of

u2

Um(X)

We wish to maximize Then

to

to

as Up+ 1

in

p.

Xp.

If

s = k2+'''+k p,

Xp+ 1

are

goes from

-~

to

+~.

p + 1 = m,

km

tracts.

of degree

s 2 = k2,

we see that k2 + k3 +

s9 = k

s~ : n+m-2. s~

Since

sign, we see that

and having

if we choose the

intervals Xp

xm

m ~:2

different to

to

Write

Then clearly for

are independent.

Taking

This

xI

xI

km"

components.

Also, when

of constant

tracts.

1 = Hm ~ ~m ~:2 s , Am

(kv+l)

kp+ 1 + 1

sign.

+i,

intervals

k 2 + k 3 + "'" + k m = n

3 ~ ~ <, m.

to

sign changes as a function of

we can find

distinct

we have constructed

k = kp+ I,

it is true for

has constant

kp+ 1 + 1 kp+l

kp+ 1

xI

Vp+ I,

kp = 2k 2 ~ = 3

Suppose

goes from

has

has

has

Thus the sign changes of

IUp+II

=

in

Rp+l : ([p+l~:l Ix~ 12)%"

]Upl > O

is true for

Suppose

Otherwise we define

is the function of Lemma 5 with Vp+ 1

tracts.

u2u3...u P

Up+ 1 = I,

We see that

2k 2

so that

Up+l(Xl,X2,''',Xp+ I) where

has

+ 1

for

We obtain the

as nearly equal as pos-

38

sible. and

This we can

m-l-q

(3.1).

of t h e

This

If

do b y

gives

m = 2,

n < m,

we have

(3.3).

This

sV

equal

(3.2)

Then

setting to

of the

p,

where

p = i,

q = 0

q = n-

true

p,

equal q

for

and

If

m > 2,

l,

k = 2n.

and

k = 2q+l

: 2 n,

p = 2,

In v i e w o f t h e a r i t h m e t i c - g e o m e t r ~

mean

theorem,

21n+m-21 m-I

is ,divisible b y x

=

m-

n+m-2 m-I

suppose

I m---m-~--;

~

1.

inequality

k

so t h a t :

e

x > 2.

we always

(3.4).

Equality

bound

if

for

k

we

set

S

:

1

e,

q m-i ~

:

holds

Then

2(x-e)(m-l)8(x+B) (m-l)a

- ~

We note

=

log

2 + (m-1){log

)8

8 )a } m - i

(i + ~

x + 8 log(l

- ~)

+ m log(1

+ ~)

that

log(l+t)

t - [1 t 2 + ~ t 3 - -

=

-

>

t-t

2

if

Thus

log k

~8(~+8)

>

log

2 + (m-l)

= ~8 = ~ ( I - ~ )

< 1

is

q = 0.

gives

log k

since

which

have

2xm-l.

a lower

,

2{x(l

This

in

To obtain

P +

n > m,

by

:

~ =

p + i,

given

when

Thi-s is t h e r i g h t - h a n d

to

are

n = m,

k

and

sV

as required.

p = n,

remains

q

.

og x

Thus

x2

Itl

<

!

2

n - 1

39

k

This

>

gives

2x m - I

the r e q u i r e d

We can a l s o Choose

n

deduce

so t h a t

-(m-l) ~ ~ j

exp

lower

i.e.,

can s

2 n-I

< k < 2 n,

suppose

(3.1)

if

n

such

if

k

k ~

is odd,

k

:

is any

in

(3.%).

i

Suppose i.e.,

+ [lo__ Llog

of o r d e r

k < 2 TM.

that

let

l 2J

n

first

" with

at

least

k

tracts,

as r e q u i r e d .

that

1 p = [ n,

is e v e n

Now

a function

~ n,

Next in

find

2xm-i ( ii m-I 1 - 4x 2}

the c o r o l l a r y .

-_

T h e n we

bound

>

2 m.

We s u p p o s e

we h a v e

q = I.

Thus

~1 (n+l) 2

integer,

first

1 p : ~ (n+l),

that

m = 3.

q : 0,

while

Then if

n

we h a v e

k

:

~1

we d e f i n e

n

to be the

or

{(n+l)2_l}

smallest

integer

that

(n+l)

T h e n we can

find

(n+l) 2 = 2k I

a function

2

>

2k.

of o r d e r

or

2kl+l ,

so t h a t

suppose

m > 3.

Let

n

having

k I ~ k.

kI

domains,

Thus

~(k,3)

largest

integer

~ n ~

where (2k) 89

as r e q u i r e d . Next

2

n

be the

(n+m m-I

<

k.

~

k,

such

that

n+l

having

Then

2 and more

so in v i e w than

k

of

(3.4)

tracts.

n+m-i m-I we can Also

i "m-I 8) find

a function

of o r d e r

40

1 8

<

1 ( 1 k)m-i

n + 1

<

(m-l)(

k) m-I

This

completes

n+m-2 m-i

1

since

m >_ 4.

Theorem mials.

5 probably

However

+ ~1 (m-l)

gives

for s.h.

- (m-3)

1 ( m - l ) ( ~1 k)m-i

<

the p r o o f of The corollary.

The right

functions

result

the r e s u l t

for h a r m o n i c is not

polyno-

in general

sharp. ie

to

Example

5.

We define

p, and

u = np r 8 9 Isin(38v/2) I ~:i v

It is evident positive 3n.

u

9

domains,

is the

rem 4. must

idea

of at most

One can then

show that

These

is that

space

i/k

results

Thus except

of

3n

Tracts

of order

for o b t a i n i n g

has

k

in a s u i t a b l e

S = i/k

circular where

the

in

R4

5.

lower

these

Tracts

growth

tracts

sense. is ob-

cone with solid

Theo-

occupies

average

The m i n i m a l

To a r i g h t

it is

to prove

Tracts

one of these

~ = 1

and order 3

[8] in o r d e r

u(x)

R TM

,

vertex

angle

of

[5].

Un-

as one.

represent m = 2

joint w o r k with or

right

in these

our estimates

if

= r e

given by T h e o r e m

for such a tract

is taken

non-overlapping

However,

8

and so at least

and solid angle

fortunately, unless

I/k.

has

Our m e t h o d

when The tract reduces

The whole

k

of the

s

be n o n - o v e r l a p p i n g

at the origin

u

found a f u n ct i o n

instead

a proportion

by

Also,

+ ix2v

at all points w h e r e

same as that used by HEINS

The basic

tained

x2~_l

is h a r m o n i c

we have

LOWER BOUNDS FOR

bounds

u

R 2P,

is s.h.

In p a r t i c u l a r

having

4.

and so

that

in

are

k = 2 circular

Two cases sharp when

S. F r i e d l a n d

we cannot cones

fill

Rm

exactly

each of solid angle

our results

cannot

m = 2, when

be sharp.

They reduce

To

41

Theorem 4 and when k > 2

and

k : 2,

m > 2

where we show that

the lower bound never differs

bound of Theorem

5 by more than an absolute

THEOREM

O < S < 1

6.

For

s

let

D

>

Ixl

: i.

For

from the upper

constant

factor.

be a right c i r c u l a r cone in

Rm

taking the form

xI

cos

where

(Xl,X 2 '''" ,xm)

48 a point in

Also,

~

that the

is so chosen

the intersection of whole of

D

with

Rm

( m - l ) - d i m e n s i o n a l surface area of

Ixl : 1

is

S

times the area of the

Ixl : i.

Next

u = r~f(8)

let

on the boundary of

= ~(S),

where

D, m

(4.1)

be harmonic and positive where

r : Ixl

is 8 u p p o s e d fixed.

s

1 inf ~

~

such that

COROLLARY.

If

in

D

x I : r cos 8.

and

and zero We write

Then

k [ Z(S 9:1

where the infimum is taken over all sets of S ,

Ixl = ([~:im x~)289

and

), k

p o s i t i v e numbers

~ S v = i.

~(S)

> r

0 < S < i,

where

r

is convex

then

(4.2)

s

~

r

Assuming the main t h e o r e ~ we deduce the corollary inequality.

whenever (4.1).

[ S

In fact, since

= I, and this yields

This technique

lest results

~(S)

is convex we have

s

is due to Dinghas

in this direction.

from Jensen's

~ r

in view of

[4] who obtained

the earl-

42

Before of

m.

going

For, a n y

Rm+l

of t h e

B(r))

when

lower

bound

s

we

for

Rm

if

t = cos

stead of mial

choose mined

that

be c h o s e n cos ~ related

in

becomes

Rm

order

with

(in f a c t

respect

to

yields

a s.h. the

function function

same

value

X m + I.

Thus

any

a lower

bound

for

4 shows

that

the

satisfies

function

6.

s : 0,

For we may However,

now

is an i n t e g e r , which

the cone

largest

y

will

but

for

z e r o of

has

=

k = ~

not

at

solid

y

(-i,i)

in

S

y = 0

If

and

e,

(4.4)

S

f m [~ (sin x ) m _ 2 0

=

is d e t e r B

has

8 = e. S

are

dx~

where

F(lm) (4.5)

fm =

1 (m-2) 2

p.

82] on

u = (sin

--d2u + J'(~+v)2 + v ( 1 - v ) " [ u de2 u

m > 2.

8)Vy,

where

v =

that

(4.6) and

r(89 setting

vanishes (For

The proof

at

l

sin 2 8

8 = 0, e

m = 2, u = 0 of Theorem

at

and

=

8 = ~'

6 is r a t h e r

0

J

is p o s i t i v e du _ 0 d8

in b e t w e e n at

long and will

in-

We must

e

then for

m

be a polyno-

function.

angle

and

equation

and write

t = i.

8 < e

is h a r -

by

[14,

of

0

in g e n e r a l

a Gegenbauer

is a n a l y t i c

8 < e

y > 0

set

r~f(e)

the differential

+ B(B+m-2)y

so t h a t

We find

in

< m.

solution

is t h e

is a d e c r e a s i n g

( l - t 2 ) y '' - (m-l)ty'

~

so t h a t

lower

~(k,m)

automatically

y = f(8)

m + i.

unless

u

constant

of Lemma

(4.3)

where

u

and

s m'

that

function

order

set

The method in

s.h.

same

when

monic

on w e n o t e

for

8 = 0. ) be published

to Thus

43

elsewhere

[3].

I should like to indicate

it depends and then to deduce method is due to Huber

only some ideas on which

some numerical

to Lemma 7 have been proved by Bandle

let

8 are the following

The first and third are due to Huber

LEMMA 6. y

Let

D

[9].

g r a d i e n t of

y

Results related

Rm

be a smooth domain on the unit sphere ~n

D

and i8 p o s i t i v e

three

[2].

be a smooth f u n c t i o n on the closure of

the boundary of

The

[9].

The main tools of the proof of Theorem lemmas.

consequences.

in

D.

Let

D

and

which vanishes on

Vy

denote

the

along the surface of the unit sphere and set

I (Vy) 2 do (4.7)

k(D,y)

~

, f y2 do

where

do

fixed

D

~(D)

is

( m - l ) - d i m e n s i o n a l surface area on

y,

and varying

when the function

subtended by

D

~(D,y)

v = IxlPy(x/Ixl)

at the origin.

(4.8)

attains

l(D)

D.

Then for

its m i n i m u m value

~ =

i8 harmonic in the cone

Here

=

U(p+m-2).

Lemma 6 means that (4.9)

in

Ay + Xy

D,

where

of the Laplace

A

is the Laplace

value of the equation rems on expansions

operator,

~

that part

along the

is thus the lowest eigen-

The result follows

of functions

i.e.,

with differentiating

The quantity (4.9).

0

Beltrami

operator concerned

surface of the sphere.

=

from standard theo-

in series of eigenfunctions

of

elliptic partial differential operators. The next result contains

the main new idea of our proof.

It is

44

based

on a s y m m e t r i z a t i o n

of the

isoperimetric

result

due

LEMMA S

7.

to E.

Among

times

cap

are

by

related

satisfying

suppose

and

final

8. D(r)

D(r)

be

together

be d e f i n e d in terms

of

sphere,

be s.h.,

the

support

xI =

boundary

the

B(r)

=

C

is a

The result problems give

a well-known inequality earlier bound

when

Suppose fine

ug(x)

R TM

a

in

u

on the of

u(x)

whose

8.

when

Here of

is

D

is

S

and

8

only

u = 0,

1 ~ = ~ (m-2) p

area

when

and

we

= ~/2a

= 1/2 S.

Rm

sphere

D(r)

of

>

u > 0

somewhere.

of radius

on the u n i t

D(r)

C exp

and

and

r,

let

sphere.

Let

p = ~(r)

{Irle

as in

(4.8)

}

ro

t

constant.

extends

q.

to

case

which

was

without

of

direction

Dg,

it was

for

is due k u

to

= 0

additional by

Heins

extension

in p r o v i n g

Theorem

Talpur D~,

which

[8]

and r e d u c e s

It is the

m = 3,

tracts (x)

proved

7 is e l e m e n t a r y

obstacle

is small,

in

any e s s e n t i a l

Wirtinger.

now that we have = u(x)

where

ease, L e m m a

our m a i n

in this

the a r e a

Rm

m = 2,

In this

inequality

result

in

use

is

u > 0 of

lemma

conditions

y = u = cos pS,

in terms

max

f r o m the

Theorem

of a s p h e r e

is m i n i m a l

Also

Ixl=r where

essential

is a f u n c t i o n

0 < e < ~.

Then

~.

makes

Ixl cos

with

projection

(4.7)

l(D)

8)~y

result

u

as in

surface

u = (sin

m = 2,

radial

itself

in the p r e v i o u s

where

for

subsidiary

be the

and

If

on the

D

of the unit

u > 0

Let

Let

domains

(4.4)

which

[13].

0 ~ 8 < ~,

m > 2.

The

LEMMA

all

(4.6)

e = o, ~,

inequality

Schmidt

the area

a spherical

argument,

gives

to to

of t h i s 6.

An

a good

[15]. 9 = 1

elsewhere.

to This

k.

We de-

function

is

45

s.h. and so we can apply the result of Lemma 8 to each By(r)

denotes the m a x i m u m of

By(r)

>

i

C exp

k

Ck B (r) v

Now the quantities

> --

~v(t)

tities for spherical

values

exp

is at least

, r0

t

r r/e

]ro

_>

max By(r) l_
>_

~v(t)dt t

are not less than the corresponding

k~,

k n By(r) v:l

~=i

quan-

The sum of the proportionate

caps is at most one and so the sum of their

right hand side of (4.1).

say, where

s

is the quantity on the

Thus

C k exp Irle ks ~dt r0

_ -

ck I~--~--! r IRk v-0!

C(e--~0)s

Since this is true for every s.h. the lower order of

rle ~vCt) dt

caps by Lemma 7.

areas of these spherical

If

uv, we obtain

--

v:l

ug(x).

u(x)

function with

is at least

s

k

tracts,

and this proves our

Theorem 6. We have thus reduced our problem to the estimation values

~

of solutions

can be done by Sturm's lower bound for bound for function

a

~

comparison

theorem.

as a function of

as a function of

in the curly bracket

which the solution

is known.

obtain an inequality S.

of the differential

for

S

~.

~,

equation

of the eigen-

(4.6).

This

We wish to obtain a or equivalently

a lower

It is enough to replace the

in (4.6) by a bigger such function Once this is done, we can use in terms of

~

or for

~

for

(4.6) to

in terms of

46

4.1.

Numerical

estimates.

We now develop

some

results

I/(2S)

is c o n v e x

in

S,

Theorem

4.

if

m = 4,

Again

y = sin(p+l)/sin 8 ,

(4.2)

However

cases

between

is p o s i t i v e

We assume have

m = 3,

in t h e

former

~ = ~ ,

which

1 ~ = ~ , case

when

increases

from

1 ~

to

-

4 ~

1-

to

m ~ 5,

of

~.

We dis-

since

in t h e

p > I,

p : i.

since

latter. we always

Then

1

e2

sin2e

,

and

function (4.6).

and negative

1 r

is

2e).

to our equation

corresponds

y = t = cos e ,

which

=

and that

seem to be an elementary

case

0 < 8 < ~

u = i,

p(S)

while

Sturm's method

the

s

i (~ _ 1 ~ ~- s i n

not

Since

1 ~ ~ k,

find that

i,

=

does

we can apply

tinguish ~(i-~)

~

direction.

yields

we

p - ~

S

In other

in t h i s

as

8

increases

from

0

In f a c t ,

0

d-~

sin2e

=

~

<

0 < ~.

n=l

Thus 1

(4.1o)

L--: e

+

1

~

I

<

If

m =

3,

so t h a t

(p+u)2

say.

The

+ u(l-u) sin 2 8

i

<

sin2e

--

i u = ~ <

,

+

1

-

~

4

--~

we deduce

(p + 21_)2 +

the

<

e

<

~

-that 1 2

C2

_

+

1 4e 2 '

equation

solution

6.3.14]),

o

12 + 1 4e ~-

d2v + (C2 + 4e-~)v de 2 has

,

~z

where

v :

el/2Jo(Ce)

Jk(X)

is t h e

(see, Bessel

--

0

e.g.,

[14,

function

p.

127,

of o r d e r

formula k.

Thus

47

Sturm's zero

comparison

of

J0(x),

> Also

in. t h i s

theorem

since

]0 , -~-

case

_

if

J0

1 : ~ 2 + ~ + ~1 - --~

(4.4)

cosec28

i 4S

that

is t h e

: 2.4048

then

S

Also

shows

C2

.2 30 ~2

>

gives =

! - cos 2

- 8 -2

~

increases

i 4 sin 2 ~

< --

i

:

8,

with

"

we

W

+ i 2

~

. 2 sln 7

deduce

0<~<

~2

that

~

'

-- 2

"

Thus .2 2

2

+ # Thus

since

~ >

>

+

4

Jo

1

+

~2 W

for

~ < ~

,

we

1

1

~2

2

>

4S

2

obtain .2

4S

4

i.e.,

( 4 . 11 ) which

B is

If

Talpur's

> [15]

30

2

m=

'

3

result.

m > 4

S

=

fm

is0 (sin x)m_ 2 dx

--> fm

=

fm ~

[e0 (sin x ) m - 2 c o s x

(sin

e)m-i

r(89 Thus

if

dm

: fm/(m-l)

= 2F( ~i) F {

d ~)m-i (

[i( m + l ) }

,

we have

1 <

1 sin ~

< -

1 + 2 ~ 1 - ~ .

dx

first

~8

Thus K

if

1 = ~ (m-3)

function

so t h a t

1 = ~ - ~

of order

(U+U)2

where

m > 4, ,

and

K.

+ ~(i-~) sin 2 8

9(l-u)

let

JK

!

0,

1 v : ~ (m-2)

where

be t h e

first

zero of Bessel's

We have

<

(~+u) 2 + 9 ( l - v )

+

:

C2 + 82

1

-C2 = --(~+~)2 +

let

9(l-v).

T

Thus

Sturm's

theorem

shows

'

that

J< C

i.e.~

C

JK --

>

,

[C%m) } m-i

>

j

K

2 +~-

--

1

and

C

=

i (m-2)} 2 - -i- ( m - 2 ) ( m - 4 ) ]%

[{~ + ~

i

Thus

(~.12) where = ~

JK

is t h e

(m-3).

first

I + i - ~-m,

zero of Bessel's

It is k n o w n

(see,

e.g.,

3K ~ K ~ I m We can also obtain For this

purpose

we

~ +

some results

function

[1],

as

m >

p.

371)

= 2

which

are

h V~''

~ 2

t

i fmlh (c o s tlm-I dr. 2/~'0

As

m

§ m

we have

of order that

independent

and

-

4 ,

m -~=.

set

e,

89 (m-2).

}

(dmlm-I 2 JK ~-I + F - 1

>

<

12

~I

of

m.

49

(COS t----Im-l~) ~

(i - ~-~t21m-l----+ e - 89 9

Also in view of (4.5)

( lm) 89 i r(~-)

m

=

( 2 ~ ) ;~.

=

I 2/U~7 (

Thus

' i

S

2

I

I h e - 89

~

dt

0

[

~ e_ 89 2 dt. h

Again U

Finally,

=

since

(sin 8)9y

cos t--]"~(m- 2 ) y --~

:

i = 7 (m-2),

e -~
we have (~+v) 2

(~+~)2 + ~(l-v) sin 2 8

cos2(t//m)

+ v(l-v)

s in 2 8 ~(i-~) =

+ (P+~)2(I - t~)

~(i + 2 ~ -

tTl + 0(i),

while d2u de 2 Thus if

m --~

(4.13)

the equation

(4.6) becomes

d2u + {~ + ~ - ! t2}u dt 2 2 4

Our solution of this equation t--~+ ~, and (4.14)

d2u m -dt- 2

:

h

It turns out [14, p. 106,

-

0.

is the one which tends to zero as

is the largest S

=

zero of this solution. i

[

(5.5.2)]

Further

~ e_ 89 2 dt. h that

u = e-

i/%t2

H (t//~),

50

where

H~(t)

gives

is Hermite's

for large

t

Now Sturm's

comparison

theorem

[14, p. 132]

h v-f

m

where

function.

<

(2~+i) 89 -

A0(2p+l)-~,

A 0 : 1.85575. We can also obtain a good convex minorant.

i

e- %h2

2~

h

=

i

I ~ e-%t2(l

(2/[~7

We note that

+ t~ ) dt

h

Thus

e,h We set sume

2~+i = x,

and assume

p > i,

so that

x > 3.

We also as-

>

i.

log ~i

<

~I h 2 + ~i log(2~)

<

~i h 2 + log h + ~i log(8.~)

<

{(21J+i) % - A0(2p+l)-*'4}2

<

x - 2A0x*~

+ ~i log x + ~I log(16~)

+ A ~ x - */3

x - 2A03~

+ ~1 log S + ~1 log(16~)

.2.^~ + AO/J

h

<

Then

2U + 1 -

.45

+ log(h +

<

)

i + ~- log(8~)

i + ~- log(4p+2)

2]~ + . 6 .

T~us

_> The inequality

89

if

is also true if i

h < i, <

h>_l.

since then

8 ~(- ~ e 89,

51

so t h a t 1 log ~

while

(4

p > i.

1 1 [ + ~ log(8~)

<_

2.12

Thus we have

15)

p

>

~. l o g

-

3,

"

0 < S < 1

~

We c a n

linear

also

obtain

1 S = ~ ,

sharp when

and have

the v a l u e

a bound

~ = i.

the right 1 ~

ity

<

of this

~

which

The

simplest

derivative dU dS

'

derivative

is v a l i d

_

be

for all

1 S : ~ .

at

if

-2.

~

such convex

1 2S 2 '

must

2

S

and

is

bound would

Since

m : 2~

Thus

our

desired

inequal-

is

p

(4.16)

This

is e q u i v a l e n t

>

2(I-S),

0 < S < I.

to

e_ 89 2

h

where

h

is t h e

largest

form

Ortiz

and

We now

define

z e r o of o u r

I [7] h a v e

proved

solution

(4.16)

In t h i s

(4.13).

by computer 9

=

2(l-S),

7 5 s _< 1

r

=

max{2(l-S),

[ log [ -

1

1

3}, 9

Further,

of

1

r

0 < S < 1 --

--

2

'

we write 1

Cm(S)

=

max

r

=

max{r

" e "t2S ;

=

2.-'S "

1

r

JK g Jo

+ ~ -

be

~ m

4

<

m

<

~

52

Here 2

j<

is

(m-3)

the

first

zero

of

Bessel's

function

of o r d e r

< :

and F( 89 d

=

2r( ~1 )r{ 1 (m+l)}

m Clearly

the

functions

0 < S < 1 (4.12),

and

(4.15)

$m(S)

2 < m < ~. and

(4.16)

are Also

that

W(S)

Theorem

is d e f i n e d

6, C o r o l l a r y

we h a v e

THEOREM

7.

We h a v e

with

s

If

m = 2

k = 2, sharp

>

we d e d u c e

s

as is s h o w n

by

the a b o v e

In o t h e r

cases

d e r of m a g n i t u d e .

for

~ 89 k

6 and

Cm(S)

this

= 1

= IXll ,

our r e s u l t s

where

7,

(4.11),

,

and

~ r189

We r e c a l l

m > m0,

for

R TM

2 ~ k < ~,

and

which

are not

as above.

Thus

2 ~ m < ~.

is a c t u a l l y this

has

sharp

result

sharp.

If

is a l s o

2 tracts

and

order

1.

but

the

right

or-

give

that

1 JK ~ < ~ ~ m,

Thus

Lemma

S

notation

for

s

u(x)

from

of

to

Cm(~),

we o b t a i n

in

r

as in T h e o r e m

leads

functions

it f o l l o w s

u(S) where

convex

m0

as

k --*~.

is an a b s o l u t e

constant,

Theorem

7

yields 1 s

>

On c o m p a r i n g gives

the

and when The

this

correct

max{

result bound

k < 4 m-l. bound

given

m((k) m - I

with

- 2),

Theorem

~ log k -

5 we

see t h a t

within

a constant

factor

A similar

conclusion

holds

in T h e o r e m

7 seems

.3,

2 - ~}.

this

inequality

both when for

to be v e r y

k > 4m - I

m ~ m 0. near

the

truth

53

when

k

m = 3

is large for a small fixed value of in T h e o r e m 7 we obtain Talpur's result

(4.17)

Z(k,3)

>

(i+o(i)) ~1 5 0 k 8 9

m.

T a k i n g for instance

[15]

as

k --+~.

To obtain an upper bound we divide the unit sphere in regions

Dv

each h a v i n g

choose a point

x

small diameter.

and project

to obtain a plane region and

A

A

r

having area

range to have

Q

r2

Av

such hexagons

H

p

If

N

we xv

is large

will be close to

Dv

4~.

We

by n o n - o v e r l a p p i n g h e x a g o n s of 1 r2/~.

Thus if first

is small compared with

Q

By joining each

Av.

[ Av

These h e x a g o n s have area

chosen large and then

Dv

N

which will be allowed to tend to zero and

cover most of the i n t e r i o r of r.

In each region

into

onto the tangent plane at

will be close t o g e t h e r and

now choose a q u a n t i t y

side

Dv

R3

>

Hp,

(l-c)

I/N

N

is

we can ar-

where 8~

r2/(27)

to the origin we obtain

Q

non-overlapping

a p p r o x i m a t e l y h e x a g o n a l cones In each is p o s i t i v e

C in

P

C . P we can construct a h a r m o n i c function

Cp,

zero on the b o u n d a r y of

Ixl Pyf

Up where

1

P

~p,

Ip

l

D

of

C

P

are related as in (4.8) with

i.e~,

Up(Up+l) Also

1

is defined as in (4.7) for the i n t e r s e c t i o n

with the unit sphere and m = 3,

Cp

u (x) which P and of the form

D

=

Xp.

a p p r o x i m a t e s to a small plane h e x a g o n of side

r

and so

a p p r o x i m a t e s to the lowest eigenvalue of the a n a l o g u e of ( 4 . 9 )

P for a plane hexagon of side

r.

This is

g i v e n by

Ir -2,

where

5~

is an a b s o l u t e c o n s t a n t s a t i s f y i n g %

<

3 J-~ ~nd

p = .89850

leads

for

is

the

sufficiently

~p

inner

radius

small

~ <

<

r

we d e f i n e

side,

then

least

Q

is

tracts.

(4.18)

of

--

p2

a hexagon of

--

<

~

(l+e)

to s.h.

Ra

to

and

u has

P

(x)

in

order

Thus we deduce that for large

s

<

one.

This

[~-Trl

be equal in

side

~ct~sl ~

~r u(x)

u(x)

<

to

~o

c~§

p If

X

(l+~

<

C and zero outP max Up and at

k

i. 219k 89

where A1

Thus

(4.17) and

large.

A l s o we should compare u,

v~

s

<

1.013. to w i t h i n

i~ o

when

k

is

(4.18) w i t h T h e o r e m 5, w h i c h yields a

such that

lul

has

k

tracts and order at

(2k) 89 = 1 . 4 1 4 . . . k %.

#Professor

[12].

(27)

(2~-~)p

(4.18) limit

harmonic function most

-

Hersch has pointed

o u t t o me t h e s e

inequalities,

which are obtained

in

REFERENCES i.

M. Abramowitz and I. A. Stegun, Handbook of Mathematical Functions with Formulas, Graphs and Mathematical Tables, National Bureau of Standards Applied Mathematics Series .55 (seventh printing), May 1968.

2.

C. Bandle, Konstruktion isoperimetrischen Ungleichungen der mathematischen Physik aus solchen der Geometric, Comment. Math. Helv. 46 (1971), 182-213.

3.

D.A. Brannan, W.H.J. Fuchs, W.K. Hayman and U. Kuran, A characterization of harmonic polynomials in the plane, to appear.

4.

A. Dinghas, Das Denjoy-Carlemansche Problem fur harmonische Funktionen in E ~ Det Kal Norske Videnskabers Selskabs Skrifter 1962, no. 7, 13 p.

5.

S. Friedland and W.K. Hayman, Eigenvalue inequalities for the Dirichlet problem on the sphere and the growth of subharmonic functions, to appear.

6.

W.K. Hayman,

7.

W.K. Hayman and E. Ortiz, An upper bound for the largest zero of Eermite's functions with applications to subharmonic functions, to appear.

8.

M. Heins, On a notion of convexity connected with a method of Carleman, J. Analyse Math. 7 (1959), 53-77.

9.

A. Huber, Uber Wachstuniseigenschaften gewissen Klassen yon subharmonischen Funktionen, Comment. Math. Helv. 26 (1952), 81-116.

Meromorphic

Functions,

Oxford,

1964.

i0.

U. Kuran, On the zeros of harmonic functions, Soc. 44 ( 1 9 6 9 ) , 303-309.

ii.

U. Kuran, Generalizations of a theorem on harmonic functions, J. London Math. Soc. 41 (1966)~ 145-152.

12.

G. P61ya and g. SzegS, Isoperimetric Inequalities i__nn Mathematical Physics, Princeton Univ. Press, 1951.

13.

E. Schmidt, Die Brunn-Minkowskische Ungleichung und ihr Sp~egelbild sowie die isoperimetrische Eigenschaft der Kugel in der euklidischen und nichteuklidischen geometric, I. Math. Nachr. I (1948), 81-157.

14

G. Szeg~, Ortho~onal vol. 23, 1959.

15.

M.N.M. Talpur, On the sets where a subharmonic function is large, Thesis, Imperial College, London, 1967.

Polvnomials,

Amer. Math.

J. London Math.

Soc. Coll. Publ.

ISOMORPHISMS BETWEEN FUCHSIAN GROUPS ALBERT MARDEN* INTRODUCTION, Let

G

be a n o n - c y c l i c fucbsian group acting in the unit disk

Then in general b(G)

A

from

p*

Q.

p

and

terminates at

d e t e r m i n e d by in

is b r a n c h e d over a nowhere dense set of points

in the Riemann surface

loop w i t h origin

y.

p

S(G) : A/G. E A

T(p*)

If

lies over for some

y c S(G) - b(G) p~

the lift

T E G;

T

mined by a simple loop in

is an element of

S(G) - b(G)

y*

y G

y

which is deter-

S(G).

t r a n s f o r m a t i o n is either h y p e r b o l i c or p a r a b o l i c

of

are conjugate

which is r e t r a c t a b l e

to an ideal b o u n d a r y component of

is a

is said to be

Two t r a n s f o r m a t i o n s d e t e r m i n e d by

A boundary transformation

S(G) - b(g)

A.

in

Thus a b o u n d a r y

(since

G

is not

finite cyclic) and every p a r a b o l i c t r a n s f o r m a t i o n is the power of a parabolic boundary t r a n s f o r m a t i o n ary t r a n s f o r m a t i o n m = •

A ~ G

(e.g.

satisfies

In a d d i t i o n if a bound-

A = BTM

for some

B E G

then

b(G)

is a

(see [6]). The group

G

finite set and

is finitely g e n e r a t e d if and only if

S(G)

b o r d e r e d surface

(e.g.

A homeomorphism

is the interior of a finitely p u n c t u r e d compact [5]). f*: A--+g

#(A) = f*Af *-I

is said to induce an i s o m o r p h i s m

9: G--+H

if

then

f*

projects to a h o m e o m o r p h i s m

b(G)

to

b(H).

for all

A E g.

If this is the case

f: S(G)--+S(H)

which sends

Alternatively a homeomorphism

f: S ( G ) - b ( G ) - - + S ( H ) - b(H) sion

[7]).

f: S(G)--+S(H)

is said to induce

and a lift

f*: A--+A

r

if there is an exten-

which induces # .

Other

choices of lifts will induce isomorphisms of the form A ~

T2~(TIAT[I)T~ I __

for

and

T 1 E G,

T 2 E H.

O b v i o u s l y such h o m e o m o r p h i s m s

*Supported in part by the National Science Foundation.

f

f*

57

preserve

boundary

to present

a proof

is a boundary

G

and

A

i8 too.

ently

less

stringent

#(A)

is a b o u n d a r y

famous

influence

which induces

on the theory

is by means

equivalent

transformation,

statement

A

[2] which,

although

of fuehsian

of their

intersecting

condition

implies

~

important groups,

in its own right

not just

Here way.

that

the

collection

tesselation

selations.

This of

of

allows

on these

axes

the more

interesting,

the h y p o t h e s i s

G.

of the

using

extension in which

of

the m a n n e r ~

A;

theorem of

us to e x t e n d

~

This of the It is

fuchsian

G

gives

a geometrically

groups

step

in the f o l l o w i n g

give A

axes

to the fixed points

of

in the proof,

theorem.

G

equivalent

points

on

~A, ~

first

and perhaps

of the t h e o r e m

This

to

tes-

a map defined

from the h y p o t h e s i s

~A.

here.

of the b e h a v i o r

is applied

between the lattice

on

to the

(which we prove

to a r b i t r a r y

of their a c c u m u l a t i o n

they are located

in

has had great

by a h o m e o m o r p h i s m .

to all of

the first

intersecting

that when

and is c o n t a i n e d

theorem

in terms

isomorphic

is to get

with the appar-

ones.

of axes

However

to set up a bijeetion then,

axes

- b(G).

Their a p p r o a c h

it applies

generated

E H

of one.

is followed

is induced

since

intersecting

A suitable

meaningful

finitely

S(G)

hypothesis

axes

a simple

~

~(A)

Furthermore,

unpublished,

groups.

latter

under

~.

in

is due to F e n e h e l - N i e l s e n

4.1) and the same path

axes

f

is a power

here as P r o p o s i t i o n gives

that when

can also be p r e s e n t e d

but a c t u a l l y

manuscript

result

is

Then there exists a homeo-

the homotopy crass of

that the t h e o r e m

The first

of this p a p e r

are finitely generated fuchsian groups

f: S(G) - b ( G ) - ~ S(H) - b(H)

We r e m a r k

theorem

H

transformation,

uniquely determines

their

purpose

i8 an isomorphism with the property

~: G - - + H

morphism

The main

of the converse.

Suppose

THEOREM.

and

transformations.

of

to

is done by using G

and

H

and

to show the preserves

the order

58

The second statement of the t h e o r e m follows from the fact, proved by d i f f e r e n t methods S(G) - b(G)

in [i] and [6], that a h o m e o m o r p h i s m of

onto itself which induces the identity a u t o m o r p h i s m of

is h o m o t o p i c in

S(G) - b(G)

G

to the identity.

The theorem originates with the c l a s s i c a l result of N i e l s e n that every a u t o m o r p h i s m of the f u n d a m e n t a l group of a closed surface is induced face.

(up to an inner automorphism)

by a h o m e o m o r p h i s m of the sum-

Nielsen's proof was given in terms of fuchsian groups but with

the m a c h i n e r y now a v a i l a b l e

in topology,

this and more general results

one can give short proofs of

(of. [ii, Lemma 1.4.3]).

Beyond this

c l a s s i c a l case the first results that a p p e a r e d in the literature were those of Zieschang

[12,13]

(see also the m o n o g r a p h [14]).

He gave a

purely a l g e b r a i c development of the subject including proofs for the case that

S(G)

is compact and fop the case

b(G)

= ~;

he has

r e p o r t e d that these methods also yield a proof of the first s t a t e m e n t of the t h e o r e m in general

(personal communication).

p r o p e r t i e s of T e i c h m ~ l l e r mappings,

Macbeath

Using extremal

[4] showed how Nielsen's

result can be used to give a short elegant proof when pact well.

(but

b(G)

# ~) and the method has m o r e

S(G)

is com-

general a p p l i e a b i l i t y as

Tukia [i0] r e d i s c o v e r e d the significance of the i n t e r s e c t i n g

axes p r o p e r t y and gave a proof of this result that covers most cases, pointing out how it implies the t h e o r e m when

S(G)

is compact.

of the work cited also contains a d i s c u s s i o n of the case that contains reflections

Much G

(in p a r t i e u l a r the F e n c h e l - N i e l s e n proof includes

this ease), a situation we ignore c o m p l e t e l y here. This is a r e v i s i o n of my proof d i s t r i b u t e d

in preprint

form in

1970 and I want to thank L. G r e e n b e r g for his suggestions and interest in this work spanning eight years.

Above all it should not be for-

gotten that the proof p r e s e n t e d here is in the spirit of FenchelNielsen and merely reflects the importance of that work.

59

1.

GEODESICS AND LOOPS 1.1.

Let

y

be a simple

and fix

p~ 6 A

over

element

Ty

G

y

is retractable

point of y~

of

=

or by

S(G) - b(G)

[5,6].

y~y0~ )

If

of

~a

Ty.

Suppose

y

(resp.

T = Tu

determined

The non-euclidean

is called the axis of

line

T.

the natural projection following elementary

~r

p

elliptic) of

y

induced

A--+S(G).

directed H ~(T)

If

If

Y1 n Y2 = r

then

(ii)

If

Y1

Y2

crosses

(resp. a

determined from

y

by

Y0

which fixed point

loop

y,

is hyper-

from

~r

toward where

~

~a is

are simple loops the

facts are clear from n o n - e u c l i d e a n

(i)

if and only if

S(G)

= w(~e(T))

YI' Y2

an

the open Jordan arc

by the simple

e(y)

p

determines

toward the attractive

~e(T)

Set

with origin

from

to a puncture

It has a natural orientation

it from the repulsive

bolic.

y~

T

directs of

S(G) - b(G)

is hyperbolic, Y is called the lift-chain of

T n .

Ty.

The lift

which is parabolic in

b(G))

~ Un=_~

p.

loop in

geometry.

e(yl ) n e(y2 ) = r exactly once so does

~(yl )

and

~(Y2 ). From [6] we see that the following holds: loop,

e(y)

is with one exception a simple

which is freely homotopic

in

in the singular

~ r

(if

b(G)

the Poincar~ metric on

A.

bounds a simply connected points

Xl,X 2 E b(G)

is an arc from

xI

More generally

loop in

S(G) - b(G)

to

metric on

S(G)

y.

in

S(G)

is a simple S(G) - b(G)

e(u

is a geodesic

induced by

The single exception region

y

to let

a hyperbolic

by another

loop

~

occurs when

containing

from y

exactly two

and these are both of order two.

determines

6

If

Then

~(y)

x 2. y c S(G) - b(G) transformation

whose lift

60

be any loop from Ty.

Then

also determines

y

p

which

can be replaced Ty

but in

60

addition

8"0

and the r e s u l t i n g

is a simple

loop

a po w e r

T

of

for all

2.

T ( G loop

S ( G

case cited

N 6" = r

for all

6

S ( G

not

is h y p e r b o l i c y c S(G)

which

with axis

- b(G)

~*(T).

if and only

is not a power

of

T,

if

with

Then

T = T

S~*(T)

Y

N ~*(T)

the e x c e p t i o n

= r

of the

THE LATTICE CONSTRUCTION

b oun d a r y assume

Primarily

in this

restriction.

chapter

Once this

remove

surface G'.

S(G) There of

transformation

G

of

S(H')

then revert

- U D

is a natural S(G')

transformation a po w e r of

T.

H)

if

reasoning,

S(G). if

T

is a

for

H @

G'

~i:

is induced

by

is no x

of

S(G)

- b(G)

of

x.

set

T

the

b(G)

G'---+G

~2:

is induced H

this

by a

that a r i s ~

is a b o u n d a r y

is one for

and by

to

parabolic

we ean r e p r e s e n t

Clearly

~l(T)

~2r162 I_ by

if

component

D c S(G)

S(G)

G'.

H'--~H

Denote the corre-

by a h o m e o m o r p h i s m

by one sending

H',

r

by

~2@r

I

S(G') and

notation. boundary

transformations

by the c o n d i t i o n

if and only

contains

isomorphism

We see that

Now we can c h a r a c t e r i z e of

of

into

group

G

that

but w i t h the same branch

to the o r i g i n a l

correspondingly

H

a small n e i g h b o r h o o d

if and only So replace

nor

an ideal b o u n d a r y

if and only

isomorphism.

S(G)--+S(H)

G

of the f o l l o w i n g

the c o r r e s p o n d i n g

sponding

the proof

is one as well we are going

for all p u n e t u r e s

from the e m b e d d i n g

H'

r

that neither

Because

is done

new group

then

If for example

is a puncture,

smaller

in o r d e r to simplify

transformation

t ran s f o r m a t i o n s .

to

S(6")

arcs.

above.

2.1.

by

if

are Jordan

Y

Suppose for a simple

if and only

6"

lift-chain

if

e*(S)

that

N ~*(T)

T ( G = r

of

G

(and

is a b o u n d a r y

for all

S ( G

not

2.2.

Fix a point

0 ~ A

H = r

as base point.

based at

0

which is not a fixed point of

A t o p o l o g i c a l fundamental re$ion

is contructed as follows.

(g = genus

S(G))

simple loops

m u t u a l l y disjoint except at

In

S(G) - b(G)

(ai,Bi),

0,

so that

I ~ i ! g,

In

S'

draw simple arcs

{u }

from

~(0)

(using the natural c o m p a c t i f i c a t i o n )

point in

b(G)

{ei,Si,Ti }

region

Let

w h i c h contains

G

cuts

S(G) - b(G)

be the closure in

0

te~m max of

G

vertices of

G

for e i t h e r

of

G

G

G

of

S(G) - b(G).

S(G)

and to each ~(0).

The

of one of the lifts of

G

RI

are called vertices of

~A.

G.

It is c o n v e n i e n t to use the

of

G

which joins two vertices,

G

where

o I U ~2

G

joins two

or via an arc of

The totality of images of rays of

will be r e f e r r e d to as rays of

will be one

of

via an elliptic fixed point of

have a vertex in common but

segment

under

Ol,a 2

which is included in the ray. under

~(0),

into a simply c o n n e c t e d

A

a) a side of

b) the union of two sides

~A

2g

either joins two vertices or joins a vertex to an ellip-

tic fixed point or to a point on

or

G

0.

The t o t a l i t y of images of G

for

is the

which are m u t u a l l y disjoint except at

totality of

A side of

from

or

to each ideal b o u n d a r y

component

R I.

S'

G

draw

N ~iSia~I6~ I

o r i e n t e d relative boundary of a planar subregion

G

G.

71 U y2

If two rays

71, 72

is not a Jordan arc,

it

(or it will be a point) after c a n c e l l a t i o n of a common

(or segments).

In w r i t i n g a union of rays it will always be

a s s u m e d all possible c a n c e l l a t i o n s are carried out. If have

G

0

contains

points

S.(0) l

P = $(0) P

vertices then there are

as an end point;

i = l,...,n,

to

n

(P

S i 6 G.

The t r a n s f o r m a t i o n s

is a vertex, the points

from a vertex

P

Q

rays of

denote the other end points by

are said to be adjacent to

adjacent to

n

implies

to a vertex

Q

Si 0.

SSiS-I(P) Q

generate

G

which

Si(0) , G.

The

More generally if are said to be a d j a c e n t

adjacent to

P).

A path in

is a finite set of vertices

G

62

P = P0,PI,...,Pk = Q of the path is

k.

such that

Pi

is adjacent to

is a path from

Property i.

Given

~

in

Ns(~)

S(P)

~ ~ ~A,

Nr(~) If

A n (Iz-~l

:

which meets in

ment

s

c

A. in

Ns(~)

A r}.

Nr(~).

Then if

P, Q

are vertices in

Ns(~) n A

from

P

to

Q;

{G(0)}

~

P

Ns(~)

S(G)

determines

G

and

S1 ~ G

a

i- 1

H

as the image under

G.

Let

H

of the set

of

H:

e

of adjacent

as the image under

~

of paths in

be a t o p o l o g i c a l fundamental region for

H

constructed

with r e s p e c t to

P r o p e r t y 2. exists

~

then

We define adjacent vertices of

0 < s < r

H

can be

S 6 G.

~(SI)(~(P)).

and paths in

Q

o.

{H(0)}

=

G

to

map

~(SI(P))

vertices of

P

which meet

onto the set of vertices

is a vertex of

S(G),

draw a line seg-

a path from

~: S ( 0 ) - - ~ ( S ) ( 0 ) , If

such that any two vertices

does not meet the r e l a t i v e b o u n d a r y of

The i s o m o r p h i s m

of vertices

then

so small that any fundamental region

c o n s t r u c t e d from the sides of those

2.3.

S 6 G,

S(Q).

s < r

can be joined by a path in

Indeed, choose

Nr(~)

to

If

set

is a limit point there exists

S 6 G,

The length

It is clear by r e f e r e n c e to the network of rays

that there is a path between any two vertices. S ( P 0 ) , . . . , S ( P k)

Pi-l"

0.

Let

~

be a limit point of

H.

Given

such that any pair of vertices of

can be joined by a path in

Nr(~).

H

r > 0 in

there

Ns(~)

63

For choose a path in H

H

which are also vertices

from

-l(p)

paths. in

to

-I(Q))

Then choose

N

between

of

H

P, Q

of vertices

(namely the image of a path in

and let

rI < r

each pair

M

be the m a x i m u m

of G

length of these

so small that for any vertex

P

of

H

(~)

all paths of length < M from P are contained in N (~). -r Finally pick s < r I so that for each pair P, Q of vertices of H r 1

in

Ns(~) ,

there is a sequence

Nr1(~)

such that

region

T.(H),I

and hence

P

2.4.

and

T.l ( H. and

Q,

Pi-i

Pi-i

~

of vertices

and

P.•

for each

S ( G

corresponding

i,

Nr(~).

to the fixed points of

that the fixed points of

in

of the fundamental

can be joined by a path in

~}(S), attractive

repulsive.

are vertices

Consequently

Extend the function

by specifying those of

Pi

P0 = P'PI .... 'Pk = Q

G

on

~A

are to correspond

to attractive,

to

repulsive

Obviously this could be done for the elliptic

to

fixed points

too but we don't need to do so. A loop

y,

S(G) - b(G)

not necessarily

can be replaced

of generators

of

~I(S(G)

a simple loop,

by an expression

- b(G))

and

of (s

then

U~

cellation

G

such that

({) Pi

if

Y0 = U y[

where

uk(yZ)

closed Jordan arc Let which

AL

by

i(T)

bolic set

y

(resp.

lie to the left

one component

because

(resp.

is adjacent Yi

fixed points of

of

to

which

lie in

i, P.

after can-

adjoined

of internal

loops is a

of

A - y*

(there may be more t[han y*

the c o r r e s p o n d i n g

F(T) = [(T) n R(T),

Pi-i

=

A.

y

interior points of

G

T

for all

be the union of those components

(resp. right)

R(T))

of

T = Ty.

P0,PI,...,Pn

Pi-i

is the ray from

in the closed disk ~R )

to

and elimination

in

in terms of a given set

of vertices

with the two fixed points

of common segments

~(0)

which also determines

Making use of this one finds a sequence T(P 0)

from

may lie on

~A).

set of vertices

~L U y*

the set of vertices

(resp.

Denote

and hyper-

[R U y*)

and hyperbolic

fixed

and

64

points lying on

y*.

Then

i = L(T)

(and

R(T))

have the following

properties.

(s

Two vertices in

(ii)

i

A path joining

(iii)

k

can be joined by a path in

P ~ L

to

contains the fixed points

point) of a n o n - e l l i p t i c for all large vertex take

(iu)

Q ~ i

If

P E k. P

i

Ikl

contains a vertex on

(resp. the attractive

S ( G

if and only if

(resp. all large

(If

S

F.

fixed

sk(p)

E k

k) and any given

is not a power of

as any vertex of

t.

T = T

Y

we can

G).

contains one fixed point of a boundary transformation,

it contains the other as well.

(u)

T

is a power of a b o u n d a r y t r a n s f o r m a t i o n if and only if

one of AR

L, R

is equal to

F.

contain limit points of

In other cases both

2.5.

T.(G), 1

The set of points in

T. E G, 1

F' = ~(r)

a c c u m u l a t e s only to the fixed points of the closure of the orbit under Set tation,

i' = ~(i), i',

Property L'

such that

s > 0

#(T)

R' = ~(R).

Ns(~)

3.

Suppose

are also in

To prove this choose

F.

is invariant under and

~(T)

~ H.

Indeed,

F'

is

~(Y0 N F).

H

as

L, R

do in

G.

is a sequence of vertices of where

~ ~ F'.

H

in

Then there exists

such that all vertices and h y p e r b o l i c fixed points of

are in

as a

We have to show that, modulo orien-

play the same role in

lim Pk = ~ ~ 3A

AL

with boundary

of the finite set

R'

{Pk }

and

G.

These p r o p e r t i e s are easily deduced by t h i n k i n g of union of f u n d a m e n t a l regions

AL

H

which

L'. r

so small that

Nr(~)

does not contain

65

a vertex on

F'.

vertices in

Ns(~)

k,

Pk E Ns(~).

Q ~ L' of

By Property

Pk

(w

If now

to

Q

Q

x E N (~) s

E L'

Q

such that any two

Nr(~).

For all large

is a vertex in

by a path in

Nr(~).

Ns(~)

but

But by property

this path contains a vertex on

for any vertex sk(Q)

s < r

can be joined by a path in

Suppose first that

Join

L

2 there exists

F',

a contradiction.

is say the attractive fixed point of

S E H

of

k.

and hence

H,

sk(Q)

E Ns(~)

~-I(s)k(~-I(Q))

implies the a t t r a c t i v e fixed point of

for all large

E L

for all large

~-I(s)

(~i)

is in

L

then

Therefore k.

This

and hence

x E L' Property

3 implies the following.

Property

3'

for all large

2.6. (a)

Suppose for some

k.

Let

Then the attractive

E'

lim Pk = ~

W E H

{Pk }

and

~

~ ~ 3A

B ~ H

lies in that closed interval on

points of

B

W

wk(p)

in

L'~

lies in

or

~A

bounded by the fixed H.

It is here,

for (b), that we are m a k i n g essential use of the fact that if

but both fixed points of

~-l(B)

and hence both fixed points of Property 4. then

E'

Xl, x 2

If

Ty

of

B

L

are in

B

is

For then not just one

(property

(~v) of

L)

L'.

is not a power of a b o u n d a r y t r a n s f o r m a t i o n SA

bounded by the fixed points

~(Ty). E'

is a closed set on

closure there is a sequence ~

is one.

are in

is a closed interval on

Indeed

then

~-I(B)

L'.

with fixed points in

yet c o n t a i n i n g no other limit points of

a b o u n d a r y t r a n s f o r m a t i o n then

E L'

which satisfy either

of vertices

(b) there is a b o u n d a r y t r a n s f o r m a t i o n L'

P E L' ,

fixed point of

be the set of points

for a sequence

and

aA.

{~n } E E'

must be a limit point of

H

For if with

and the

~

lies in its

lim ~n = ~" ~n

If

~ ~ ~'

can be taken to be

66

limit points as well. in

i'

with

Property that

Thus we can find a sequence of vertices

lim Pk = C'

a contradiction.

3 and the r e q u i r e m e n t

Z' - {Xl,X 2}

closed intervals bounded by we use the h y p o t h e s i s that formation. Z'

For then

R'

On the other hand

(b) in the d e f i n i t i o n of

is an open set. xI

So either

and

x2,

{Pk }

or

Z'

Z'

imply

is one of the

Z' = ~A.

At this point

T

is not the power of a b o u n d a r y transY contains limit points of H ~ {Xl,X 2} so

is an interval. As a c o n s e q u e n c e of Property 4 we can i d e n t i f y

the c o m p l e m e n t a r y closed intervals of points of that

i

r and

The c o r r e s p o n d i n g R

~A

that the fixed points

T.

lies in

i'

Suppose

3A

is

that lie to

directed toward

corresponds to of

G

i

S E G

in the sense lies in

i

(resp. a t t r a c t i v e fixed point) of

T : T

Then the axis of

only if the axis of

2.7.

with

In p a r t i c u l a r we have:

Property 4'. transformation.

i'

~*(T)

(attractive fixed points)

if and only if the fixed points

i'

situation with respect to

can be i d e n t i f i e d with intervals on

the a t t r a c t i v e fixed point of

and

d e t e r m i n e d by the two fixed

the left and right r e s p e c t i v e l y of the axis

~(S)

R'

Property

r

Y

is not a power of a b o u n d a r y S E G

crosses that of

crosses that of

5.

If

A ~ G

T

if and

r

is a b o u n d a r y t r a n s f o r m a t i o n so is

~(A).

For suppose

~(A)

is not

(and hence not a power of one either).

Then if the axis

~*(V),

V E H,

Property 4' that

#-I(v)

must be a power of a boundary transformaticn.

Before finding such a mations. H0

V

For o t h e r w i s e

of finite index,

crosses

observe that H

H

e*(#(A))

it follows from

contains b o u n d a r y t r a n s f o r -

w o u l d have a torsion free, normal subgroup

isomorphic to the f u n d a m e n t a l group of a closed

67

surface.

The c o r r e s p o n d i n g group

G O -- ~-I(H 0)

would give rise to

a compact c o v e r i n g surface, c o n t r a d i c t i n g the fact that no finite sheeted c o v e r i n g of

S(G)

is compact.

boundary t r a n s f o r m a t i o n s V1

lie to the left of

VI, V 2

of

~*(#(A))

V + = VIV n n2

about a fixed point of a fixed point of so that and

e*(V +)

~-I(v-)

U i = ~-I(v i) let

GO

V2

V I. and

lie to the right.

large

n,

the ele-

map the exterior of a small circle

That is, we can find e*(V-)

cross

(arbitrarily large)

e*(#(A)).

Hence

n n n = UIU 2

#-I(v+)

n -n = UIU 2 are powers of boundary t r a n s f o r m a t i o n s and the are already known to be.

A/G 0

is a

a b o u n d a r y element in may suppose that elements in

V2

onto the interior of a small circle about

be the group generated by

the surface

so that the fixed points of

for sufficiently

n -n V - = VIV 2

and

H

while those of

Using an argument of Tukia [i0], ments

C o n s e q u e n t l y we can choose

GO

UI, U 2.

3-holed sphere. GO .

UIU 2 are

But this is impossible:

Replacing

GO

by

is a b o u n d a r y element. UI, U2, UIU 2

ring to the a b e l i a n i z e d group,

is a free group, and

Either

U2

UIU 2

U21 ,

or

UIU21

is

if necessary, we

Then the b o u n d a r y

and their conjugates.

it is clear that

For

n -n UIU 2

By refer-

is not a

b o u n d a r y element or a power of one.

Property

2.8. 3&

5'

Properties

We will call

d e t e r m i n e d by

~

L'(T)

T = TY

o r i e n t a t i o n p r e s e r v i n g if the interval on lies to the left of the axis of

d i r e c t e d toward its attractive o r i e n t a t i o n reversing.

4 and 4' hold for any

fixed point,

otherwise

r

Note that the q u e s t i o n of w h e t h e r

#(T ) Y is called r

is

o r i e n t a t i o n p r e s e r v i n g is c o m p l e t e l y d e t e r m i n e d once one knows the location of the a t t r a c t i v e fixed point of whose axis crosses that of

Property choice of

y

6.

If

and

T

~

T.

r

for some

S 6 G

Our d e f i n i t i o n makes sense because of:

is o r i e n t a t i o n p r e s e r v i n g with respect to one it is o r i e n t a t i o n p r e s e r v i n g with respect to

6~

every o t h e r choice.

To prove fixed point. The axes

this

orient

Suppose and

T = Ty

of

T

fixed point

of

U

of

T

lies

in

~(U).

to

T

then the a t t r a c t i v e and h e n c e

right

a*(~(U)).

of Case

not powers axis

2.

If

~

That

fixed point of

is

# T

and

U

T

Take

V

joining

and

U.

with

attractive

about

v2

large

onto

fixed points Case loop

6

2.9.

LEM~

T = T

orient

its a t t r a c t i v e

2.9.

Suppose

lemma

fixed

G

e*(#(T

1 to

simple G

such

of both

near

~i"

of a small

about

are

whose

Case

of

vi

U

v I.

Thus

T For

circle the

~2"

transformation. loop.

Take a simple

The d i r e c t i o n

of

m*(#(T))

)).

summarizes

our results. of

G

Of any h y p e r b o l i c

and

We no longer H

are hyper-

transformation

point.

is finitely

@(B)

circle

41'

y6

and

V ( G

the axis

the e x t e r i o r

transformations

the axis

such that if

Then in fact

to

T

the

~2

fixed point

of a small

is a figure-8

all b o u n d a r y

lies to the

yet

~I'

them crosses

is a b o u n d a r y 7 y6

#(T)

we again apply

maps

be close

from that of

As above

isomorphism ~(B).

will

The f o l l o w i n g

need to assume

toward

V

such that

can be d e d u c e d

bolic.

the interior

of

3.

n -n V = VIV 2

n,

to the left of

then we can apply

limit points

line

sufficiently

fixed point

with respect

If we can find U

the n o n - e u c l i d e a n Vi E G

lies

are disjoint

and

To find

[10,1.4].

i.

is orientation preserving withrespect to U.

transformations.

result.

preserving

%(U)

Case

the a t t r a c t i v e

the a t t r a c t i v e

fixed point of

that

Find

for example

and hence

the a t t r a c t i v e

of the a t t r a c t i v e

are two choices.

is o r i e n t a t i o n

the axis of both

of T u k i a

U = T6

Assume

[(T)

The axes of

the desired

argument

in

in the d i r e c t i o n

and

cross.

of b o u n d a r y

crosses

obtain

U lies

a*(#(T))

all axes

B ~ G

generated and

i8 a boundary

is a boundary

r

G--~H

transformation

transformation

is an so is

if and only if

6g

B

is.

of

Moreover

~(S),

the axes of do.

~(T)

For all such

S, T

fixed point of

S

lowing occurs.

Either the attractive

U 6 G

3.

T

fixed point of

One of the fol~(S)

or it lies to the right.

t(T)

and in the latter case,

~(U)

lies to

And if all

lie to the left of the axis of

former case all fixed points of ~(T)

such that the attractive

lies to the left of the axis of

the left of the axis of fixed points of

cross if and only if the axes

S,T E G

T

then in the

lie to the left of the axis of

lie to the right.

CONSTRUCTION OF THE HOMEOMORPHISM 3.1.

In this chapter we will use Lemma 2.9 to piece t o g e t h e r

a homeomorphism

(orientation r e v e r s i n g if

S(G) - b(G)---*S(H) - b(H) will write

e*(T)

fixed point of

that induces

for the axis of

T

and

~(T)

versely).

~.

is) of If

T

is hyperbolic we

oriented toward the attractive

for the natural p r o j e c t i o n

with o r i e n t a t i o n d e t e r m i n e d by transformation,

T

~

e*(T).

or a power of one,

~(T)

H o w e v e r if

T

~(~*(T))

is a b o u n d a r y

may be parabolic

(or con-

It causes no p r o b l e m and simplifies n o t a t i o n if we do not

d i s t i n g u i s h this case

(after all,

a(~(T)) may then be r e g a r d e d as the

limiting case of a simple loop). If

T

is d e t e r m i n e d by a simple loop on

is a simple loop on limiting case that

S(G) - b(G) ~(T)

S(G) - b(G)

is a simple arc between two points of

(see w

~(T)

except for what can be r e g a r d e d as a

of order two; we will not d i s t i n g u i s h this case either. properties of geodesics

then

b(G)

From the

and from Lemma 2.9 we obtain the

following result.

Property if and if

~(e)

7.

e(T)

~ ~(}(T))

~2 = ~(T2)

is a simple loop in

is a simple loop in

are simple loops then

%(el ) N %(a 2) = ~,

and

(b) ~I' ~2

S(G) - b(G)

S(H) - b(H). (a) a I n e2 ~ %

If

if and only ~i = a(Tl) if and only

cross at exactly one point if

70

and only if serving,

~(el ), #(~2 )

e2

so crosses

crosses %(el )

do; furthermore if

el

#

is o r i e n t a t i o n pre-

from left to right if and only if

(the reverse is true if

#

%(~2 )

is o r i e n t a t i o n

reversing).

COROLLARY.

nents of

S(H)

card b(H)

~ card b(G).

3.2. b(G)

number of ideal boundary compo-

genus S(H) = genus S(G),

: corresponding number for

Case i.

The t h e o r e m is true if

S(G)

is closed and

= r Choose a set of

S(G)

such that

otherwise desics.

ai

2g

~i n ~. : ~, 3 The result

for

A:

under

in G.

~*(Tj )

RI

R* I 8g - 4

8g- 4

contains a side of

sary we may assume

R* I

of

The

cut

S(H)

~. 1

S(G)

RI

i ! i < 2g,

along the

ei

T. ( G 3 By r e p l a c i n g

lies to the left of

is simply

such that the axis Tj

by

T?] I

if neces-

~*(T.). ]

7 that the geodesics

into a simply c o n n e c t e d region

Doing that we see there is a lift

R 2.

R2

A

B i = ~(~i ) However

of

R2

bounded by

Bj = e*(r

t o g e t h e r to give an o r i e n t a t i o n of

With respect to this

o r i e n t a t i o n preserving)

DR ,

R2

SR~.

which fit

lies either to the left

f*: R~--+R *2

is consistent w i t h the i d e n t i f i c a t i o n s of paired sides under f*

(%

or to the right.

C o n s e q u e n t l y we can construct a h o m e o m o r p h i s m

~.

it

and a p p l y i n g

one segment of each of the oriented axes

p a r t i c u l a r o r i e n t a t i o n of

but

sides a r r a n g e d in pairs equivalent

is easiest to follow the orientations by looking in Lemma 2.9.

on

may be taken as geo-

elements * R I.

{el )

serves as a fundamental region

It is clear by a p p l y i n g Property S(H)

simple loops

exactly once,

of cutting

has

There are

ai+l

Ii - Jl > 2. --

RI

A fixed lift

G

(g = genus S(G))

crosses

connected.

in

and

S(G),

can be extended by the action of

G, 9,

and

H

G

to map

which and A---+A

71

and is o r i e n t a t i o n r e v e r s i n g if and only if f: S(G)--~S(H)

3.3.

~

is.

The p r o j e c t i o n

is a homeomorphism.

Case 2.

The t h e o r e m is true if

S(G)

b(G)

is a triply

c o n n e c t e d plane region. A s s u m e first that all ideal b o u n d a r y components of are puneh/z~s.

Then

S(G) - b(G)

and the isomorphisml r three punctures.

and

S(H) - b(H)

We refer to Maskit

are tmiply punctured spheres

r

[8, Lemma 7] for an e l e m e n t a r y

is induced by a conformal or a n t i - c o n f o r m a l

map between these p u n c t u r e d spheres.

mation

In fact

r

is of the form

for some p o s s i b l y o r i e n t a t i o n r e v e r s i n g MSbius transfor-

A.

Suppose now that some of the ideal b o u n d a r y components of and

S(H)

determines a one-to-one c o r r e s p o n d e n c e between the

g e o m e t r i c proof that

T - - - + A T A -I

S ( G ) and

S(H)

are not punctures.

Sew on to these once punctured disks.

After doing this one obtains new groups Ii: G - ~ G ' ,

I2: H--~H '

homeomorphisms

G', H'

w h i c h are induced by

f~: A--~A

S(G)

and isomorphisms

(orientation preserving)

o b t a i n e d by lifting a h o m e o m o r p h i s m

1

fl: S ( G ) - b ( G ) - - ~ S ( G ') - b(G')

and similarly for

thermore the isomorphisms

preserve b o u n d a r y t r a n s f o r m a t i o n s which

I.

H,

i = 1,2.

Fur-

1

in the case of

G' , H'

find a h o m e o m o r p h i s m 12~I?l:i G ' - - H '

3.4.

are now parabolic. h*: A - - ~

The map

Case 3.

w h i c h induces the i s o m o r p h i s m

f~-lh*f~:z~ ~ - - A

S(G) - b(G)

on

1 < i < 2g,

S(H)

on

S(G)

is what we are looking for.

has one ideal b o u n d a r y component.

Construct the system of geodesics g = genus S(G),

Hence from above we can

el'

1 J i ~ 2g,

and the c o r r e s p o n d i n g

as was done in Case i.

system

Recall from w

none of these geodesics pass through the points of Then the results

RI, R 2

of cutting

are doubly c o n n e c t e d plane regions.

S(G),

S(H)

Fix a lift

8i'

b(G)

or

that b(H).

along these curves R* 1

of

R1

and find

72

as in Case i the corresponding RI

determines

an orientation

lift

R *2

of

R 2.

of the relative

boundary

so that

composed of segments

of a finite or infinite number of axes

that of

~0R[.

segment of

T

so that the orientation

Then in turn

~0R~

~

on whether

Next

R[

also maps

~

to

set

~i c ~0R~

finite union of sides of end

draw a simple arc

from

A(p*)

Let

y~

be the lift of

bound a wedge

aI

in

R *1

the action of

GO

in

R* I"

y

which

r r

0)

consisting

maps the negative

to the positive

end.

as a fundamental construct

q*

a homeomorphism

to a h o m e o m o r p h i s m

of

under A--~A

r

0)

f*: ~ G

which

G

~2

GO

consisting

end point

p*

A

of

of a

of

~i

is

GO.

In

RI

b(G),

p*.

The arcs

Yl*

or to

~2 c ~0R~

sides of

and

A(y~)

region for

for the action of

R *2.

Automatically

(in the orientation wedge

in

R *2.

Consequently

~2

that on the boundary under

of

02

H.

30R2 )

serving

we can sends f*

exten~

is what is needed.

the proof of the theorem.

on

which

in its natural compactifica-

set

does not fall into one of the previous yl,...,yn

of

there is a connected

to points equivalent

We can now complete

joint simple loops

GO

And there is a corresponding

region for

points equivalent

3.5.

end

of each

or not.

serves as a fundamental

of the corresponding

e*(T)

agrees with

to the point of

from

Likewise there is a fundamental

is

from segment to

by a generator

S(G)

RI

lies to its left or right

subgroup

p = ~(p*)

of

of

30R *I

e*(T)

preserving

The negative

a point on the ideal boundary of tion.

R~

In particular

R[.

Now

an orientation

for the action of

mapped to the positive YI

~0R2 ;

by a cyclic

onto itself.

of

2.9 is consistent

is orientation

is preserved

~oRI

fundamental

determines

which by Lemma

segment giving an orientation depending

lies to its left.

~0RI

(which is connected)

and we may choose

RI *

The orientation

categories.

S(G) - b(G)

Assume

Draw mutually

such that

S(G) dis-

73 (i)

YI

is the relative boundary of a subregion

R I r S ( G ) - b(G)

which is compact of genus equal to the genus of this is positive,

(ii)

Yi

and

Yi+l'

components

otherwise

is a triply connected

i j i j n - i,

in

S(G) - b(G)

Yi

separates

(iv)

Yn

bounds a triply connected

We may assume each that

Y1

and

lift

R~ 1

of

Yn

Yi

for each

along an axis lying over preserves consists

R~ l"

Yi"

There

group

is contained

2.9).

* Si+ 1

S~ 1

in

S~

Let

be the subgroup of

Gi

corresponding

to

is too.

S~ . l

Because

An element

G

preserves

Gi

of

boundary

#: GI--~H I.

from

R~.

fl' f2

f2:R2"--~$2

where

~(T)

S1

while

R~ l G

in

that A

in that the axis

~*(#(T))

20 S*i

c ~0 S

(Lemma

is preserved

is a boundary

by

transfor-

if it is also a boundary t r a n s f o r m a t i o n

H..

and

~0R~

Consequently

~: G2--~H 2. We must match

and S2

for

the i s o m o r p h i s m

transformations. s homeomorphism

Let

R2

e*(T)

fl" f2

to obtain homeomorphisms R1

and similarly

fl: R --~S 1

Using Case 2 there is a h o m e o m o r p h i s m

which induces

Project

Fix a

and their orbit under

T ~ G.

Using Case 2 or 3 we can construct

R 1*

1

to

first

1

the boundary transformations

separates

of

R~ 1

if and only if

or if its axis lies in

f2:R2-'~$2

~0Ri

to

with respect

induces

is adjacent

Yi+l'

to

which

and assume

Ri+ 1

mation with respect

~: G.--~H. 1 1

(see w

so that

l

to

Rn+l c $(G) - b(G).

(the others are always).

one over

~0Ri

is adjacent

'

Ri+l,

G.. l

is a region

~*(T)

~(G i) = H i

subregion

Thus the relative boundary Yi'

boundary

Yi+l'

is a geodesic

i

of an axis over

the infinite

from

are non-degenerate

R~ 1

are the relative

if

region.

of a triply connected subregion

(iii)

Yi-i

S(G)

be the axis that

across

e*(T).

fl: RI--+SI'

are adjacent along the geodesic

are adjacent

along

~(~(T)).

Fix a closed

74

annular

neighborhood

the

lift of

K

f2

restricted

adjacent

R 2 - R 2 fl b(G) to

e*(T).

of

~K

(b) the h o m o t o p y

class

lift

to

h*

of

also agrees on

RI ,

of

~

h with

g = h

to

of

on

h

RI U R 2

and

a homeomorphism H

to e x t e n d

g*:

f*

to all

Yn

map

4.

R --+S n n

THE

Then

The

first

Fenchel-Nielsen

Sn

[2].

AXES

over

to

Ri+ 1

Finally

fl

8.

RI

so that the

on

~*(T)

Define

g = fl

Let

g*

be the lift

and

f~

on

ending

R 2 - K.

at

R ~ with n use the a c t i o n of G and

Yn'

for i n s t a n c e ,

is d e g e n e r a t e

of o r d e r two in

b(G).

We can

to an arc and the r e q u i r e d

THEOREM

statement There

Tukia

independently

The s e c o n d

4.1.

of the

case a p p e a r s

Suppose

follows

- b(G)

~(T I)

f: A - - ~ A

and

p r o o f due to L.

in M a s k i t ' s

the r e s u l t w i t h

paper

[8].

some r e s t r i c -

f rom [i] or [6].

that ~(T 2)

which induces

is uniquely

is due to

w h i c h uses c o m b i n a t i o n

there is an isomorphism

fuchsian groups with the property

homeomorphism

proposition

unpublished

communication)

rediscovered

statement

axes if and only if

following

is a n o t h e r

of a s p e c i a l

S(G)

(a) is c h o s e n

on

too r e d u c e s

A proof

in

Extend these boundary

fl

two p o i n t s

theorems.

PROPOSITION

give h o m e o m o r p h i s m s

R2 - K9

on

R e. i

case that

(personal

tions.

~K

4.

Keen and B. M a s k i t

[i0]

8K*

and

is clear.

INTERSECTING

4.1.

g = f2

from

is an arc b e t w e e n R n = Yn"

regard

on

that agrees w i t h

9 --+ U S I.

U R~

In the e x e e p t i o n a l then

)

~(T) c ~K

K*

w i t h the r e s t r i c t i o n

with property

that a g r e e s w i t h

In this m a n n e r p r o c e e d

of

and d e n o t e by

fl

~f2(K).

on the part Of K,

~(T)

h: K--~ f2(K)

K*--~f2(K

f2

8

to t h o s e of

(a) a h o m e o m o r p h i s m

of

The map

to the o t h e r c o m p o n e n t

of the c o m p o n e n t s maps to

K c

determined.

TIJT 2 ( G

also do. ~.

9: G - - + H

between

have intersecting Then there is a

The homotopy

type of

f

75

4.2. loop

An element

T = T

of G which is d e t e r m i n e d by a simple Y is c h a r a c t e r i z e d as a b o u n d a r y t r a n s f o r m a t i o n

y c S(G) - b(G)

either by its being parabolic, for all

S ~ G

or by the p r o p e r t y that

not a power of

T.

e(T) Q ~(S) = r

For this reason when

G

is

finitely g e n e r a t e d the p r o p o s i t i o n is a special case of the theorem.

4.3.

To prove the p r o p o s i t i o n w h e n

recall that {~n ]

of

b(G)

S(G)

G

is a discrete set in

with

~n c ~ n+l

and

is not finitely generated

S(G).

Select an e x h a u s t i o n

~ n c S(G) - b(G)

satisfying

the f o l l o w i n g properties.

(a)

~

- b(G) N ~

(b)

Each component of

~

bounds a c o m p o n e n t

of

n

has finite Euler characteristic.

n

is compact in

n

S(G)

-

b(G)

-

S(G) - b(G) a

of

n

infinite

and Euler

characteristic.

U ~

(c)

n

= S(G).

We can also assume each component of Fix lifts 80 ~

in

Gn

A

{~}

of

{~n }

~

is a geodesic.

n

so that

~*n c ~*n+l.

The relative

and

lim ~*n = A.

is a union of axes lying over

8~ n

be the finitely generated subgroup of G n c Gn+ 1 .

then

S(Gn) - ~n

Moreover

S(G n )

G

contains

ary t r a n s f o r m a t i o n of

G1

G1

and

80~ ~.

f~: A--~A

which induces

is o r i e n t a t i o n preserving.

mined by the p r o p e r t y that ~*(r

and in fact

c ~ 0 ~ *.

Then

b(Gn).

H 1 = ~(GI).

The h y p o t h e s i s on

sends these to b o u n d a r y t r a n s f o r m a t i o n s homeomorphism

~*" n'

A bound-

is either a boundary t r a n s f o r m a t i o n of

or its axis is c o n t a i n e d in

fl*

~n

is a union of annuli c o n t a i n i n g no points of

Start by a p p l y i n g the t h e o r e m to

assume

that preserves

~*(T)

S(H I) - ~

c 80~

in

r

H I.

~*

~

G

implies that

Thus there is a

GI--+H I.

Let

Let

For d e f i n i t e n e ~ ,

be the region deter-

if and only if

is a union of annuli c o n t a i n i n g

76

no points of

b(Hl).

Hence we may assume that

We will proceed by induction. orientation

preserving

~: Gn_l--~#(Gn_l) ~'~ ~i"

is from Let

n

- ~

be a component

that preserves Because

n-l"

property

~n'~%

Here

it.

show that preserves

is determined

g*

~n - ~n-l*

So

An

is orientation

easily seen by applying under

this case in

~

corresponding

G'n c Gn

of the axes of

G'. n

the orientation.

free of points of

b(G n')

G n.

A'*'n under

Therefore

Hn_ 1

We have the map

to

~* n-i

(i e. "

axes

Now

To

and likewise

to

~n*

~ * - i - - ~n*l

and

for the

g*: A*n--~A'*n

of

~n - ~n-l"

exactly as the

1

the result of adjoining

(in

S(G')n - n(A:)

corresponding

~'*n gives a region

fn-l:

it.

Recall that

Thus we can also assume

A '~'~ are situated adjacent n

are involved).

axis

This is most

of merely one pair of intersecting

to

of

that induces

Lemma 2.9 to the full group

are situated adjacent

~I'*

be the

has the intersecting

Carry this process out for one lift of each component

A* n

as

preserving we have to know whether

S(H n) - n(An{~).

The regions

~n

f~n : ~"n ~"--~ ~n'*"

to

and let

g~{: A--~ A

G 1 c G n) determines

is a union of annuli

from

lies over a component

~: G ' - ~ H ' = ~(G') n n n

the relative orientation

the behavior

"~ fn-i

of

there is a h o m e o m o r p h i s m

is an

which induces an isomorphism

We will show how to extend

An

subgroup

= Hn_ I.

f~'~ * "-~ ~n-i ,r n-l". ~n-i

Assume

homeomorphism

fl: ~i--+~i

axes under

the orbit of the

~'*n corresponding g*: ~ * - ~n-i

n

-

to

~*.n

n-l"

It remains

only to patch these maps together across the common bound-

ary axes

But this is carried out as in w 9

4.4.

The following

2.9 to Teichm[ller COROLLARY 9

5, giving "

Assume

n

of the theorem and Lemma

theory. %: G--~H

erated fuchsian groups between boundary

is an application

f~%.

is an isomorphism

such that

transformations

~

between finitely

gives a one-to-one

in which parabolic

gen-

correspondence

transformations

77

correspond

to parabolic

the same Teichm~ller

4.5. direction. #: G--~H

transformations.

Sorvali

[9] has made an interesting

G.

#: G--~H

to a h o m e o m o r p h i s m determined by

~

preserves

h*:

conformal

h*

3&--~ SA

belong to

study of isomorphisms and cross ratios of

which induces

Lehto

is quasisymmetric if

answered the question affirmatively out torsion when the distortion

f* G

h*

~ ~

then the

can be extended there,

f*

is

is uniquely determined

can be chosen to be quasiis not finitely generated)

and

for groups of the first kind with-

caused by

h*

is not too great.

of the ergodic properties

which is a deep and beautiful

we cannot deal with here.

axes property

[3] has raised the question

point us in the direction

the boundary mappings

elements

that induces

only up to homotopy but

(this is non-trivial

These results

parabolic

f*: A--~A

is of the first kind). if

H

in a different

If in addition to the intersecting

resulting h o m e o m o r p h i s m

whether

and

space.

in terms of their effect on multipliers

an isomorphism

G

G

We close by citing some related results

fixed points of

(if

Then

of

subject but one

REFERENCES i.

L. Bers and L. Greenberg, Isomorphism8 between Teichm~ller spaces, in Advances in the Theorey of Riemann Surfaces, Ann. of Math. Studies, vol. 66, P r i n c e t o n Univ. Press, 1970.

2.

W. Fenchel and J. Nielsen, Motions~ to appear.

Discontinuous

Groups of n o n - E u c l i d e a n

0. Lehto, Group isomorphisms induced by quasiconformal mappings, in C o n t r i b u t i o n s to Analysis, Academic Press, N.Y., 1974. A.M.

Geometrical realisations of i8omorphisms between plane groups, Bull. Amer. Math. Soc. 71(1965), 629-530. Macbeath,

A. Marden, Helv.

On finitely generated Fuchsian groups, Comm. Math. 42(1967), 81-85.

A. Marden, Math.

On homotopic mappings of Riemann surfaces, Ann. of 90(1969), 1-8.

I. Richards and B. Rodin, Analytic self-mappings of Riemann surfaces, Jour. d'Anal. Math. 18(1967), 197-225.

A. Marden,

B. Maskit, On boundaries

of TeichmSller spaces II, Ann. of Math.

91(1970), 607-639. 9.

The boundary mapping induced by an isomorphism of covering groups, Ann. Acad. Sci. Fenn. AI 526(1972).

T. Sorvali,

On discrete groups of the unit disk and their isomorphisms, Ann. Acad. Sci. Fenn. AI 504(1972).

10.

P. Tukia,

ii.

F. Waldhausen, On irreducible 3-manifolds which are 8ufficiently large, Ann. of Math. 87(1968), 56-88.

12.

H. Zieschang, Uber Automorphismen Nauk SSSR 155(1964), 57-60.

13.

H. Ziesehang,

14.

H. Zieschang,

ebener Gruppen, Dokl. Akad.

Uber Automorphismen ebener diskontinuierlicher Gruppen, Math. Ann. 166(1966), 148-167. E. Vogt, H.D. Coldewey, Fl~chen und ebene diskontinuierliche Gruppen, Lecture Notes in Math. no 122, Springer-Verlag, New York, 1970.

ON A COEFFICIENTPROBLEMFOR SCHLICHTFUNCTIONS ALBERT PFLUGER

1.

The p r o b l e m of maximizing

solved for

k - m-i 2m

later for all real

'

m = 1,2,...,

k

by J.A.

p r o b l e m of maximizing Re{a2r_l-ka~},

la 3- ka$1

~ E~

Jenkins

[2].

i f(i/z)

S,

and

the coefficients

= z + b 0 + bl/Z + b2/z2

which contain expressions result

r > i.

and more generally,

A motivation

f(z)

for doing

= z + a2z2 + a3z3 + ...

/f(z 2) = z + c3z3 + ...

+ ...

are polynomials

of the form

a 2 r - l - ka2"r

or of

in the

aj

A preliminary

is given by

PROPOSITION

i.

If, for

R e { a 2 r _ l - la~} cients

among

a2,...,ar_ I

~ < r/2

M. Schiffer

[5].

real,

then

case

There,

Let

S(a2,...,ar_ I) a2,...,ar_ 1 assumed

S

in

ar,...,a n

Re F

within

:

(minimizes)

which have the coeffi-

one knows the extremal

real.

functions.

i is based on the following result r < n < 2r- 1

be the class of functions

S(a2,...,ar_ I)

maximizes

this p r o p o s i t i o n was proved by

be an integer > I,

f(z) maximizes

in

f

has all its coefficients

k = 0

as the coefficients

that

polynomial

r

f

of course,

The proof of Proposition

THEOREM A.

(I > r/2),

those functions

In the particular

2.

of

was

We consider here the

Re{a 5 - ka~}

this might be seen in the fact that if lies in

S

by Fekete and Szeg~ in 1933, and

(minimizing)

where

within the class

of

in

S

which have

z 2,...,z r-I

i8 not empty.

and let

Let

respectively;

it is

F(ar,...,a n)

be a

and suppose z + a2z2 + ... + anzn + ... the class

S(a2,...,ar_l).

Define

80

3F Fj : 3--~ (at ..... an ), e = 0

j : r,...,n,

n < 2r- 1

for

respectively.

n

(2.1)

Re{

for all

f~

f*(z) If

This

theorem

ficient

Theorem

([5]).

This

with

1~

B.

f(D)

f

result,

arzr*

3~

for

([1],[3])

which

slits,

C

which

Q(w)

Note Schaeffer Theorem

that

not i n f l u e n c e d a 2 , . . . , a r _ I.

n

+

....

equality

Coef-

of M. S c h i f f e r

by v a r i a t i o n a l

methods

as S(a2,...,ar_l),

in

is b o u n d e d

then

by finitely

many

piece-

A~_ 1

the c o e f f i c i e n t s

= f~(z)

(2.2)

coincides

that

such

Q(w)dw 2 < [

that

W

and

a!V)zJ ]

and S p e n c e r

B states

a*z n

of the E x t e n d e d

A1 - An-1 + + n+l "'" -~

A~-l

j=v

+

slits,

(2.2) ~

,. ,

and a r e s u l t

is o b t a i n e d

Re F

maximizes

v = r, .... n

where

0

n = 2r-i,

W

theee

+

a combination

may be stated

is a f u n c t i o n

along

<

= f.

Jenkins

in

+

in the caee

r

is e s s e n t i a l l y

latter

analytic

there

fr

of J.A.

If

= a

r

if

i8 a domain

wise

2~

a

side c o n d i t i o n s ,

THEOREM

+ ... + a r _ i z r - I

or if

(2.1) o n l y

in

n = 2r-i,

for

Then

r ~ 2 - a g Fn(ar-ar) }

Fj(a~-aj)

z + a2z2

n < 2r-i,

occurs

Z

j=r

e = 1

S(a2,...,ar_l) ,

in

=

and 8et

([5], for

p.

=

n

[

aCV)~.

j:v

and

with

J

F.

are

]

(at .... ,an- ).

part of the formulas

36) if no side c o n d i t i o n s

v = r,...,n,

by the side c o n d i t i o n

the c o e f f i c i e n t s

fixing

by

]

3F

] : ~

determined

g i v e n by are present. Av_ 1

the c o e f f i c i e n t s

are

81

For the p a r t i c u l a r Jenkins'

THEOREM

Extended

Let

C.

case

General

r

of a s i m p l y

Coefficient

connected

Theorem

be an integer > 1

and

domain

takes

the

in

following

r ~ n ~ 2 r - i.

origin be c o n t a i n e d in the simply c o n n e c t e d domain

~

C, form.

Let the

in

C

which is

bounded by finitely many p i e c e w i e e a n a l y t i c slits such that for some

_ An-i

Q(w)

- --~

A1 + -~

+ ...

W

the quadratic d i f f e r e n t i a l

negative along these slits. for

n < 2 r - i.

Let

Re{

be

1

for

n = 2r- 1

n r [ A j _ i c j - e ~ An_ 1 c } j=r

for all c o n f o r m a l mappings : w + c rw r + ...

in the case

e

and zero

Then

(2.3)

V(w)

is non-

Q(w)dw 2

W

V:

+ Cn w n

n : 2 r - i,

0

such that

f(D)---~C

If

+ ....

~

or if

n < 2 r - i,

equality occurs in (2.3)

only if

Cr : 0 V

is the

identity mapping.

Proof and

of T h e o r e m

let

satisfy

f*

A:

Let

be in

f

maximize'

S(a2,...,ar_l).

the h y p o t h e s e s

of T h e o r e m

Re F Then

C.

The

within ~ = f(D)

relation

S(a2,...,ar_l), and

V = f*of -I

f* = Vof

then

implies

*

a.

aj and

in p a r t i c u l a r

=

]

a

- a r

n [ A _ic 9 ~:r An_ 1 = F n a* - a r

and

= c r

a(~)c J

= c . r

n [ j:r

:

U

,

From

and

f*

= Vof,

j = r,...,n, (2.2)

it f o l l o w s

that

r

(

i ~:r

a( ]

~)c)

A n _ i C r2 = Fn(a $ _ a r )2 ' the

F. ] hence

uniqueness

:

n [ Fj(a~ - a ) j:r ] 3 (2.1) part

holds.

Since

of T h e o r e m

A follows

r

at once

from Theorem

is thus

complete.

3.

i

~=r

In w h a t

follows

C.

the

The r e d u c t i o n

uniqueness

part

of T h e o r e m

of T h e o r e m

A to T h e o r e m

A will

play

B and

a

C

82

dominant tion,

role.

As

it t a k e s

it m a y b e d e s i r a b l e

course

sufficient

quite

a simple

form,

in the p r e s e n t

to g i v e

a simple

proof

for

to do t h i s

From the way one

proves

Extended

Coefficient

that

General

equality

occurs

the

in

for the

uniqueness

fundamental Theorem

(2.3)

only

part

inequality

([1],[3]),

if the

it.

It is of

of T h e o r e m

(2.3)

C.

in t h e

it f o l l o w s

following

situa-

immediately

two conditions

are

fulfilled:

1~

With

respect

to t h e

function

A1 = An-l-- + .. + n+l " -~

Q(w)

W

V

satisfies

the

differential

(3.1)

2~

equation

Q(V(w))(V'(w)) 2

the

domain

There fying

V(~)

are

is d e n s e

functions

the conditions

n : 2 r - i.

=

For

in

=

2~

Q(w),

C.

for which

1~ and

V(w)

with

Q

(3.1)

and having

w + c w r + ... r

instance,

if

admits

at

0

solutions

satis-

an expansion

+ c w n + ... n

Q(w)

= w

solution

the mapping

W

1 2r '

r > I,

such a

is V(w)

=

w

=

i

c r w + r--~T w

+

...

.

(l_cwr-l) r-I But t h e r e

is a s i m p l e

criterion

mapping

as a s o l u t i o n .

LEMMA.

Let

to

(3.1)

It is g i v e n

r < n < 2r-2,

having

at

0

for

the

V(w)

and

(3.1) by the

let

V(w)

to h a v e

only

the

identity

following

be a h o l o m o r p h i c

solution

expansion

:

w + c wr +

....

r

Then

V

The p r o o f

must

be

of t h e

the

identity

lemma

mapping.

is m o d e l e d

after

the p r o o f

of L e m m a

XXIV

in

[4]

83

as

far as that p r o o f

deals with uniqueness.

Let

(3.2)

where,

~

=

V(w)

for the m o m e n t ,

Q(w)

k

:

w + c_+~wK •

k+l

may be any p o s i t i v e

An_ ! AI n + ~ + "'" + - ~ '

=

+

W

9

m

m

~

integer,

An-i

and lei

~ 0

W

be such that (3.3)

Q(z)dz 2

Clemrly, branch

we may a s s u m e

of

(Q(w))i/2

(3.4)

1~

first

An_ 1 = I.

=

Close to the o r i g i n a s u i t a b l e

that

n

where

G0 :

ml

= ~' m

tion.

Hence,

taking

square

~)W ~

expansion:

n+l 2 (i + a i w + e ~ w 2 + ...).

w

I (Q(w))i/ 2 dw

W -m

Q ( w ) d w 2.

has the f o l l o w i n g

Q(w) I/2

We a s s u m e

=

is odd:

n = 2m + I.

Then

=

w -m [ e w ~ + e log w + c, O

of

(3.4)

roots

+ O

and

in

Z -m

:

0

c

is a c o n s t a n t

(3.3)

and

~ (~) z ~

integrating

+

log

of i n t e g r a we obtain

Z

0

Or

z TM

(3.5)

+ cwmz m

=

wm ~ a

0

But

(3.2)

z(w) ~

zv

+ awmz m log

w ~

'

0

implies

=

w~(l

= 1,2,...,

+ ek+lWk

+ ...)~

=

w ~ + ~ek+lWk+~

+ o(wk+B+l),

and

log z(W)w

=

log(l

+ C k + l W k + ...)

=

o(wk).

8~

Substituting

~ wm+U(l

in

(3.5),

we o b t a i n

+ m C k + l wk + o ( w k + l ) )

+ cw2m(l

+ o(wk))

u:0 e ~ w m+9 (i + ~ C k + l W k + 0 (wk+l )) + O ( w 2 m + k ) , ~:0 finally , cancelling

and

1 s0 = - ~

[0= a w m+9

the a s s u m p t i o n s

2 r - 2,

so t h a t

for all

k > 0;

2~

n

Let

+ cw 2m + O(w m + k + l )

of the

k > m.

V(w)

n o w be even:

e0

=

fact

implies

c = 0

: w,

thus

n = 2m.

Then

=

Taking

1 1 m - -2

O.

k + 1 ~ r

This

hence

:

we h a v e

w -m+l~

proving

square

and

and

the

n : 2m + 1

then

lemma

Ck+ 1 = 0

for odd

n.

[ e w~ + c 0

we

and s u b s t i t u t i n g

lemma

Q(w) I/2 dw

with

sides,

gives -Ck+lWm+k

Under

on b o t h

u

roots

in

(3.3)

and

integrating,

see t h a t w -m+l~

~ euwU 0

+ c

=

z-m+~2

~ e zv 0

Or

0 Since

all

terms

that

c = 0;

but

cz m-~2

hence,

Z

0 are h o l o m o r p h i c

squaring

2m-i ~

in the

tions

previous

z = V(w)

solution

V(w)

case

~wU

:

~

= w.

This

it f o l l o w s

W

2m-I ~

~ ~

~Z

9

0

of odd

= w + c2w2

w = 0,

yields

0 As

at

+

D

it t u r n s

n, 9

O

proves

~

this the

out

equation

lemma

that

among

admits

for e v e n

n.

only

all the

func-

85

Now,

if

n < 2r- 2

from

the

lemma.

the

If

uniqueness

n = 2r- 1

part

and

c

in T h e o r e m

C follows

= 0,

(3.2)

then

at once

holds

with

r

k + 1 = r + i. r + i

4.

Since

instead

Consider

of

now

n : 2r - I,

corresponds mizes

and

function

of m a x i m i z i n g

the and

f~

~4.1)

within

- an)

- 21ar(a

real .

are

reduces

Yr

One

occurs

with real.

and

Let

of

Then

S f

f

which maxi-

from Theorem

A it

the o t h e r

part

If the

F the

(4.1)

of if

f

and

this

the

the

inequal-

an

and

l

ar : ar .

:

A it

follows

Hence

f : f,

i.e.,

la r2

-

F : -(a n - la~)

and

gets

i.

"'"

is r e p l a c e d

+ Irar

all real,

analogous

to P r o p o s i t i o n

if

maximizes

(minimizes)

coefficients

a r*

one maximizes

+

are

implies

all real.

~ > r/2,

an + l l a n - i

real.

because

" a]. = x 3. + lyj,

and

from Theorem

result

the

~,

2

of P r o p o s i t i o n

I. ]

is

0

~)Yr <- 0.

and

are

functional

=

f*

J

to

in

Similarly,

such

f~ = ~

With

vanishes

coefficients

cients

f.

r (a* - ~ r ar )2}

- at)

E S(a2,...,ar_i).

~ < r/2,

equality

have

len~na w i t h

subclass

of

S(a2,...,ar_l) ,

r

f

the

a2,...,ar_ 1

(~-

where

a2,...,ar_ 1

S ( a 2 , . . . , a r _ I)

$

a 2 ,. ..,ar_ 1

the

the

R e { a n - AaS}

coefficients

to the c o e f f i c i e n t s

Re{(a

f o r all

If

we can a p p l y

that

(4.1)

ity

the p r o b l e m

Re{a n - la~}

follows

- 2

r.

r > I,

be an e x t r e m a l

n < 2(r+l)

i,

by

n = 2r - i,

- la$, the

i.e.,

same if

method

of p r o o f

I < r/2

(l > r/2)

Re F

among

those

functions

a2,...,ar_ 1

real,

then

f

has

gives

all

in

S,

a

and which

its c o e f f i -

86

In the p a r t i c u l a r says that (I > i) real.

Re{a 3 - la~}

Thus the e x t r e m a l set

functions Using points

differential

of this r e g i o n ,

is r e d u c e d

+ a3z3

r = 2) in

Proposition S

for

i

~ < 1

w h i c h h a v e all t h e i r c o e f f i c i e n t s

in the real

= z + a2z2

Schiffer's

(hence

( minimized)

functions

problem

{(a3,a2)}

f(z)

n = 3

is m a x i m i z e d

only for s c h l i c h t

the p o i n t

made

c a s e of

xy-plane

+ ...

equation

one finds that

to the i n v e s t i g a t i o n

in

corresponding

S

of

to the

w i t h real coefficients.

corresponding it is b o u n d e d

to the b o u n d a r y by a J o r d a n curve

up by the two arcs:

for

0 < t ~ I

AI:

a3

=

2 a 2 - i,

A2:

a3

:

i + t2(l

a2

:

•

and

{a 3 - l a ~ } ,

m(1) for

0 < I < ~,

A2

at two s y m m e t r i c I ~ i).

outside

points For

and

(3,-2),

which

and

k (z) = -k(-z),

straight line

through

it c o n t a i n s point

Let

( A I U A 2.

if

0 < I < i,

I < 0,

To

through

~ > i,

the same p o i n t s

the are

region

through

If

~ < i,

for

I < i

(3,2)

and

point (3,2)

=

z (l-z) 2 the

(3,-2).

a 3 = la$ + m(l)

(3,-2),

it t o u c h e s

k(z)

corresponds

the p a r a b o l a

(3,2) and

A I.

~ = 0

to

(it t o u c h e s

the p o i n t s

to the Koebe f u n c t i o n s

passing if

is t a n g e n t

and at the s i n g l e

it p a s s e s

respectively.

a 3 = M(0)

a 3 = la~ + M(1)

of the c o e f f i c i e n t

correspond

On the o t h e r hand, passes

(a3,a 2)

the p a r a b o l a

a n d lies o t h e r w i s e

if

t = 0.

min

A2

(i,0),

for

max :

Then,

+ (i - log t) 2)

- log t)

(a3,a 2) = (1,0)

M(1)

-2 ~ a 2 ~ 2

while AI

for

~ = 1

at the single

(-i,0). It f o l l o w s

(minimized)

in

that S

by e x a c t l y

two

(I > i) (real)

Re{a 3 - la~} functions.

is m a x i m i z e d

87

9

The

case

minimized

(in

f(z) and that

~ : I S)

by the

h(z) l-ah(z)

:

-

In c o n n e c t i o n already

equation

As

in 1936

functions

shown

The

I~

Let

Theorem which the

in

are

A.

Then,

maximizes

had used

h*(z)

section,

we

LSwner's

{(la2r_ll,larl)}

-

z l-z 2 "

should

remark

differential

for

(r-l)-

r = 2,3, . . . .

4, T h e o r e m

A gives

n = 2r - i, other

good

within

information suitable

if o n e

subclasses

of

as d e f i n e d

in

examples.

be a n o n - e m p t y

I > r/2

a

is

z = l+z2

h(z)

where

Basilewitsch

some

if

R e { a S - a~}

functions

of this

(minimizes)

coefficient

b y the

the material

S,

S ( a 2 , . . . , a r _ I)

that

where

-2 < a < 2, ---

'

R e { a n - laS},

following

S)

the region

in S e c i i o n

extremizes S.

J.

to d e t e r m i n e

symmetric

5.

with

-2 _< a _< 2, (in

h*(z) l-iah*(z)

One knows

functions

'

it is m a x i m i z e d

f(z)

that

is c l a s s i c a l .

subclass

(I < r/2) R e { a n -la2}r

there

of

S

is at m o s t

__in S(a2,

one

..,ar_ I)

f a n d has

real. r

In f a c t ,

(4.1)

Re{(a*n and

if

la .2) r

f*

is a l s o

(I- r/2)(a

- ar)2 ~

a r* = at, < r/2

c a n be w r i t t e n

so t h a t

-

(a n

function

this

inequality

0.

I > r/2

and

* ar

Thus

equality

if

occurs

-Re{a

solution

in

(4.1)

which

reduces

implies

f*

to

then = f.

If

- la2}. r

to the p r o b l e m

a 2 = ...

0,

is r e a l ,

is k n o w n

in t h e p a r t i c u l a r

of

(5.1)

< --

a maximizing

one maximizes

A complete

form

- ~a 2) + (I - r / 2 ) ( a * - a )2} r r r

n

2~

in t h e

-- ar_ I -- 0,

r > 2.

case

88

The

s o l u t i o n was o b t a i n e d

Extended

General

Coefficient

to be e x t r e m a l w i t h o u t here

by J.A.

Jenkins

Theorem

([2]) by a p p l y i n g

to f u n c t i o n s

using variational

extremal

functions

Let

f

maximize

S(0,...,0), Then

i.e.,

(4.1) h o l d s

for the

almost

a r = ar

and

R e { a n - ka~},

an = an.

A it f o l l o w s

used

one to d e t e r m i n e

n = 2r -i,

functions

for all f u n c t i o n s fe(z)

The m e t h o d

the

immediately.

among those

functions

which were guessed

methods.

is b a s e d a g a i n on T h e o r e m A, w h i c h a l l o w s

the

f*

: e-lf(ez), Hence

for w h i c h in

lel

equality

within

the

(5.1)

S(0, .... 0), : I.

holds

If

c r-I

subclass

is s a t i s f i e d . in p a r t i c u l a r = I,

then

in (4.1) and f r o m T h e o r e m

that

f(z)

This r e l a t i o n

=

implies

2hi e : er-i

e-lf(ez),

a. = 0 ]

if

j - i

is not a m u l t i p l e

of

r - i,

i.e., f(z)

There

=

z(l + a z r-I + 2(r-l) r a2r-lZ

is a f u n c t i o n

F

in

F(z)

+

3(r-l) a3r-2Z

+

) ....

S,

=

z + A2z2

+ A3z3

+ ...,

such that f(z)

(5.2) Indeed,

=

I F ( z r - l ) r-I

=

z

+

=

~A2r

z(l + A 2 z r - I r-~

~r-2 (A 3

! + ...)r-i

+ A3z2(r-l) 222r-i

z

+

-

A

=

z(l + a r _ i Z

+ a2r_2Z

)z

+

....

define 2 F(z)

. . . )r-l. + 1

F

is h o l o m o r p h i c

If 1,2,

F(~ I) = F(~2), then

in

D

and n o r m a l i z e d

a n d if

f(z I) = f(z2),

zk

at

is c h o s e n

hence

zI = z2

0,

and

such that and

f(z)

= F(zr-l) r-l.

r-i ~k = Zk '

E1 = ~2"

This

k = shows

89

i that

F

then

f

is in

S.

Conversely,

~ S(0,...,0).

PROPOSITION

If

2.

S(0,...,0),

a2,...,a n

vanish,

maximizes

then

f i

9

9

n = 2r - i,

S

maximizes

within

whose

coefficients 1 f(z) = F(zr-l) r-l,

where

the real part of - r/2 + i,

~=

r-i

S.

Conversely, a function

in

is of the form

A 3 - ~A~,

in the class

f(z) = F(zr-l) r-l,

Re{a n - la2},

of functions

+ A3z3 +

F(z) = z + A2z2

and

This proves

f

the class

F E S

if

f

any such function maximizing

similar proposition

F

produces in

Re{a n - la~}

holds for functions

by

F(zr-l) r-I = f(z)

S(0, .... 0).

minimizing

Of course,

a

in

Re{a n - la~}

S(O,...,O).

3~

Let

a2,...,ar_ 1

aje j-I = aj

be such that

for

j = 2 , . . . , r - i,

2wi where

e = e k

and

k

is a divisor of

j - 1

is not a m u l t i p l e of

k.

f(z)

S(a2,...,ar_ I)

i.e.,

a. = 0 ]

if

By an argument similar to that of

Example 2, one shows that a function this subclass

r - i,

f

maximizing

Re{a n - %a~}

in

satisfies the functional equation

: e-lf(ez). Thus a similar r e d u c t i o n as in the previous example is possible.

Remark.

In the two foregoing examples,

the extremal function

IF

2hi satisfies the equation k > i,

f(z)

the extremal domain

nents in

C,

differential vanishes~

= e-lf(ez), f(D)

e = e k

Consequently,

for

has at least two boundary compo-

and this fact implies that in the c o r r e s p o n d i n g quadratic Q(w)dw2

An_ 1 A1 = ( - - n ~ + "'' + - ~ ) dw2 w w

the coefficient

By the work of Jenkins it then follows that

f

A1

is extremal

also in the w i d e r class of those functions w h i c h are univalent and

90

meromorphie

in the

zr + z + ar

r+l

unit

r = 2

having

at

0

the

expansion

+

ar+lZ

If

disc

.... (k = I, n =

3) the

situation

is d i f f e r e n t .

have

A1

hence

the a b o v e

=

remark

a~ 2 )F 2 + a~ 2 )F 3

=

applies

I = i.

only

if

2a2(i-

I);

By

(2.2)

we

REFERENCES i.

J.A. Jenkins, An extension of the General Coefficient Theorem, Trans. Amer. Math. Soc. 95(1960), 387-407.

2.

J.A. Jenkins, On certain coefficients of univalent functions II, Trans. Amer. Math. Soc. 96(1960), 534-545.

3.

J.A. Jenkins,

4.

A.C. S c h a e f f e r and D.C. Spencer, C o e f f i c i e n t Regions for Functions, Amer. Math. Soc. Coll. Publ. vol. 35, 1950.

5.

M. Schiffer,

On certain extremal problems for the coefficients of univalent functions, J. A n a l y s e Math. 18(1967), 173-184.

Univalent functions whose n 329-349.

r e a l , J . A n a l y s e Math. 1 8 ( 1 9 6 7 ) ,

first coefficients are

ON INCLUSION RELATIONSFOR SPACESOF AUTOMORPHICFORMS CH , POMMERENKE

INTRODUCTION

i.

Let

F

be a F u c h s i a n group,

that is a d i s c o n t i n u o u s group of

Moebius t r a n s f o r m a t i o n s of the unit disk F

be a f u n d a m e n t a l domain of

and

1 ~ p ~ ~,

analytic

in

let

D

A~(F)

F

D

with area

onto itself,

and let

8F = 0.

q = 1,2,...

For

denote the space of functions

g(z)

that satisfy

(I.I)

g(r162

q : g(z)

(r ( r )

and

f(1 - Izl2)pq-21g(z)IP d x d y

(1.2)

< |

if

i < p < |

if

p : =.

F

sup

(~

-

i~I~qlg(z>l

<

-

zEF The integral is i n d e p e n d e n t of the choice of the f u n d a m e n t a l domain F;

the s u p r e m u m is not changed if we r e p l a c e

AI(F) q

and

A~(F) q

(q = 2,3,...)

A~(r)

(for instance [3],[8],[10],[6])

r A~(r)

D.

The spaces

[1],[2],[4].

c o n s i d e r e d the p r o b l e m w h e t h e r

(1.3)

by

are of p a r t i c u l a r interest in the

theory of F u c h s i a n and K l e i n i a n groups Several authors

F

(l ~ p < |

R a j e s w a r a Rao [I0] has shown that,

for

Hence it suffices to c o n s i d e r the case

q > i,

p = i.

have

93

J. Lehner

[5] has recently proved

exists a constant

y = y(F) > 0

inf d(z,r z~D where

d

such that

I Y

for all hyperbolic

denotes the n o n - e u c l i d e a n

results on universal particular,

properties

it follows that

(1.3) for the case that there

distance.

He uses A. Marden's

of Fuchsian groups

(1.3) holds

r E F,

if

F

[7].

In

is any subgroup of a

finitely generated group. We shall show that on

(1.3) is not true without

some r e s t r i c t i o n

r.

THEOREM

There exists a F u c h s i a n group

I.

A12(r)

(1.5) and therefore

such that

~ A~(r)

that

(1.6)

A~(r) r A~(r) To s e e t h a t

(1.5)

g E A~(r)\A~(F).

from (1.4).

F

(1 ~ p

implies

Then

(1.6)

|

we c h o o s e a f u n c t i o n

g2 E A ~ ( F ) \ A ~ ( r ) ,

and

It is a pleasure to acknowledge

Marden and L.Greenberg

<

(1.6)

follows

conversations

with A.

on this counterexample.

In the last section we establish a generally valid inclusion relation different

2.

(with

q = I)

A~(F)

is replaced by a somewhat

space.

THs C O U N T s 1 6 3 The function

in

in which

is called a Bloch function

f(z)

D

and satisfies

(2.1)

sup

z~D

(1

-

Izl

If'

z)l

<

|

if it is analytic

94

We shall need the classical c h a r a c t e r i z a t i o n disks on the Riemann image surface LEMMA.

(see,

in terms of schlicht

e.g.,

[9]):

An analytic function is not a Bloch function if and only if,

for every

p < +%

of radius

p.

it maps some domain in

It will be convenient q = i,

becomes

g(r162

(2.2)

f(z)

is not,

to integrate

in general,

= g(z).

I;

:

g(~) d~

D

one-to-one onto a disk

condition

(l.l) which,

for

The function (z E D)

automorphic

but has periods

c(r

that

satisfy (2.3)

f(r

= fCz) + c(r

It follows from (1.2) and (2.1) that derivatives

of the Bloch functions

Theorem i is contained THEOREM

2.

(r E F). AT(F)

for which

consists

of the

(2.3) holds.

Hence

in the following theorem.

There exists a Fuchsian group of the second kind and a

non-Bloch function

(2.4)

f(z)

satisfying

[I,f'(z)l 2 d x d y

(2.$) for which

< |

F

The Fuchsian group points on

aD

is a bordered convergence

F

is of the second kind if the set of limit

is of measure

zero;

Riemann surface.

this is true if and only if

In particular,

our group is of

type.

We give first an outline of our construction.

(2.5)

B :

D/F

Let

U {0 < Rew < 3-n, 2n < Imw < 2n+l} U U {0 < Row < 3-n, Imw : 2n}. n=l n=2

We attach a different

copy of

B

suitably translated

to each

95

vertical

side of

free v e r t i c a l

B. T h e n

we a t t a c h a new t r a n s l a t e d copy to each

side of the r e s u l t i n g surface~

simply c o n n e c t e d R i e m a n n surface over each h o r i z o n t a l

(2.6)

V

of w i d t h D

onto

2n. R

that contains

We o b t a i n a

schlicht domains

strip

= {2 n < I m w

n

R

and so on.

< 2n+l}

(n = 1,2,...)

Hence the Lemma shows that the f u n c t i o n

is not a Bloch function.

Furthermore

R

f(z)

mapping

is invariant

under a group of "translations" which c o r r e s p o n d s to a F u c h s i a n group F

in

D,

onto

B.

and some f u n d a m e n t a l domain Hence

Proof.

F

of

F

is m a p p e d o n e - t o - o n e

(2.4) is satisfied because area

Let

T

B < ~.

be the free group g e n e r a t e d by

Thus each element of

T

(~n)n=l,2,...

can be u n i q u e l y w r i t t e n as a reduced word

kl.. g k t

(2.7)

T : anl

where

t

9 nt

is the length of

(2.8)

R :

of "copies" of w = (w,T)

B.

with

(2.9)

Given

R e w 0 ~ 0,

= w +

~0 = (W0'T)

s u f f i c i e n t l y small)

= •

T.

We define

k +1 ~ - k R

if

n

= n +l )

as the d i s j o i n t union

U (B,~) TET

Every element

w ~ B

p(~)

(k

and

T

w ~ R

of the form (2.7),

t -n 9 [ k 3 ~=i E R,

can be u n i q u e l y w r i t t e n as

we set

and we define

(w E R).

D O = {lw - w01

and define a n e i g h b o u r h o o d of

< 6} w0

by

(6 > 0 (D0~T)

and by

(2.10)

(D O N B~T) U ([D O + 3 -n] N B,rqn I)

if

2 n < I m w 0 < 2 n+l,

R e w 0 = 0,

Then

n e i g h b o u r h o o d s o n e - t o - o n e onto a disk in

p(~) C,

maps each of our and

R

together with

if

9B

the global p a r a m e t e r by

p

is a Riemann surface w h i c h we denote a g a i n

R. We show now that

R

is simply connected.

(piecewise smooth) closed curve on intersects

(B,id).

R;

Let

T

be a reduced w o r d

m a x i m a l length such that

C

crosses the b o u n d a r y

Then

only on points on

~(B,T)

onl...qnt_l.Since

T' =

B

B(B,T).

Now the r e d u c e d word

process we see that

C

C'

8(B,T)

~(B,T')

homotopic to

T'

has length

C

(of the form (2.7)) of

C

of

(B,T).

where

is simply c o n n e c t e d and

is c o n n e c t e d we can find a curve

be a

we may assume that

Let

C crosses kI kt_ 1

C

~(B,T) n B(B,T')

that does r~tcross

t-l.

is h o m o t o p i c to a curve in

R e p e a t i n g this (B,id)

and

t h e r e f o r e to a point. Given

~ ~ T

we d e f i n e a c o n f o r m a l s e l f m a p p i n g

w = (w,T)

+

~

isomorphic to

~

Since

R

is a f u n c t i o n R.

If

R

by

= p(w)

+

-m

~ j 3 ~=i

T.

Also,

]I.. ~

(~ = ~ml

by (2.9),

J~ "~m~ ).

is simply c o n n e c t e d and has free b o u n d a r y arcs there h(z)

that maps

D

c o n f o F m a l l y and o n e - t o - o n e onto

he E T ~ then

(2.12)

maps

~

p o ~(w)

of

~*(w) : ( w , ~ ) .

These m a p p i n g s form a group

(2.11)

~

r = h -I 0 k~ o h

D

o n e - t o - o n e onto

transformation.

D

and is analytic,

hence a Moebius

Thus

r

: {r : ~

E T ~} O

is a F u c h s i a n group w i t h f u n d a m e n t a l d o m a i n

F = h-l(B,id).

F

F

has free b o u n d a r y arcs on

kind.

8D

the group

Since

is of the s e c o n d

97

The c o m p l e x - v a l u e d f u n c t i o n satisfies,

c = c(r

f(z)

:pol

oh=po

is a constant.

m a p s some d o m a i n in

in (2.6).

is a n a l y t i c in

D

and

by (2.12) and (2.11),

foe where

f = p o h

Since w i d t h

not a Bloch function.

V

D n

h+c

Thus

= f+c

(2.3) is satisfied.

o n e - t o - o n e onto the strip

= 2n § =,

Finally

By (2.9), V

defined

n

the Lemma shows that

f(z)

maps

F

f(z)

o n e - t o - o n e onto

is B,

and it follows that

fF~If'(z)12dxdy

= areaB

=

( )

< =.

n=l

AN INCLUSION RELATION

3.

We p r o v e now a d i f f e r e n t v e r s i o n of (1.3)

(with

q

=

1).

I want

to thank the r e f e r e e for his helpful comments.

T H E O R E M 3.

D

Let

r

be a Fuohsian

group.

Let

f(z)

be analytic

in

and

(3.1)

If,

f(r

= f(z)

for some fundamental

(3.2)

+ c(r

domain

(r ~ F).

Fj

d = diam f(F) < =

then

f(z)

is a Bloch function.

Our new c o n d i t i o n tion (2.4). q ~ 2.

(3.2) n e i t h e r implies nor is implied by condi-

It is not clear w h e t h e r T h e o r e m 3 has an a n a l o g u e for

The q u a n t i t y

d ~ a m f(F)

remains u n c h a n g e d if we r e p l a c e

Proof. the Lemma,

Suppose that

f(z)

there exists a domain

depends on the choice of F

by

~(F)

but

(r E F).

is not a Bloch function. ~ c D

F

Then,

by

m a p p e d h o m e o m o r p h i c a l l y by

98

f(z)

onto a disk

{lw - w01

< 5d}.

H : f-l({[w

Note that

H,

homeomorphic

the relative to

and satisfies

(3.3)

K n H ~ ~

then

K c H.

and so

f(K)

dicting

(3.3).

Let mental

z0

closure

contains

K

a point of

be the preimage

domain for

< 4d})

of

H

under

~.

n

in

D,

f(z).

is compact

Also,

if

and

K c D

is

f(K) c {lw - w01 ~ 3d}

For otherwise,

F

(3.4)

which

of

must contain 8f(H)

w0

satisfies

a boundary

= {[w - w01

in

H,

= 4d},

and let

(3.2) and

point of

F

H~

contra-

be a funda-

z 0 ( ~.

Clearly

f(~) c {lw - w01 ~ d}.

Thus

~

satisfies

Let (3.1), and

- w01

{lw - w01 ~ 4d}

connected

Let

el(F) (3.2),

(3.3), (r

E F)

and

S =

Then

S

cannot

c(r

phism on adjoining

H, F

c H.

=

U nEZ

an infinite

{]w - w01

F c H.

domain adjoining

f(r

Since

is univalent

strip of width at most

< 4d} = f(H).

Since

fundamental

for which

f(r

~ S.

Im [c(r

/ c(r

el(F)

and

on

H,

If(F) + nc(r

f(z) domain

As before,

we clearly have

Because

By

c {]w - w01 ~ 2d};

f(z)

we can find another

(3.5)

F.

Let

U f(r nEZ

lies within cover

we see that

r

# 0.

that

be a fundamental

(3.4)

(3.3) then yields

we must have

and we conclude

r

adjoin

# 0. F,

the set

d

and so

is a homeomorr c(r

(r ~ 0;

E F) and

99

A : ~ U el(F) U r is connected. Thus,

U r162

U r162

A moment's thought meveals that

by (3.3),

A c H.

Since

f(z)

f(A) c {I w - w01 ~3d}.

is univalent in

H,

we

conclude from f(r that

= f(z) + c(r I) + c(~ 2) = f(~2Or

~io$2(z) = $2o~i(z)

for

z E H.

But then

(z E H) ~io~2 = ~2o~i

by

the identity theorem. Thus the subgroup of hence cyclic [6; p.14].

F

generated by

and

$2

is abelian,

It follows that its homomorphic image

{nlc(r I) + n2c(r 2) : nl,n 2 ( Z} (3.S).

r

is also cyclic,

and thus contradicts

REFERENCES

i.

L. Bers, Automorphic forms and Poincar~ series for infinitely generated Euchsian groups, Amer. J. Math. 87 (1965), 196-214.

2.

L. Bers, Uniformization, moduli, and Kleinian groups, Bull. London Math. Soc. 4 (1972), 257-300.

3.

D. Drasin and C.J. Earle, On the boundedness of automorphic forms, Proe. Amer. Math. Soc. 19 (1968), 1039-1042.

4.

I. Kra, Automorphic Forms and Kleinian Groups, W.A. Benjamin, Reading, Mass., 1972.

5.

J. Lehner, On the boundedness of integrable automorphic forms, Iii. J. Math. 18 (1974), 5 7 5 - 5 8 4 .

6.

J. Lehner, A Short Course on Automorphic Functions, Holt, Rinehart, and Winston, 1966.

7.

A. Marden,

8.

T.A. Metzger and K.V. Rajeswara Rao, On integrable and bounded automorphic forms II, Proc. Amer. Math. Soc. 32 (1972),

Universal properties of Fuchsian groups in the Poincar~ metric, Annals of Math. Studies 79 (1974), 315-339.

201-204. 9.

Ch. Pommerenke, On Bloch functions, J. London Math. Soc. 2 (1970), 689-695.

i0.

K.V. Rajeswara Rao, On the boundedness of p-integrable automorphic forms, Proc. Amer. Math. Soc. 44 (1974),

(2)

278-282.

QUASICONFORMAL MAPPINGS OF THE DISK WITH GIVEN BOUNDARY VALUES EDGAR REICH*

CONTENTS I.

Introduction.

II.

Quasiconformal mappings

2.

Extension method of Beurling and Ahlfors

8.

The extension method of Ahlfors and Weill

4.

The extension method of Ahlfors and Weill

5.

Existence of extremal mappings

The Main Inequality

III.

IV.

i.

(continued)

(M)

i.

Introduction

2.

Trajectories of quadratic differentials

3.

Proof of

4.

Teiohm~ller mappings

5.

Affine stretches

in the disk

(M)

The Necessary Condition for Extremality i.

An auxiliary inequality for mappings agreeingon a subset of

2.

The functionals

3.

Example - affine stretch of angular domains

4.

Example of an argument based on normality

I[<],

Q[K]

~U

and the necessary condition

Proof that the Necessary Condition is also Sufficient i.

Another auxiliary inequality

2.

The characterization of extremal

dilatations

From Section II on these lectures are based largely on published and unpublished work undertaken jointly with K. Strebel. The major published reference is [14]. The proof arrangement in Section III.l is based on a written communication from K. Strebel. It is more efficient than the proof in [14]~ w in that the basic inequality (M), instead of a new inequality is used.

102

3.

Alternative characterization of extremaldilatationsbymeans of Hamilton's functional

4. V.

Direct interpretation of the functional

H

Further examples i.

Afflne mappings

2.

An extremal problem for functions analytic in the disk

References.

103

I,

INTRODUCTION

i.

Q u a s i c o n f o r m a l Mappings.

For the purpose of these lectures we

can define a plane q u a s i e o n f o r m a l m a p p i n g as a sense p r e s e r v i n g homeomorphism,

w = f(z),

of a r e g i o n

first order partial derivatives f

If

= K(z)f

k = ess sup Im(z) I = IIKII= < i,

l+k K = l--~k~

[D(x,y) + D - ~ ]

in

f

~.

is K - q u a s i c o n f o r m a l ,

-If-12z > 0

a.e.

(f = u + iv),

J = ux2 + uy2 + x ~ + ~Y =< (K + ~)J,

at all points of total d i f f e r e n t i a b i l i t y , D(z)

L2

zE~ I. Such a m a p p i n g has the p r o p e r t y that the J a c o b i a n

J = UxVy - UY vx = IfzI~ (I)

p o s s e s s i n g locally

satisfying the Beltrami equation a.e.

z

~,

is the d i l a t a t i o n at

z,

l+I<(z) I (D(x,y) = l_i<(z) l , z = x+iy), that is,

sup D(z)

and

almost everywhere.

the maximal d i l a t a t i o n of

f.

Since q u a s i c o n f o r m a l m a p p i n g s of Jordan domains can be extended to h o m e o m o r p h i s m s b e t w e e n the closed domains,

and since the regions

can be n o r m a l i z e d by m a p p i n g them onto disks it makes about the e x i s t e n c e of mappings of with given b o u n d a r y values.

U = {Izl < i}

sense to ask

onto

U = {lwl < i}

We start off with a survey of two methods

of c o n s t r u c t i n g q u a s i c o n f o r m a l m a p p i n g s with given b o u n d a r y values. [Note:

R e c e n t l y a n o t h e r general m e t h o d has been found by L. C a r l e s o n

[7]).

2.

E x t e n s i o n Method of Beurling and Ahlfors.

e x t e n s i o n method of Beurling and Ahlfors

[4] we c o n s i d e r the boundary

c o r r e s p o n d e n c e between the real axes of the of the unit circles,

To explain the

z

and

w

planes instead

and we look for an e x t e n s i o n of the b o u n d a r y

values to a q u a s i c o n f o r m a l m a p p i n g

f(x,y)

= u(x,y)

+ iv(x,y)

b e t w e e n the upper half planes. Let

u = h(x)~

-~ < x < ~

denote the g i v e n b o u n d a r y

104

correspondence, -~ < x < ~

assumed to be monotonically increasing,

homeomorphically onto

-~ < u < ~.

provides a naive way of extending onto

h(x)

and to map

The following

to a mapping of

{Im z > 0}

{Im w > 0}

(2)

{ u(x,y) : 189

+ h(x-y)]

v(x,y) : ~[h(x+y) - h(x-y)] Calculating

Thus

D(x,y)

using (I),

we find

the "naive" method (2) gives a quasiconformal extension if and

only if

h

is absolutely continuous and there exists a number

M > 0

such that 1 ~ s h'(x) ~ M.

(3)

The naive method does not always provide an extension even if one exists.

Example:

h'(x) = 21x I

w = zlz I

is 2-quasiconfoPmal in the plane,

but

does not satisfy (3).

The great idea of Beurling and Ahlfors was to replace (2) by

I u(x,y)

: ~i [ ~ h ( x + t y ) dt +

~ h ( x - t y ) dt ]

(4)

v(x,y) = ~1 [ flh(x+ty) at 0

-

flh(x-ty) dt ] 0

They showed that (4) provides a quasiconformal extension if and only if (3) is replaced by the "quasisymmetry" condition

(5)

1

and that (5) is,

h(x+y) - h(x) < C ~ h(x) - h(x-y) = ' in fact,

necessary and sufficient for the

existence of a quasiconformal extension.

105

3.

The Extension Method of Ahlfors

theoretically mappings of mapping

and Weill.

We next describe a

significant method of constructing U

due to Ahlfors and Weill

is constructed

[i].

fairly explicitly

quasiconformal

The quasiconformal

(the most transcendental

operation being the solution of a linear differential

equation)

so as

to have the boundary values of a given eonformal mapping defined for {Iz[

> i}.

A certain hypothesis

on the conformal mapping

is necessary

for the method to succeed~

and in this brief exposition we make an

additional

is not really required.

a s s u m p t i o n which

We start with the conformal mapping be normalized

f(z)

#(z)

= ~

+ 0(

),

'

~(z) : ( ~ )

1 f" 2

- y(N)

This quantity was encountered

,

of

f,

Izl ~ 1.

by Schwarz

[15] in connection with the

problem of mapping a circular polygon conformally Schwarz was

assumed to

z ~ |

denote the Schwarzian derivative f"

cs)

[z[ ~ i,

so that

(6)

Let

f(z),

faced

onto a half plane.

with the problem of determining

f,

given

r

i.e. of solving the non linear third order differential

equation

He credits Weierstrass

observation:

Let

~l(z)

with essentially

= z + 0(~),

D2(z)

the following

= 1 + 0(i),

z § ~,

(6).

satisfy the

linear equation

n" + ~e(z)n : 0,

Izl ~ I,

then n

f(z)

:

(z)

*

To obtain the A h l f o r s - W e i l l

extension of

f

to

IzI ~ i,

let

106

Tk(Z)

Izl <

: ZDk(1),

i,

k

=

1,2,

and define (7)

~(z)

(z) = 2x -*- ~ - ,

I zl =< i,

where Ik(Z)

: ZTk([)

+ (i - IZI2)T{([),

~, r~e is ) - f(e le) ,

Clearly

o<e<

2~.

k = 1,2.

A formal

computation

shows that =

If

sup ~ Izl
1

/ ~Zl

~

,

< 1

Suppose

=

then f(z)

-

i-~(1

~

- z[)2r

Izl

is quasiconformal.

: z + 0(~),

z + ~,

<

i.

This means

is schlicht

the

and

Z

holomorphic

for

Izl ~ i,

and the holomorphic

function

r

defined

by (6) satisfies 1 k : ~ sup [(Izl 2 - 1 ) 2 1 r l,l>j

(8)

Then

~(z)

as defined by (7) provides

extension of

f(e i8)

t_s

U.

7 (9)

~(z)

Note:

According

always

satisfies

between

~(z)

(i- Iz12) 2z ~

- ~

k <= 3.

derivatives

a boundary

Method

value problem

values of a conformal

Izl < i.

the quantity

and differential

of Ahlfors for

mapping

[8].

and Weill

k

of (8)

of the relation

equations,

the Ahlfors-Weill

version by Duren and Lehto

The Extension

~(_),

See [5] for an exposition

recent work of Becket generalizing

4.

l+k l-k - quasiconformal

has comple• dilatation

z to a theorem of Nehari

Schwarzian

the sharpened

a

< 1.

also the

condition,

and

See also Ahlfors

[2].

(continued).

To pose

{JzJ < i}

in terms of the boundary

given for

Izl ~

1

at first

107

seems artificial, it.

but there is a natural point of view which leads to

Instead of specifying the boundary values of

us merely postulate that

f

quasiconformal mapping of K(Z),

Izl < i.

f

explicitly,

let

has the boundary values of a U

onto

U

with a given complex dilatation

Let

lzl

f < (z) = z + 0(~),

> 1

be a homeomorphic solution of <(z), az

Let

r

Izl
0

Iz] >l.

be the Schwarzian derivative of

hypotheses of

w

IZl > i.

If the

are satisfied then this gives rise to a

quasiconformal mapping

~K

of

U

such that

there exists a linear transformation Izl < i.

fK(Z),

T

~<(e i8) H fK(e i8 ) .

such that

Now

f = T o fK'

Hence = To~

is a quasiconformal mapping of

U onto

values as

~<

f.

We may refer to

quasiconformal mapping of values

fI~ U.

The mapping Example I.

U

with the same boundary

as the Ahlfors-Weill canonical

onto

U

corresponding to the boundary

(Of course it will exist only if ~

has the complex dilatation

Suppose

K(Z) ~ ~,

0 ~ ~ < I.

z § ~, f~(z)

The function

U

r

=

~ ~,

~K )

~ < ~.

~(z),

satisfies (8)).

as given by (9).

We get

Izl < 1 Izl > 1

is easily calculated,

is satisfied if and only if the canonical mapping

Z +

CK(z)

and one notes that (8)

The mapping

has complex dilatation

~K

(and hence also

108

~ Example

2.

( -i Ill2 1 l ~7 ~ /

Suppose

is a parameter,

K(t,z)

II~II= < =,

one can calculate r

= tv(z)

Izl>

Iz I > i~

= 0

as

o(t).

for all functions

where

t ~ 09

and its Schwarzian

within an error of

ff v(z)g(z) d x d y

0 < t < 1,

lle(',t)ll~ = o(t)

f(t,z),

1

+ e(z,t),

t

In this case derivative

Suppose

g

holomorphic

in

U

for

U

which

Ilgll = f[

Ig(z)l d x d y

< ~.

It turns out [3,10] that

U

II~(',t)ll~ = o(t), same boundary

t § 0;

values

as

i.e., f,

the mapping

5.

1

of Extremal

Mappings

be the boundary

homeomorphism h

of

U

onto

to quasiconformal

smallest maximal

U.

W(Zl) ,

chordal

distance

f E Qh i}.

of

be extended Choose

that

<(z),

values Izl < i.

h(e i8)

Qh

of this phenomenon

,

0 ~

8

<

2~,

U

be the class of extensions

onto

U.

If

i_nn Qh'

i9

for normality

is that each mapping w(z3)

Qh

of

is nonempty

a mapping with

Zl,

of the family assumes

> d > 0,

z2,

z3

arbitrary

example

such that the

i,j = 1,2,3,

to the sphere by reflection

(the earliest

Qh

of a family of

at three fixed points

(w(zi),w(zj))

on

{Izl=

is the Strebel

4 we will from now on not usually

explicitly,

but instead

This determines

specify

the mapping

i ~ j [ii].

about

may contain more than one extremal

As in paragraph boundary

dilatation 9

of a sense preserving

mapping

condition

w(z2) ,

It is known example)

Let

9

Let

mappings

mappings

values

{Izl=

has the

dilatation.

A sufficient

quasiconformal

Let

explanation

values

then there exists an extremal

Proof.

U

IV.4.

Existence

lh(eie) I =

of

but a much smaller maximal

We will come back to an illuminating in Section

;~

i}. "chimney" mapping 9 specify

a complex

function

f

the

dilatation (and hence

109

the boundary values of that mapping function) mapping.

Since composition by a conformal mapping does not affect

the maximal dilatation, Qf

whether or not

depends only on the function

(for the boundary values class

Qf,

mappings K

f~/fz = <'

if

Qh

<*

<.

f

is extremal

We say that

induced by it) Izl < I.

if

f

K

in its class,

is extremal

is extremal

in the

In view of the existence of extremal

is nonempty we see that to a given complex dilatation

there are associated

i.e.

up to a conformal

one or more extremal complex dilatations

is the complex dilatation of an extremal mapping

class of mappings with the boundary values complex dilatation

<.

<*~

in the

induced by a mapping with

The number

k* = H<*ll. is~

of oourse~

completely determined

not be unique. class.

K s - l+ke l_k~

The lectures

extremal

K's

~

need

is the smallest maximal dilatation

in the

by means of an equivalent

i.

THE MAIN

INEQUALITY

Introduction.

information

about the complex dilatations of

U

on the next page.

onto

U

that the boundary values of

closed subset

S

of

3U = {Izl

ideas we will restrict

If l~(z)l d x d y

< ~.

(M)

provides

of two quasiconformal

= i}.

f

and

~

under

agree on a

In order to focus on the main

the proof to the case

theorem then states that

The inequality

(and of the inverse mappings)

the assumption

with

problem for anal~tie

In this section we will derive the fundamental appearing

~

extremal

of

(M)

(M)

f,

even though

interest.

inequality

mappings

<,

that follow will discuss the c h a r a c t e r i z a t i o n

functions which also has independent

If,

by

holds for any

S = {IzI r

= i}.

The

holomorphic

There is no loss of generality

in

U,

in

U

restricting

r

to be analytic

for

{Izl ~ i}

and to have at most

ii0

/

I-M II

,-I i!

%M ,-% 9

N

.I~

'0

0

0 H

m

N -e-

--

|1

V

O 4-'

U

O

rn

gl O K'-I

>h.J --~ (3 144 Z

§

0 .H 4~

O

4~ 9H

N

~

N

,-,

V

)4

9

N

N

H v

S

~

9,-I

m

~L

O C_)

.,

'-."

8

~ 0

N

0

--

~

..~

-,-I

I

||

~

v

N

vii 9H

N

~

,,., II

,-4

'~

hl

~

"0-

..

~H ii

N V

QI

~4

,-4

'

11

II

II

v

N

~

N

D

0 H

v 7

~

0

i!i

simple zeros.

The general result follows by a simple approximation

procedure. 2.

Trajectories

trajectories r

and orthogonal

Differentials

traOectories

in the Disk.

of holomorphie

play an important role in the interpretation

A trajectory of differential orthogonal i.e.

of Quadratic

r

is an arc in

displacement

on which

functions

and proof of

r

2 ~ 0

in the direction of the arc).

trajectory of

a trajectory of

U

The so-called

~

-~.

is an arc in Conceptually,

U

on which

the simplest

(dz

(M). is a

An r

2 ~ O;

situation

is that in which

r

= f$s

is both single-valued

de

and univalent.

In that ease,

trajectories

orthogonal

trajectories

are merely inverse images under

horizontal

and vertical

lines,

this situation continues point

z0

for which

three trajectories

r

= z

~(z O) ~ O.

for

(3 strip

domains )

of

In the general ease,

to hold locally in the neighborhood

meet at

trajectories

respectively.

9 -I

z0

If

r

of any

has a simple zero at

(under equal angles).

trajectories with

and

for a

2 simple

r

zeroes

(4 strip domains)

Zo,

112

trajectories for where right.

~I(U)

r

: r!

2

i8 region shown at

(5 strip domains).

The heavily drawn trajectories are the

pre-images of the horizontal dashed lines, The following decomposition

r

is possible [17]. Up to a n set of Lebes~ue 2-dimensional measure O, U : U Zk' where {Zk} k-i are disjoint simply connected "strip" domains. Each Zk is swept out by a family of trajectories single-valued

is horizontally

#k(Zk )

the intersection

convex,

many

Zk

U

r

Ck(Z)

Zk

= f/~Tdz.

there exists Each region

i.e. if a horizontal

r

is holomorphic

so that countably many,

can occur.

and in each

line intersects

consists of a single open interval.

it is merely assumed that {Izl ~ i},

of

schlicht branch of

@k(Zk )

of

of

under the mapping

for

{Izl < i},

(In [17] instead

instead of merely finitely

Actually,

in our use of the strip domains the

advantage of limiting ourselves

to finitely many is purely didactic.)

The above sketches

indicate

several possible configurations.

The

strip domains are bounded by the heavily drawn trajectories. 3. case

Proof of (M). S = {Izl = i}.

r = ~,2 ~(U)

We now proceed with the proof of

where

is schlicht and holomorphic

convex.

unnecessary.

for

Izl ~ 1

and

This occurs when there is merely a single

strip domain for the orthogonal the discussion,

for the

To clarify the ideas we will first assume that

#(z)

is vertically

(M)

trajectories

the concept of trajectory

of

r

For this part of

is quite trivial and

113

We introduce and

~

domain of

{I~I <

l}

planes as indicated in the diagram below. ~

and the range of

The holomorphic function

~

fl

applied twice~

//~ I

~

,..

:

~

once to

1},

and the

Note that the

have been changed to

{I~I < i}.

is to be thought of as "arbitrary",

satisfying the assumptions mentioned above.

~I

{Izl<

as anothem copy of

{Izl< i},

The same function

and once to

~

{I~I < i}.

E:~ .:~~Y~<~,..~" ~ ./ 4-

~g<'/

~L

"~'

'"''"

V /.<

~

~

lwl
9

Hypothesis:

fCe i8) = ~ceiS), O<

e

<

2~r.

=

is schlicht

Let

F~

be a vertical segment forming a cross cut of ~(U), with

fixed abcissa

~.

By means of the given mappings

f,

fl ~

~,

we

is

i14

trace the image of

F~

as indicated in the diagram through the

w,

~,

~

have the same endpoints.

z,

~

planes arriving at ~ = ~ o fl 0 f(F~). In view of the hypothesis f(e i@) ~ ~. rte i0 ), 0 ~ e < 2~ it is clear that F~ and Hence

length F~ ~ length ~ .

(9)

Inequality

(M)

is an integrated form of (9),

to an application of Schwarz's inequality. d~ : idn = ~'(z)dz, d

somewhat weakened due

Namely,

dw : p(z)dz + q(z)dz,

writing

d~ : Pl(W)dw + ql(W)dw,

: #' (z)dz,

inequality (9) takes the form

Ir~

(i0)

dn < =

Ir~

~

.

I

I

(plp + ql~) _ (plq + qlg ) ~'(z___~) dn. ~'(z)

Using the area expression Sf d~d~ = Area ~(U) = SfIr ~(u)

and i n t e g r a t i n g

(11)

both sides

~/< I~, 1

u

of (10) with respect

f f l ~ ( z ) l dxdy ~ SSI~'(DI. u

Now

~ dxdy = ffi~(z) I dxdy,

u

I '(z l 9 ICplp+ql

: ~ea

-

I

u

I,~'('~)l ~ ~d~'

to

~(U~

:

ffl~(z)

dxdy,

~,

we o b t a i n

(Plq§ also.

I~(z)l

dxdy

To take

U

advantage of this relation one squares (ii) and applies Schwarz's inequality appropriately to the right side of (ii).

(12)

<

dx dy,

u il~II =

ffi~Cz)[ dxdy. U

The result is

(IP112- Iqll2)(Ipl'-

Iql ~')

Next it is necessary to express the coefficient

115

of

Ir

in the integrand of (12) in terms of the various complex

dilatations.

This yields the expression on page

We must next consider the situation when U :

U Zk , k=l

n > i.

Instead of again using the

C

and

~

planes it is conceptually

better to introduce the Teichm~ller metrics lr189

in the disks

The place of {Iz] < 1},

r~

the number

r

F~ = Ck(y~)

and

and

{I~] < i}

respectively.

is taken by the orthogonal trajectory ~

the trajectory normal to merely

{Izl < i}

lr189

y~.

(In the case of a single strip,

is a vertical segment as before.) ~

in

indicating a real parameter measured along

In the case of several strips,

the same endpoints as

y~,

= fl o f(y~).

if

y~

y~ ( E k,

Evidently

y~

is

then has

The place of (9) is taken by

the inequality

f

(13)

Inequality

Ir189

(13) is trivially equivalent to (9) when

same strip domain if

y~

~ ~/Ir189

Ek

as

y~,

but it holds,

extends into other strip domains,

[17] that trajectories

is in the

more generally,

even

in view of the known fact

of functions holomorphic

global geodesics in the Teichm~ller metric.

y~

in the disk are

(This fact is quite

intuitive if one looks at some typical pictures, as those on page .) Zk

Integrating with respect to

is covered,

over

{Izl < I}

;fJr Zk

dxdy~

~

so that a single strip domain

then transforming the integrals so that they extend we obtain,

fflr189

by analogy with (Ii),

]r

r 89I(plp+qlq) - (plq+qlp) [r

Idxdy'

Zk

k = 1,2,-.-,n

116

Summing over

k,

This completes

we come back to (ii) and again obtain

the proof of (M) when

In the case when reflection of

y~

shows that

S

is a proper closed

(13) continues

lie on a c o m p l e m e n t a r y

the requirement

that

4.

Mappings.

preliminary

r

2

U

subset of

aU

a little

S, provided that c o m p l e m e n t a r y

trajectory of

r

This leads to

be real on the c o m p l e m e n t a r y

arcs.

We conclude this section with a

application of (M) by proving a known

the so-called Teichm~ller mappings. onto

= I}.

to hold even if the endpoints

arc of

arc is a trajectory or orthogonal

Teichm~ller

S = {Izl

(12) and (M).

A mapping

is called a Teichm~ller mapping

[17] result about

w = f(z),

of

U

if it is a quasiconformal

mapping with complex dilatation (14)

K(Z)

: k I~(z)l

The function

r

is a constant, r

f

will be assumed 0 ~ k < 1.

Except

is locally of the form

conformal,

and

is a globally (in function

$(z)

A k = z + k~ {lwl < 1})

to be holomorphic

in

in the neighborhood

~-l o A k o #, is affine.

where

U,

and

of a zero of

~,z = r

The function

k

W

is

~(w) = W'(~ 2

single valued holomor~hic function induced by the

and the value of

k, (0 < k < 1),[17];

f-l(w)

has complex

dilatation ~(w) : -k ~ . l~<W)l The norms

IIr

simultaneously

= iI~ /~~ _ r ~

dxdy,

finite or infinite,

II~II =

KII~II,

and

~(w),

lwl < I,

are single-valued

respectively.)

•

l~(w)l du

dv

are

and are connected by the relation l+k K -- y : ~

(All these facts are immediately obvious also

j

li~ll = I I 1

.

in case

schlicht functions

r for

and hence Izl
117

THEOREM. K(Z)

[17]

Suppose f(z)

is a Telchm~ller map with complex dilatation

il~ll < ~.

= k~(z) / l~(z)l,

Then

f

is the unique extremal

mapping for its boundary values. Proof.

Apply

inequality

(M)

(M) a c c o r d i n g

ko--r~/lo(z)l

~(z)

~(z)

e(w)

-k~--C~"/ I~,(~o) I

K(z)

K~(z)

= complex

extremal map !

table.

Now

<(z)

K

to the f o l l o w i n g

(,,,).

dilatation

f*,

Kl(~)

= complex

f,-l.

(Hence

of an

ll<*ll= = k*

dilatation

l]tC].Iloo

of

= k*.)

We o b t a i n

< I;li(lk)'1_k.1 K'("

dy< i k liCl§

i - i<,(~)i2

U

<

l - k l+k ~ 11011.

=

l+k

U

dxdy<

(i-:< (w)l2)

l_k ~

Henc e,

1 + k < 1k_____~ + ~ _1- - ~ = 1 _ k~ 9

i.e.

f

k = k~

is e x t r e m a l , one f i n d s

.'. k ~ k ~.

Working

backwards

.'.k = k ~,

f r o m the a s s u m p t i o n

that

that

K1(w) : - k

~-rjY

l~(w) l so t h a t

5.

f~ = f.

Affine

Stretches,

m e a n s of the a f f i n e

Let stretch

s

be a s i m p l y c o n n e c t e d

region.

By

118

w = FK(Z)

= Kx + iy

is m a p p e d onto a r e g i o n mapping,

w i t h domain

the T e i c h m ~ l l e r m a p p i n g and

~

~,

(K > 1),

~K"

FK(Z)

and range

and only if

(Clearly f

is itself a T e i c h m ~ l l e r ~K'

or it can be related to

f = ~-i o F K o ~

are conformal m a p p i n g s of

respectively.

z = x + iy,

FK

U

of

U

onto

onto

~

and

U,

where

~K

is extremal for its b o u n d a r y values

is extremal for its b o u n d a r y values.)

if

Later on

~ e e t i o n V.I) we will o b t a i n a n e c e s s a r y and sufficient c o n d i t i o n for

~

to have the p r o p e r t y that

FK(Z),

z E ~,

m a p p i n g for the b o u n d a r y values it induces. the f o l l o w i n g sufficient c o n d i t i o n

At this point we state

[16] which is a c o r o l l a r y of

Strebel's theorem of the last section: z E ~

is an extremal

If

Area ~ < ~

then

FK(Z),

i__ssu n i q u e l y extremal for its b o u n d a r y values. It is an open q u e s t i o n w h e t h e r or not every q u a s i s y m m e t r i c

boundary correspondence between a T e i e h m ~ l l e r mapping.

~U

and

itself can be r e a l i z e d by

It is d e f i n i t e l y known that not every

q u a s i s y m m e t r i c b o u n d a r y c o r r e s p o n d e n c e can be r e a l i z e d by a m a p p i n g where the known

$

of

(14) has finite norm

[6,18,12,13]

area for w h i c h

FK

II$II- This follows from

examples of simply c o n n e c t e d

~'s

of infinite

is u n i q u e l y extremal.

In the theory of q u a s i c o n f o r m a l m a p p i n g there is a general "principle" that if one looks for extremal m a p p i n g s having minimal m a x i m a l dilatation)

(i.e. m a p p i n g s

subject to specific side

c o n d i t i o n s then the extremal m a p p i n g will be a T e i c h m ~ l l e r mapping. For our problem, values,

w h e r e the side conditions are the set of b o u n d a r y

the principle

is false in the sense that one knows of

examples w h e r e there are extremal m a p p i n g s unique)

that are not T e i c h m ~ l l e r mappings.

by finding simply c o n n e c t e d regions

~

(however they are nonThe examples are o b t a i n e d

[18,12]

for w h i c h

FK

is

119

extremal,

but not uniquely extremal.

Such

fl,

of course,

must

necessarily have infinite area.

Ill.

i.

THE NECESSARY CONDITION FOR EXTREMALITY,

An Auxiliary Inequality for Mappings Agreeing on a Subset of

LEMMA.

f(z)

Let

K(z) = f /f .

be a quasiconformal mapping of

Let

z

S

be a closed subset of

fs(Z)

exists a Teichm~ller mapping

ks

-[-~-~"

such that

llCslI < ~,

fs(e i8) = f(e i8)

and such that ft

(is)

II%ll = ]]l%(~)l~dy U

Proof.

~U.

U,

Suppose there

= real for

Then

1 + K(z) Cs!z!

2

1-ks

I%~z~l

l-I
l+ks

z E ~U\S,

e i8 ( S.

whenever

~ ]]l%(z)l

onto

with complex dilatation

Cs(Z)dz2

f@

U

~U.

~

~

dy.

U

First apply inequality (M) according to the following

table.

(M)

Now

f $ (z)

f(z)

K(z)

ks ],s(z) 1

~(w)

-k s

~(z)

f(z)

<,(w)

R(w)

r

(0s(W) is induced by as in II.4)

Cs(Z)

This yields the following

(16)

L%

I%(z)ldxdy <__ I%Cz)l 1-k~ U

1-

dx

I~Cw)l ~

s

U

Next interchange the roles of replacing

0s(Z)

K(w)

by

<(z)~

w

~s(W)

and by

z -r

in

(16). and

This results in Cs(Z)

by

-~s(W).

120

Since

$ (w) = F o r (z) l+k s s s we have Ks = l-k '

locally,

F = horizontal

stretch by factor

s

(17)

[r

Idu dv

Since

Cs(Z)dz2

holomorphic r

l+k _ l-k s iCs(Z)[d x d y , s

is real on

on

aU\S,

and

~U~S

it follows

Cs(Z)

exists

is real or imaginary.

corresponding {lwl

= I}

points;

corresponding

This fact establishes Carrying

it out,

Inequality

Re

i.e.

Hence

Cs(W)dw2

the validity

locally

{Izl

= l}\S

+

for

1-

under

S

dilatation

IZkl = i:


k = 1,2, ....,n,

(15).

therefore,

I<(z) lz lCs(Z) I dxdy => m-ks

Zk~k+l ,

mapping

fact,

f (z) n

values

fn(Zk)

suppressed of

Zk,

I~nCZ)l, llCnll< ~, Cn(Z)dz~

is the extremal = f(zk),

the fact that

k = 1,2,...,n,

Summarizing

such that = real

Zn+ 1 = Zl) , mapping

of

k = 1,2,...,n. fn'


not just on

those aspects

kn'

fn'

points

with

on each open arc

onto

U

r

[18].

we have

depend on the values

n.)

of the above

(In

with the given

In our notation and

n

fn(Zk) : f(zk),

k = 1,2,...,n. U

fs(Z) required

namely a set of

There exists a Teichn~dller mapping

( a r g z k < argzk+l,

JlCsll"

s

for which the T e t h e r

and such that

fs(Z).

U 1 - IK(z) I~

in the hypothesis of the Lsmsa is guaranteed to exist, {Zl,Z2,-..,Zn},

at

process.

U There is a speelal set

~i(w)dw

to read as follows.

U

of the Lemma,

is

there and

of the interchange

(15) can be rewritten

l<(z)I 2 d x d y

r

= real on the arcs of

and making use of (17) gives

Under the hypotheses

1-

that

the same holds

to those of

/Jl - l<(z)[2 dxdy = ~ u

(18)

(w = fs(Z)).

facts that will prove

121

relevant

later:

U,

= f_/f . z z

K(Z)

IZkI = i. U,

Let

f(z)

Let

be a quasiconformal

{Zk},

k = 1,2~...~n

There exists a quasiconformal

with complex dilatation

such that

fn(Zk)


= f(zk),

onto

fn(Z)

of

= knCn-~--~/ ICn(Z)l,

U

onto

IICnll < =,

and

[<(z)12 kn 1 - I<(z>l = [~n(Z)ldxdY~l---------~ II~nll"

(19) U

2.

U

be a set of points,

mapping

k = 1,2,-..,n,

K(Z)r

mapping of

U

The Functionals

ILK], QEK]

a given complex dilatation the two n o n - n e g a t i v e

I[K] =

and the Necessary

<(z),

numbers

Iz[ < i,

I[K],

Q[K],

sup

Condition.

IIKII= = k < 1

For

we define

as follows.

dx dy

II~ll<=l

1-

[K(z)] 2

U

Q[K] =

sup If II~II<_I l - l]K(z)]2 < ( z ) l ~ l~(z)Idxdy. U

The allowable

I1~11

:

functions

r

in each case are holomomphic

in

U,

f f [ ~ ( z ) l dxdy. U

By (19) we obtain k n l-k

n

On the other h a n d ,

< IlK] + Q[K]. =

trivially

I[K] < =

k l-kn

(20)

Q[K] !

k l_k

2 '

-

k2 l_k

. Hence, ~

k =< I[<] + Q[<] =< l-k " n

As in 1.5 we now introduce the extremal <(z);

i.e.,

extremal

k ~ : II<~ll~, where

quasiconformal

<~(z)

dilatation

k~

determined

is the dilatation of an

mapping with the same boundary values as

by

122

corresponding

to

K(z).

Let the number of points

zk

increase to

in such a manner that

max Izi+ 1 - zil § 0. The mappings f l~i~n n can be extended to the whole complex plane via reflection. Since {fn }

is a normal family~

converging implies

there exists a subsequence

uniformly on

that

{k n}

{Izl ~ i}

to a mapping

Then

f0

sequence, l+k 0 is l_k0

the same boundary values as

f.

In view of (20),

i+~ k ni = k 0 lim

{fn },

exists.

and hence also

values of and

k

f.

by

THEOREM.

z

k *.

Thus

k*

k 0 ~ k ~,

f

II<*ll.

i.e.

quasiconformal

and has

k 0 ~ k.

Now

using only the boundary

k 0 = k*,

is a quasiconformal If

<*(z)

and

f0

K

by

is an

mapping of

U

onto

U,

then 9

The above now yields the following necessary c o n d i t i o n for to be extremal

K*

is eztremal for the same boundary

k* k l-k* =< I[K] + q[<] =< ~

(21)

we can also assume that

(20) remains valid if we replace

IIKII = k.

=

are determined

(20)

We have proved the following.

Suppose

= K(z),

values,

f0

Therefore

extremal mapping.

f~/f

is a bounded

{fn.} l f0(z). Since

(i.e. for

f(z)

to be extremal

K(Z)

for its boundary

values). THEOREM.

If

(22)

i8 eztremal

<(z)

I[K] :

Proof. immediately I[~]

-

Since

then

k l-k 2

k = k*

we have equality

in (21).

This

implies k l_k-----f,

q[<]

_

k2 l_k z 9

Later on we will see that this same condition but also sufficient.

As we will also point out(w

is not only necessary, condition

(22) is

123

equivalent to the known n e c e s s a r y c o n d i t i o n of R.S. H a m i l t o n

3.

Example - A f f i n e Stretch of A n g u l a r Domains.

formal mapping f(z), domain

~,

<(z) = %/fz'

[9].

If the given quasicon-

is defined on an arbii-Pary simply connected

the question of whether

f

is extmenml for its boundary values can be

reduced to a mapping defined on the disk by means of an auxiliary conformal mapping.

Equivalently,

d i r e c t l y for

~.

all the p r e c e d i n g work can be carried out

Since none of our w o r k a c t u a l l y m a d e use of any

special properties of the disk,

one translates

each result for

extremal m a p p i n g s of the disk into one for extremal mappings of by m e r e l y f o r m a l l y r e p l a c i n g

U

by

~.

(More generally,

~

need

not to be simply c o n n e c t e d or even schlicht if the n e c e s s a r y facts about t r a j e c t o r i e s of quadratic d i f f e r e n t i a l s can be generalized. See [19].)

Thus,

if

~l (~(z) with

1-

holomorphic In[K],

in

Q~Er].

I<(z)l ~

e,

Helle =

IIr f/lr

I~:(z)l ~

dy),

In the special case that

then (21),(22) hold f(z)

is the affine

mapping f(z) = FK(Z)

one gets

I~[K] -

= Kx + iy = K+I 2

k sup 1-k 2 II~11~1

K-I k : k-~'

(z + k~),

Iff~(z)

dxdyl,

K > i

and c o n d i t i o n

(z ( ~ )

(22),

w r i t t e n in n o n - n o r m a l i z e d form becomes

ff(fz) (23)

dx dy

sup ~ r holom, ins ffIr

That is,

= i. dy

as a result of our last t h e o r e m we know that (23) is a

n e c e s s a r y c o n d i t i o n on

~

for the affine m a p p i n g

FK(Z)

of

~

to

be extremal for its b o u n d a r y values. Let

fla

be the angular region

0 < arg z < m,

(0 < a < 2w).

124

As a n i l l u s t r a t i o n

of the application

we w i l l p r o v e that

(23) does not h o l d for

that the a f f i n e v a l u es. [16].)

s t r e t c h of

~

(This was o r i g i n a l l y Let

A : ffr n

E = {w dxdy

of the n e c e s s a r y ~ ,

and thus c o n c l u d e

is not e x t r e m a l

for its b o u n d a r y

p r o v e d by an e x t r e m a l

I 0 < Im w < ~}.

condition,

length method

Then f(w)

: ff e - 2 i V f ( w ) d u d v , E

= e2Wr

(w : u + iv).

and

B = fflr

dxdy

fflf(w)ldudv

=

=

E Since

B < ~,

is a n a l y t i c in

f_|

f~dvf 0

du

exists

lf(u§

du.

-~

for a.a.

v,

and since

E, co

f ( u + i v ) du =

c

=

const

c e -iS s i n e ,

B ~

for

a.a.

Hence, A = c~e

- 2 i v dv

=

Icl~.

Therefore,

[ABI <

sinai

that is

sup r helem.in

~'fr (z) dxdy I ~a

ffI~(z)[ dx dy

in

<

~

<

i.

v.

f(w)

125

(Note:

is actually the exact value of the

The quantity

supremum.)

4.

Example of an Argument

Based on Normality.

Another type of

argument that can sometimes be used to prove that a particular function

m(z)

is not extremal

{r I r holomorphic

in

U,

pose

IK(z)I ~ k

for

Then

<(z)

is based on the fact that

IIr ~ i}

is a compact normal family.

0 ~ Izl < p,

and

is not extremal unless

Proof.

If

<(z)

~(z) = k

<(z) ~ k,

is extremal then,

sequence of functions

Cn(Z),

p < Izl < i.

Izl < i.

by (22),

holomorphic

for

Sup-

for

there exists a

Izl < i, ll~nll ~ i,

such that

1 -I~(z)l We can assume that

~

1-k~ o<]zl
lim~n(Z) n§

: ~0(z)

1-k~

locally uniformly for

Izl < i.

Therefore

fj

f;

Izl<,~

I~.l
Now~

(25)

<•z

Cn(Z) dxdy : ~(l-p2)r

for

n : 1,2,..',

and also for n: 0.

~II
ff

~n(Z) d x d y

--~

~
f/

r

dxdy.

By (24),

therefore,

p
~(z)

dxdy

1 - TJ .)I 2

k l_k 2

I zl
ll~oll=

1

and

K(Z) : k

I~o(~)1

c,

Icl

= 1.

126

r

Since

is holomorphic

for

Izl < i,

-F~TTFF o : l ,

r

and

p < i:l

< l,

is a constant.

Note:

The above type of argument

hypotheses.

The main technical

escape of the mass

IV,

problem

fen(Z) I d x d y

above by the identity

can be adapted

to more complicated

is to be able to control

to the boundary.

the

This was achieved

(25).

PROOFTHAT THE NECESSARY CONDITION IS ALSO SUFFICIENT,

!.

Another Auxiliary

(M)

Inequality.

(M) as follows

Now

f(z)

f(z)

K(z)

f_/f Z

S

Z

{Izl : l}

~(z)

f*(z)

r

arbitrary

Since

We apply

= extremalrnappingwithsamebouundar~values

IK(z)l = l~(w) l

following

apparently

inequality

functiOn

holomorphic

at the point

weaker

to the numerator

in

w = f(z),

inequality

U,

Izl
< ~

the triangle

1 - Ir Z (~)l 2 9

<=

dxdy,

1 - i<(z)l2

f(z)

we obtain the

by applying

above the term

l}r

as

( w = f(z) ).

1 - I%(~)I 2

f~l
is an extremal

mapping

il
and

f!(w)

the inverse mapping,

127

Ii - K(z) ~

I f IzI
dxdy < l+k*

lr

= l_k ~

2

I~,(z) l, ~ d y ,

If I~(z)l Iz1<1

1-

l,~(z)[ ~

or equivalently,

Re

[f K(Z)(~(Z) dxdy< k* !I< lr idxdy+ II izJI
= l+k*l

In v i e w of the a r b i t r a r i n e s s of

Izl
1

-

I~(z)l ~ [r I~(:)1 ~

I dxdy.

r

<(~.)~(z)c~dyI < #

I<(z)l'

I--I
Ir

dx dy

17.1<1 < ~

k*

IN, II + IN, II QZK].

Thus, k~

(26)

Note That i n e q u a l i t y (21), possible T w o - s i d e d estimates of

T o g e t h e r w i t h i n e q u a l i t y (26) make k*

in terms of the given f u n c t i o n

K(z). 2.

The C h a r a c t e r i z a t i o n of Extremal Dilatations.

that The c o n d i t i o n

We can now show

(22) w h i c h was shown to be n e c e s s a r y is also

sufficient:

THEOREM.

Let

K(Z) = f~/fz'

f

f(z)

be a quasiconformal

ll~II~ = k.

A necessary

mapping of

and sufficient

is an extremal mapping for its boundary

I[~] - k l_k 2 '

values

U

onto

condition

is that

U, that

128

By ( 2 6 ) ,

Proof of sufficiency. k

:

I[K]

ke

<

l_k 2

k 2

+ Q[<]

= l+k~

l+k ~

l-k ~

Therefore, k

k9

<

l+k Hence, x(z) 3.

k < k ~.

l+k ~

But since

k~

is the extremal value,

k ~ = k,

i.e.

is extremal. Alternative

Hamilton's

Characterization

Functional.

functional

of Extremal Dilatations

R.S. Hamilton

[9] considered

of the complex dilatation

<

by Means of

the following

(when specialized

to the

case of a disk): H[<] =

sup III holom, in U IN)ll <__ 1 u

<(Z) r

dx dY I 9

Hamilton proved that the condition

(27)

H[m]

= k :

is necessary for extremality

]l~il=

of

<.

that (27) is necessary and sufficient Proof. gn(Z),

(28)

If

HI<] = k

holomorphic

in

U,

By our last theorem it follows for extremality.

then there exists a sequence of functions such that

I If ~(z)gn(z) dxdyl

§ ll~II~,

(IIgnll : 1).

u

A little thought

(29)

shows that (28) implies that

ff Ign(Z)idxdy I~(=) I~k'

-)" 0

129

for every

k'

0 < k'

II

(30)

< k,

and

~:(z) 1-

(28)

and

gn(Z) d x d y

imply that

(28)

I§

{K(z){ 2

,,K,,o 2 1-

-

}l~({}.

k l-k ~ "

U

Therefore I[<]

:

k

l_k 2

Conversely, llgnll = i, holds,

4.

if

I[<3

such that

and

H[~]

k there exists a sequence {gn(Z)}, l_k 2 ' (30), and hence also (2g) hold. T h e r e f o r e (28) :

k.

:

Direct I n t e r p r e t a t i o n of the Functional

can be interpreted as a derivative.

Suppose

is a complex d i l a t a t i o n d e p e n d i n g on t

§

(31)

H.

t

The functional

K(z,t),

t > 0,

H z E U,

in the f o l l o w i n g m a n n e r as

O:

KCz,t)

Let

K~(z,t)

and

kS(t)

= t~(Z)

+ ~(Z,t),

(32)

If

Substitute

H[~]

= 0

V,

FURTHER

1.

A f f i n e Mappings.

K(z,t),

Then

kS(0)

Comment.

l]~ll. < "

be an extremal d i l a t a t i o n c o r r e s ~ o n d i n ~ to

= IIK~(-,t)II .

Proof.

ll~(',t)ll~ = o ( t ) ,

= lim k~(t) t§ t t>O

- HI9].

431) into (21) and

then (32)

426).

implies that

k*(t)

: o(t).

EXAMPLES

Refering to III.3 and IV.2,

that a__nnaffine stretch of a simply c o n n e c t e d r e g i o n for its b o u n d a r y values if and only i f

423) holds.

we can now state ~

is extremal

W h e t h e r or not

130

the affine

stretch

is extremal

is independent

of the dilatation

K

and of the direction of stretch. We illustrate

the condition by a brief new proof of the fact

[16] that an affine 7 is extremal Proof

!

stretch of = { z ]Re z >

0,

0 < Im z < i}

for its boundary values. Let

9

f (z) = -i- e-Z/n n n

,

z

E

X

One gets

I{? lim n§

: i.

fflfn(Z)Idxdy E,

2.

An Extremal

be a measurable For functions [Ig[[ =

ff

Problem for Functions Analytic complex-valued

g(z)

function of

holomorphic

Ig(z)I d x d y

< ~,

in

z,

{Iz] < i},

consider

in the Disk. ]z] < i,

Let

v(z)

0 < []v[[~ < ~.

with

the question of whether the

Izl
ff~(z)g(z) H[v]

= sup

dxdy

[ z
f / I g ( z ) l dxdy

g

Izl
I[~[]~ or is less than

]]91[~

From the point of view of analytic to depend~ arg g(z)

roughly to

speaking~

- argg(z)

is not very small.

functions,

on the degree of a p p r o x i m a t i o n

in the neighborhood

For some

the answer

of points where

in view of the results of IV.2 or IV.3 a complete

priori

possible by means of quasiconfommal

]g(z)]

h~(z)

However,

answer is, mappings~

seem to have nothing to do with the problem;

H[~] = l[~I[~ if and only i f

of

~Ts this appears to be a difficult

question to resolve by purely complex analytic methods.

principle~

seems

is an extremal

in whieh a-

namely, complex dilatation

131

for any or every constant possibilities THEOREM. U

h,

with the following U = {Izl < i},

Let

= U n {Re z < 0}.

Proof.

We illustrate

the

example:

U+ : U n {Re z > 0},

Then

sup g holom.in U

(33)

1

0 < lhl <

< i. fflg(z)l d x d y U

The proof is provided by showing that the complex

dilatation

Z(Z)

is not extremal

: I +k,

z ( U+

-k,

Z ( U_

for any choice of the constant

k,

0 < k < i.

suffices to choose a single value for

k.)

quasieonfor~nal m a p p i n g

has complex dilatation

F(x+iy)

The image

F(U)

F(z),

z E U,

The following

x+iy ~ U+

Kx + iy, x + iy,

=

CK= x+iy

(It

~:

l+k

yz~)"

E U

is the union of the two half-ellipses

F(U+),

and

F(U ). We will now indicate how a ~ - q u a s i c o n f o r m a l mapping

U

onto

F(U),

~ ( e i8) = F(eiS),

can be constructed,

such that

0 < 8 < 2~,

~ < K.

In the figure on the page following,

=

= i,

A2

i A7 = -- + 2" points

0',

O

is the point

~ -i,

A 3 = +I,

For

j = 1,2,3,6,7

A~,

A~

~(z),

with

(34)

A1

mapping

A4 =

1 + 89

we take

i

A 5 = -~ + 7' = F(Aj) A.' ]

are still to be specified;

z = 0,

~

i

A8 = - -2- + = ~(Aj).

The

they will be the

132

At

A2

A3

"{ I~I < i)

\ Ai

A6

lw

F(z)

A;

A;

I

A2

K--2

133

images of define ~(z)

O,

~(z)

A4,

A5

under

~.

We will explain,

in the upper half of

U.

will be defined by symmetry:

A3 = (O'A4'AI)'

~(~) =

~(z).

j = 1,2,3,4,

be determined as soon as imagine that

A~

[AI,A~].

unprimed quantities.) O'

triangle

moving

A~

w = 0

onto the triangle

A~

A~

are specified. [AI,A~],

(AI,A~,O'),

A~

onto the triangle K.

(A~,AI,O')

We next perturb

Im A~ > ~, F(z)

sufficiently

and the affine map of the

A 4'

will have A 5'

and

slightly downward along the line

The function

and

then the affine map of the triangle

slightly upward along the line

(35)

At first

A computation which we leave to the reader

dilatations less than by moving

These four affine mappings will

is a point on the real w-axis,

(A3,AI,O)

z E Aj,

(This is precisely the case with the

close and to the right of (AI,A2,0)

A~,

~(z),

is the midpoint of the segment

the midpoint of

shows that if

0',

The figure has

g 2 = (0,AI,As) ,

we will define

as an affine mapping.

U,

K = 2.

A 1 = (O,A5,A2) ,

A4 = (O'A3'A4)

how to

In the lower half of

been carefully drawn to scale for the case In each of the triangles

below,

slightly,

[AI,A~]~

[AI,A~];

and by

i.e.

Im A' 1 4 < ~"

is now determined in

Aj,

j = 1,2,8,4,

!

and it is clear that if the perturbations of

A4

!

and

A5

are

sufficiently small then the dilatations of the four affine mappings will be less that Let

K.

Ol,...,o4,

circular arc,

each bounded by two straight segments and one

be the regions shown in the diagram:

Say,

aj(y) ~ x ~ bj(y) o j:

The values

j = 1~o,,~4.

[

cj ~ y ~ dj

~(ajCy) + iy)

the boundary values of

and

~(z)

~(bjCy) + iy),

cj ~ y ~ dj,

on the left and right sides of

that is oj

are

134

already determined We now require of

x

by (34) and the d e t e r m i n a t i o n

that

~(x

for each fixed

defines

~(z)

homeomorphism

+ iy)

y,

everywhere satisfying

~(z)

U A.. J=l 3 linear function

be a c o m p l e x - v a l u e d

x + iy in

of

U,

E Aj,

j = 1,.--,4.

on

This now

and it is clear that

~(z)

is a

(34).

It remains to verify that ~(z) has 4 maximal dilatation < K in U A.. This is done by finding first j=l ] the d i l a t a t i o n at each point, by means of formula (i), and then the supremum of the dilatation occurs on the boundary of

Aj).

lengthy but straightfoward, here that condition Note.

a triangle

A. ]

is,

simple ~eometrie

construction,

To calculate

traingle

A:(A,B,C)

(A',B',C")

Construct

is rather

below;

it is

into play.

onto

K0

(A,B,C),

of the affine mapping of Solution.

Find

K0 + 1

is the perpendicular

distance distance

C"

so

and similarly oriented. bisector of

Then

K0 - 1

by a

as follows.

A':(A',B',C').

A'B'

of

just a matter of solving a system of

the dilatation

is similar to

P' so that

which

The d i l a t a t i o n can also be calculated

Problem.

that

The computation,

always

of the dilatation of the affine mappings

of course,

two linear equations.

as may be expected,

will not be r e p r o d u c e d

(35) comes

The c o m p u t a t i o n

(which,

from from

C' C'

to to

C" P'

p,

i

C

C"

P'C".

135

ADDs

After the preceding

variable" method for obtaining

lectures were delivered a "complex theorems

like (SS) was developed,

but even with this new method the proof is still non-trivial; fact,

the method contains hidden elements of quasiconformal

in mappings.

REFERENCES

i.

L. Ahlfors

2.

Lars V. Ah!fors, Sufficient conditions for quasi-conformal extension, Proc. 1973 Conference on Discontinuous Groups and Riemann Surfaces, U. Maryland, Annals of Mathematics Studies 79, Princeton, 1974, 23-29.

3.

Lipman Bers, Extremal quasiconformal mappings, Advances in the Theory of Riemann Surfaces, Proceedings of the 1969 Stony Brook Conference, Annals of Mathematics Studies 66, Princeton, 1971, 27-52.

4.

A. Beurling and L. Ahlfors,

5.

L. Bieberbach, Theorie der gewShnlichen Differentialgleichungen, second edition, Springer Verlag, 1965.

6.

Eugen Blum, Die Extremalit~t gewisser TeichmSllerscher Abbildungen des Einheitskreises, Comment. Math. Helv. (1969), 319-340.

and G. Weill, A uniqueness theorem for Beltrami equations, Proc. Amer. Math. Soc. 13 (1962), 975-978.

The boundary correspondence under quasiconformal mappings, Acta Math. 96 (1956), 125-142.

44

7.

Lennart Carleson, The extension problem for quasiconformal mappings, Contributions to Analysis, A collection of Papers Dedicated to Lipman Bers, Academic Press, New York, pp. 39-47.

8.

P.L. Duren and O. Lehto, Schwarzian derivatives and homeomorphic extensions, Ann. Acad. Sci. Fenn. A I. 477 (1970), I-Ii.

9.

R.S. Hamilton,

10.

S.L. Krushkal', Extremal quasiconformal mappings, Sibirsk. Mat. Zh. 10 (1969), 573-583; Siberian Math. J. I0 (1969), 411-418.

ii.

O. Lehto and K.I. Virtanen, Quasiconformal second edition, Springer Verlag, 1973.

12.

Edgar Riech and Kurt Strebel, On the extremality of certain Teichm~ller mappings, Comment. Math. Helv. 45 (1970), 353-362.

13.

Edgar Reich,

Extremal quasiconformal mappings with prescribed boundary values, Trans. Amer. Math. Soc. 138 (1969), 399-406.

Mappings

On extremality and unique extremality of affine mappings, Proc. Colloqu. on Mathematical Analysis, Jyv~skyl~ 1970, ii pp. Scheduled to appear in Lecture Notes matics, Springer Verlag.

14.

in the Plane,

in Mathe-

Extremal quasiconformal mappings with given boundary values, Contributions to Analysis, A

Edgar Reich and Kurt Strebel,

Collection of Papers Dedicated to Lipman Bers, Academic Press, New York, 1974, pp. 375-391. 15.

H.A. Schwarz, Ueber einige Abbilduneaufgaben, J. Reine Angew. Math. 70 (1889), 105-120.

137

Zur Frage der Eindeutigkeit eztremaler quasikonformer Abbildungen des Einheitskreises, Comment. Math. Helv. 36 (1962), 306-323.

16.

Kurt Strebel,

17.

Kurt Strebel,

18.

Kurt Strebel,

Zur Frage der Eindeutigkeit extremaler quasikonformer Abbildungen des Einheitskreises If, Comment. Math. Helv. 39 (1964), 77-89. Uber quadratische Differentiale mit geschlossenen Trajektorien und extremale quasikonforme Abbildungen, Festband zum 70. Geburtstag yon Rolf Nevalinna, Springer Verlag, 1966, 105-127.

19.

Kurt Strebel, On the trajectory structure of quadratic d i f f e r e n t i a l s , Proc. 1 9 7 3 Conference on Discontinuous Groups and Riemann Surfaces, U. Maryland, Annals of Mathematics Studies 79, Princeton 1974, 419-438.

A DISTORTIONTHEOREMFORQUASICONFORMALMAPPINGS M, SCHIFFER AND G, SCHOBER

i.

INTRODUCTION,

by analytic

To study one-to-one

functions

f,

mappings

one often introduces

of a plane domain

D

the auxiliary

expression (i.i)

U(z,~;f)

= log

f(z)

- f(~)

z

-

By this means the one-to-oneness of analytically in

D

iff

U

f

can be converted

useful property.

Indeed,

is analytic

D • D.

in

f

lim [f(z) z,~+z0ED does not vanish for conformal mappings. Assume now that a domain

D

properties Virtanen

of

[i].

K-q.c. Since

however,

- f(~)]/(z

is a K-quasieonformal

into the complex plane

H~Ider continuous,

(1.2)

f

is one-to-one and analytic

This,

strongly on the fact that

mappings K-q.e.

C.

into an

depends

- ~)

(K-q.c)

exists and

mapping of

For the definition and

we refer the reader to Lehto and

mappings are well known to be bi-

i.e.~

% I z - ~IK ~ If(z) - f(~)l ~cElz- ~Il/K,

uniformly on each compact

subset

E

of

D~

0,%~-,

one is led by analogy

to the expressions

(1.3)

UK(z,~;f)

Of course,

= log

f(z) - f(~) (z - ~)I/K

one cannot expect

UK

and

and

VK(Z,~;f)

VK

= log

f(z) - f(~) (z - ~)K

to be analytic,

This work ~as supported i n p a r t by A i r Force c o n t r a c t F 44620-69-C-0106.

but the

Q

139

situation

is still worse

not single-valued and

Im V K

for two reasons.

even with one variable

need not be bounded

Both of these problems

even if

ape eliminated

First, fixed, z § ~

UK

and

VK

and second,

are Im U K

on a single branch.

if we consider

only

and

#K(Z,C;f) = Re UK(Z,~;f) : log If(z) - f(~)l

Iz - =IlI~ (1.4) for

~KCZ,~;f) : Re VKCZ,~;f) : log IfCz)Iz_ C -fi~K)I

Then

@K

and

~K

(i.5)

lim suP#K(Z,~;f) z,~ § z 0

We shall define Let

CK

A(z0,r)

the area of

are single-valued

and the HSlder

and

<

and

~K

continuity

implies

lira inf ~K(Z,~;f) ) -~. z,~ § z 0

for

z : ~

= {z : Iz - z01 < r},

f(A(z0,r)).

z ~ ~.

also. and denote by

A(r;zo,f)

Then we define

/A(r; z0,f)/~ ~K(Z0,z0;f) : log llm r+0

and rl/K

(1.6) ~K(zo,Zo;f) = log lim

r

r~o [~(r;f(~o),f-l)/~]K It will be a consequence easily verifies

of Lemma 1

that these

limits

exist.

One

that

lim i n f ~ ( z ~ ; f ) z,~z 0

~ #(Zo,Zo;f)

and

~ lim sup#(z,~;f) z~§ 0

(1.7) liminf~(z,~;f) z,~+z 0

so that if either corresponding are in

[-|

~

function

~ ~(Zo,Zo;f)~

or

~

lim sup~(z,~;f) z,~§ 0

has a limit at z 0,

continuously.

and the values of

In general, ~K

are in

we have defined

the

the values of

#K

(-~,~].

140

The purpose of the present note is to prove the following theorem: THEOREM. in

Izl < R

and

L-q.c.

n = i, "'" ,N~

real,

f: C

Suppose

N

in

is a h o m e o m o r p h i s m

]z I > R.

N

If

[ x n = O, n=l

with

[If(z ) - f(Zn)l

[ ~logl-

C

+

~

I~

that i8

]Zn] < R

and

x

K-q.c. is n

then

(L - K)/[K(L + K)]]

J

_,

(1.8) If(z m) - f(Zn) I

N <

Here,

7. XmXnlOg m,n-i

of course,

for

] zm

m : n

-

[Ra

ZjnlK(K-L)/(K+LI.

-

z n I "~

we interpret

~(Z n) - f(Zn) I

6A(r,Zn,f)/~ as

and

lim r+0

Izn - Zn II/K

rl/K

(1.9) If(z n) - f(Zn) I

llira ~ 0 [~(I~ f(Zn),f-l)/~ ]K

as Iz n - Zn IK If we set following

L = K notable

COROLLARY

and let

R § ~

in the above theorem,

we have the

special case: i.

Suppose

f

is a K-q.c.

mapping

of

C

onto

Then

N (1.10)

If(%) - f(Zn) L

[ XmXnlOg .,n=~

~ If(%) - f(z) I _ znil/K < 0 < 7 XmXlOg 1 90 = = m,n-1 J ~ - ~nl ~ N

for all

Zl,''',z N ( C

and real

Xl,''',x N

with

7. x n

n=l We may also send COROLLARY

2.

L § ~ Let

for the hyperbolic f

be f-q.o,

in

case:

Izl < i.

Then

=

0.

C.

141

f(Zn) l l l -

I f ( z m) -

N

Zm~n II/K < 0 <

Z XmXnl~ m,n-i

Iz m - Zn [I/K

(1.11) If(z m) - f(Zn) I

N [ XmXnlOg m,n=l

<

Izm - znIKIl

- Zm~n IK N

IZnl < i,

whenever

2.

xn

(n = I,-..,N),

is real

SEMICONTINUITY OF FUNCTIONALS, Denote by

homeomorphisms for

f: C § C

Izl > R.

defined

that are K-q.c.

We shall consider

Z x n = 0.

and

F

for

the f a m i l y of

Izl < R

the functionals

r

and

and

L-q.c.

~

on

F

by N [m XmXn~(Zm'Zn;f)

(2.1) ~[f] :

and

N [

~[f] :

XmXn~(Zm, Zn;f)

m,n=l

m,n I

N

for fixed

[Znl < R

and real

Xn,

n : I,.--,N,

with

[ xn m-i

=

0.

We ask for the

(2.2)

sup r F

Since

r

and

~

and

are invariant

we may restrict

our attention

f(0)

= i.

= 0,

normal

family, lim r n§

(2.3)

and

f(1)

fn,g n

inf ~. F

under the transformation

to functions

Since the aggregate

there exist

sequences

] = sup r F

converge

The purpose

of this

ous and

~

is lower

(2.4)

r

locally section

= max r F

in

F

f + Af + B,

satisfying

of such functions

fn,g n ( F

forms a

such that

lim ~[gn ] = inf ~, n§ F uniformly

to

is to show that

semicontinuous and

on

F

f,g

E F,

r

respectively.

is upper

so that ~[g]

= min ~. F

semicontinu-

142

LEMMA i.

Suppose

r-2/KA(r;z0,f) Proof.

f

is

K-q.c.

is a non-decreasing

jf = Ifzl

Let

may represent

in a neighborhood

function

If l 2

of

of

z 0.

Then

r

be the Jaeobian

of

f.

Then we

[i]

(2.5)

A(r)

= A(r;z0,f)

:

;f

~

~ d0 de

A (zTr) so that

(2.6)

A'(r)

f2~ J f

=

r de

0 for

a.e.

r.

Since

f(Iz - z01 = r)

is

q.c.,

is finite for

quaslconformality

L(r)2

f

a.e.

and Schwarz's

: (Iz-~01-

the length

of

A condition

inequality

for

K-

implies

= ' 0

(2.7)

2~ 2~K~

for

r.

L(r)

a.e.

r.

Jf 9 r2d8

We use The isoperimetric

= 2~KrA'(r)

inequality

4~A(r) ~ L(r) ~ .

Then

(2.8) for

[log r-2/KA(r)]'

a.e.

r,

and The conclusion

Remark.

follows.

On The basis of Lemma 1 it is evident

in (1.6) and

(1.9)

LEMMA 2. ~

convergence

on

~

is upper semicontinuous

is lower semicontinuous P.

Write

That both limits

exist.

The functional

functional

Pmoof.

: 0- ~A ' (- r~ 2-)~

and the

under locally uniform

143

(2.9)

r

N [ XmXnr m,n=l m#n

=

+

N [ x~r m~l

Zm;f).

The first sum is clearly continuous under locally uniform convergence~ so it is sufficient to consider the second sum. x m2 > 0,

r

= log lim r§

Suppose therefore that s > 0.

exists an

Since r0 > 0

(2.11) The

curves

Cn

converge to

for all

and

I = lim r-2/KA(r;Zm,f) r§ such that

fn

A(r;Zm,f)

as

Iz - Zml : r 0

n § ~.

sufficiently large.

there

onto Jordan

but have zero area ([i]).

Iz - Zml = r 0,

the areas A(r0;Zm,f n)

Therefore

r 0 2 / K A ( r 0 ; Z m , f n ) < r02/KA(r0;Zm,f) n

locally uniformly.

1 < k + ~e.

carry the circle

uniformly on

fn + f

exists by Lemma i,

that need not be rectifiable,

fn + f

(2.12)

JA(r;Zm,f)/~ I/K r

fn,f E F

r02/KA(r0;Zm,f)

q.c. mappings

Since

since

it is sufficient to consider only

(2.10)

Let

In fact,

+ is

Now the monotonicity of Lemma 1

combined with (2.11) and (2.12) implies (2.13)

for all function If

lim r r§ n

-2/K

A(r;Zm,f n) ~ r 0 2 / K A ( r 0 ; z m , f n) < k + e

sufficiently large. logV~-/~

fn * f

uniformly also.

By composition with the increasing

we have the upper semicontinuity of locally uniformly,

then

f-ln § f-i

locally

Therefore the lower semicontinuity of

applying a similar argument

#(Zm,Zm;f).

~

follows by

in terms of the inverse functions.

that the areas A(r0;fn(Zm)'f-l)n

still converge to

Note

A(r0;f(Zm),f-l).

144

3.

PROOF OF THE THEOREM,

2

We o b s e r v e d in Section

f u n c t i o n s for the problems

(2.2) exist.

Let

f

that extremal

be an extremal

f u n c t i o n for the first p r o b l e m

(3.1)

max ~. F

We shall make use of the variational a l g o r i t h m in [2~3,4]. functional

~

The

has the v a r i a t i o n a l d e r i v a t i v e N

(3.2)

A(w;f)

=-

n i

1 xn

w - f(z n)

1

'

and we denote N

(3.3)

J(w;f)

= [ A(~w;f) dw = i [ XnlOg [w - f(Zn)]. n-I

If we define a =

(3.4)

J o f -

( K1 ~) Kl l- 1

J o f

in

Izl < R

~L - i) J o f

in

Izl > R,

and

(3.5)

b = J o f

-

then it follows f r o m [2,3,4] that in

Izl > R,

finite at

~,

and

b a

is s i n g l e - v a l u e d and a n a l y t i c is locally a n a l y t i c in

except for l o g a r i t h m i c s i n g u l a r i t i e s at

z 1,''',z N.

Izl < R

We may t h e r e f o r e

represent N

(3.6)

a(z)

= n=l

where

g

- L log(RZ a n [ l o g ( z - z n) + K KYL

is s i n g l e - v a l u e d and analytic for

Izl < R.

Z~n)] + g(z)

We have added

the second logarithmic t e r m in the sum for our convenience. c o m p e n s a t e d f o r b y the a r b i t r a r y nature of If in

z

describes a cycle a r o u n d

(3.4) and (3.6) we find

z n,

It is

g. then by c o m p a r i n g periods

145

(3.7) It follows from (3.4) and

(3.8) on

2i K -+ 1

=

an

-

x

(n

n

=

I,-'-,N).

(3.5) that

(K + I)(K + L)a + (K + I)(K - L)a : 2K(L + l)b

Izl

= R.

(3.9)

Therefore

a(z)

L - K a(R2/~) + 2K(L + l) b(z) : L + K (K + I)(K + L)

defines an analytic

continuation

of

a

into

Izl > R.

By

N

substituting

(3.6) and

(3.7) into

(3.9) and using

[ x n = 0, n=l

we

find that

(3.10)

g(z) =

N Zn L - K g(R2/z) + 2K(L+I) b(z) -8iKL 7. XnlOg(l -~-) + L - - ~ (K+L)(K+I) (K + L)~(K + i) n=l

defines a single-valued that is finite at

~.

analytic c o n t i n u a t i o n of By Liouville's

(3.11)

g

into

Iz] > R

Theorem

g(z) ~ constant.

To determine a relation for the extremal (3.4) and (3.5) for

f

we solve

J o f: K~KI

(3.12)

function

[(K + l)a + (K - l)a]

for

Iz I < R

L + 1 [(L + l)b + (L - I)~] 4L

for

Izl > R.

J o f =

Then it follows

from (3.3) and (3.6) that

N[ XnlOg[f(z)

-

f(Zn)]

=

(K2~I)

n=l

N_-[iXn[l~ z

- z n)

+ ~K l- oL g

(Ka- ,,z~)]

n

(3.13) (K&I)

for

Izl < R.

N n~iXn [i~

Similarly,

"

K- L Zn) + k - ~ l~

-

by using also the identity

ZZn)]

+

(3.9),

constant

we find

146

N

i

[

7 XnlOg[f(z) - f(Zn)] : ~ [ ( L n-i

+ i) [ XnlOg(z - zn) n-i

N

(3.14)

]

- (L - i) 7. x~ log(z - zn) n-1 for

+ constant

> R.

We write the real part of (3.13) as

N

(3.15)

If(z) - f(Zn)l

Clearly the limit as z = Zm,

min ~ F negative of that in (3.2).

logIR'

- ZmZnl

the variational Therefore

derivative

in (1.8). is simply the

the second inequality

in (1.8)

in an entirely analogous manner.

Remark.

Since the above proof is based on variational

the inequalities

from (3.13) and inequality

(1.8) are n e c e s s a r i l y

(8.14) that an extremal

in (1.8) is of class

zn

Similarly, is of class behaves

XmXn

This proves the first inequality

For the problem

points

- Z~nl + constant.

exists. We may therefore substitute n N and sum on m to obtain (since 7. x n = 0) n-i

1 IK-L = K ~K + L]m,n.l

r

in the extreme case.

ations,

= K\K + LI n~iXn l~

z § z

m u l t i p l y by Xm,

(3.16)

follows

Z~E - L~ N

n~lXn log Iz _ znll/K

where it behaves an extremal C~

like

C~

like

We also note

function for the first

except on

Izl = R

and at the

An(Z - Zn) IZ - Zn I(I/K)-I + B n.

function for the second inequality

except on

Izl = R

and at the points

in (1.8)

zn

where

A~(z - Zn) IZ - Zn IK-I + B'.n The extremal functions

defined by (3.13) and (3.14) provide interesting homeomorphisms

sharp.

consider-

of

C

onto

C.

examples of

q.c.

it

REFERENCES 1.

O. Lehto and K.I. Virtanen, Quasikonforme .Abbildungen, Verlag, Berlin-Heidelberg-New York, 1965.

2.

H. Renelt, Modifizierung und Erweiterung einer Schifferschen Variationsmethode f~r quasikonforme Abbildungen, Math. Nachr. 55 (1973), 3 5 3 - 3 7 9 .

3.

M. Schiffer,

4.

M. Schiffer

Springer

A variational method for univalent quasiconformal mappings, Duke Math. J. 33 (1966), 395-412. and G. Schober, An eztremal problem for the Fredholm eigenvalues, Arch. Rational Mech. Anal. 44 (1971), 83-92, and 46 (1972), 394.

QUASIREGULAR MAPPINGS URI SREBRO

i.

INTRODUCTION

i.i. in

Quasiconformal, R n,

n e 2,

quasiregular and quasimeromorphic mappings

seem to be reasonable generalizations of conformal,

analytic and meromorphic functions, of these mappings

respectively;

and the theory

is in many respects complementary to the geomet-

ric theory of functions in

r

Furthermore,

complex analysis are not applicable

since most methods of

in general for quasiconformal,

quasiregular and quasimeromorphic mappings

in

R n,

the proofS in

the theory of these mappings a~e usually more direct and mostly of a geometrical nature;

in many cases,

they give better insight

into various phenomena connected with these mappings as well as with analytic functions in In this paper,

r

which is partly expository,

I shall survey

some of the elementary properties of quasiregular mappings,

illus-

trate the use of the main methods in this theory by proving several distortion theorems for quasiregular mappings and conclude with the introduction of the concept of conformal measure with an application to a two constant theorem for quasiregular mappings.

Further

properties and applications of the conformal measure will appear in a forthcoming paper. For more information about quasiregular mappings the reader is refered to V~is~l~'s 1972 expository report raphy at the end of that report, this note. know, (9)

IV4],

to the bibliog-

and to the list of references of

Several open problems are listed in [V4].

three of them have been answered: by S. Rickman [Ri 2] and

(13)

(8)

by O. Martio [M2],

by T. Kuusalo

the 1973 Analysis Colloquium in Jyr~skyl~,

As far as I

(announced in

Finland).

149

n

1.2.

N o t a t i o n and terminology.

For

x { Rn

we w r i t e

x =

'

where

el,---,e n

is an o r t h o n o r m a l basis in

R n.

r > 0

we denote

Bn(a,r)

Bn(r)

B n = Bn(1),

sn-l(a,r)

~B n.

The closure

sets

A

Rn.

in

D

2.

the b o u n d a r y

~n = R n U {=}

is a d o m a i n in

QUASIREGULAR

sn-l(r) 8A

x.e. i

a E Rn

I

and

= Bn(0,r),

= ~Bn(r)

and

S n-I =

and the c o m p l e m e n t

CA

of

will always be t a k e n w i t h r e s p e c t to

f: D § R n

continuous and that

I x - a I < r},

= ~Bn(a,r),

A,

By w r i t i n g

that

= {x:

For

[ i=l

or

Rn

f: D + Rn

or in

Rn,

we shall always assume

respectively,

that

f

is

n a 2.

MAPPINGS

2.1. A mapping

f: D + R n

qr,

f(D)

if either

is said to be quasiregular,

is a point in

Rn

or else

f

abbreviated

has the fol-

lowing properties: (i) sets in

f

is open

Rn),

for every (it)

y

(i.e.,

discrete in

f(D))

f E ACL n

f

(i.e.,

maps open sets in f-l(y)

D

onto o p e n

is a d i s c r e t e set in

D

and s e n s e - p r e s e r v i n g ;

(i.e.,

f

is locally a b s o l u t e l y c o n t i n u o u s on

almost all line segments p a r a l l e l to the c o o r d i n a t e axes and its partial d e r i v a t i v e s belong to belongs to the Sobolev space (2.1.1)

(iii)

for some

K E [i,~).

Here

L~oc(D) ,

or in other words

1 Wn,loe) ,

If'(x)l n c KJ(x,f)

f' =

~

\

a.e.

2.2.

D

f,

If'(x)I

,j=l

denotes the s u p r e m u m norm of the linear o p e r a t o r J(x,f)

in

is the formal d e r i v a t i v e of

]

f

f'(x)

and

= det f'(x).

A mapping

bmeviated

qm,

f: D + Rn if either

is said to be q u a s i m e r o m o r p h i c , f(D)

is a point in

Rn

or else

ab(i) -

150

(iii) hold where

(ii) and

(iii) are checked at

~

and at

by means of a u x i l i a r y M~bius t r a n s f o r m a t i o n s w h i c h map

2.3. A m a p p i n g ated

2.4.

qc,

f: D + Rn

if

f

Conditions

is

is said to be q u a s i e o n f o r m a l ,

(i) - (iii) are not independent. see R e s h e t n y a k

is not a constant and satisfies

open,

~

into

R n.

abbrevi-

qm and injective.

(though not so easily, f

f-l(~)

One can show

[Re i] and [Re 2])

(ii) and (iii),

then

that if f

is

discrete and sense-preserving. Condition

(i) says that,

branched covering map.

locally,

Conditions

a bounded d i l a t a t i o n in

D.

f

is a s e n s e - p r e s e r v i n g

(ii) and (iii)

This is why

qm

say that

f

has

m a p p i n g s are some-

times called m a p p i n g s of bounded d i l a t a t i o n or of bounded distortion.

By (ii) the partial

exists

a.e.

in

D.

d e r i v a t i v e s of

Moreover,

is d i f f e r e n t i a b l e

0

onto a set of m e a s u r e zero and by Martio, J(x,f)

> 0

a.e.

is d i f f e r e n t i a b l e and where now every ball

f'(x) B c D

in

in

J(x,f)

mappings

by R e s h e t n y a h

f

[MRV i]

a.e.

qm

D,

D.

f: D § Rn

[Re i] and [Re 2],

and maps every set of m e a s u r e Riekman and V~is~l~

At points

> 0,

f(x+h)

x

in

D

where

f

= f(x) + f'(x)h + o(lh[),

is a n o n - s i n g u l a r linear o p e r a t o r w h i c h maps onto an e l l i p s o i d

E

and

If'(x) In/j(x,f)

is the ratio b e t w e e n the volume of the ball w h i c h c i r c u m s c r i b e s and the volume of 2.5.

Let

dilatation

E

E.

f: D + Rn

be a n o n - c o n s t a n t

K O = Ko(f,D) ,

the maximal d i l a t a t i o n

qm

mapping.

the inner d i l a t a t i o n

K = K(f,D)

K I = KI(f,D)

are defined by If'(x)l n

K0

:

ess sup x(D

KI

=

J(x,f) ess sup xED ~(f'(x)) n

J(x,f)

The outer and

151

K

=

max

(Ko,K I )

w her e

s

(x))

inf If'(x)hl. lhl : 1

K 0 ~ K n-i I ,

By linear a l g e b r a

analytic

and

conformal

functions

C

qr

qc

K = I.

self-evident

3.

are subclasses

These

subclasses

The g e o m e t r i c

by v i r t u e

K-qm,

of

qm,

K-qr

then

The classes of m e r o m o r p h i c ,

and

is

n = 2

to the case.

respectively.

f

if

according

in

we say that

and

If

n = 2

= K

n-i KI < _ K0

K 0 = K I.

pings,

K(f,D)

:

are o b t a i n e d

meaning

of the last r e m a r k

of

K0

or

and

K-qc

map

by letting

and

KI

are

in 2.4.

THE MULTIPLICITY FUNCTIONS, THE LOCAL TOPOLOGICAL INDEX AND THE BRANCH SET OF OPEN DISCRETE MAPPINGS

3.1.

Suppose

serving. c G

f(BD)

nent of

if

of a point

with

x ( G

r > 0

defines

~ c G,

f-l(Bn(f(x),r)).

Then

U(x,r,f)

is a normal

whenever

0 < r ~ r

o"

with

then

D

~ c G

domain

for

[MRV i] there neighborhood

denote exists

and

o

compo-

and

Rn

small-

[MRV i].

such n e i g h b o r h o o d s .

the r

Fur-

neighborhood

f

in

if

d o m a i n and

has a r b i t r a r i l y

way to c o n s t r u c t

f

is a normal

is a c o n n e c t e d

complement

U(x,r,f)

for

D § f(D).

is a normal

x E G

connected

let

D

sense-pre-

domain

is said to be a normal

is a standard x E G

D,

is a normal

with

and

a closed mapping

E v e r y point

and

discrete

is a d o m a i n and

D

= {x}.

there

A domain

D r G

if

neighborhoods

Moreover,

f

A domain

n f-l(f(x))

For

if

is open,

is said to be a normal

= Bf(D).

D' r f(G)

f-l(D')

= D'.

normal

D c G

if and only

thermore,

f(D)

f: G § R n

A domain

and

domain

that

> 0

CU(x,r,f)

x-component such that is c o n n e c t e d

of

152

For N(f,A)

A c G

and

y E Rn

= sup N(y,f,A)

local topological i(x,f).

Since

over all

inde~ of

f

let

f

is open,

N(y,f,A) y E Rn

= card f-l(y)

and

at a point

N(f)

x 6 G

n A,

: N(f,G).

The

is denoted by

discrete and sense-preserving,

i(x,f)

may be defined by (3.1.1) where

i(x,f) U

:

is any normal n e i g h b o r h o o d

N(f,U) of

the same for all normal neighborhoods [MS I].

With this d e f i n i t i o n

is open,

of

f,

In fact,

x.

N(f,U)

For more details

one can show [MS l] that if

discrete and sense-preserving

domain for

x.

and

D c G

is see

f: G +

Rn

is a normal

then

(3.1.2)

i(x,f)

=

N(f,D)

<

xEf-l(y) for all

y ~ f(D).

This includes the case

D : G : Rn.

We shall also need the notation of the minimal m u l t i p l i c i t y M(f,C)

[MI]

which is defined for compact

(3.1.3)

3.2.

M(f,C)

is open,

discrete

x E D\Bf, n-2;

see

[V!].

If,

m(Bf)

= m(f(Bf))

= 0.

qm

mappings

it will be denoted by

and sense-preserving,

1 < i(x,f)

see

< ~

for

in

D

f

and is

For more information [MRV 2]

f: D § Rn Bf.

then

x E Bf

in addition,

[MR],

C

by

inf I i(x,f) y6R n xEf-l(y)nC

The branch set is the set of points where

a local homeomorphism;

of

=

sets

and

If

i(x,f)

is not

f: D § Rn = 1

for

dim Bf = dim f(Bf) qm

then

[MRV i]

about the branch set [Sa].

153

4.

MODULUS AND CAPACITY INEQUALITIES

4.1.

The o n l y t o o l s ,

condition families

so f a r ,

which are based on the d i l a t a t i o n

(2.1.1) are the inequalities

for the modulus

and for the capacity of condensers.

We quote now a modulus

inequality and two capacity

which will be used later in this note. ities of this type see [P], 4.2. R n,

The modulus i.e.

each member

y

the n-modulus,

M(F)

of

F

into

of

F

M(F)

[Ml],

Let

IV3]

r

p: R n + E 1

with

I

p ds

[Sr].

be a path family in

The modulus,

continuous

func-

or more precisely

is defined by =

inf pEF(F)

[ pn dm~ J Rn

denotes the set of all non-negative

F(F)

and

is a non-constant

Rn.

inequalities

For proofs and more inequal-

[MRV I] ~

of a path family.

tion from a line segment

where

of path

~ 1

Borel functions

for all r e c t i f i a b l e

y E r.

Y 4.3.

We shall use the following

Let

rI

and

r2,

in the sense that each

belongs to

r2

Let

Then

F(A,B,D)

A

and

4.5.

B

compact

then

D c Rn

in

e

A

in Rn.

in

rI

be a domain~

A

If

rI

is m i n o r i z e d ~'

For more details

and

B

disjoint

by

which

see IV2].

sets in

5.

will denote the family of all paths which connect D.

sense,

c.f.

By a condenser

[Sr]) we shall mean a pair

is a proper open subset of subset of

defined by

property of the modulus.

has a subpath

M(F I) 5 M(F2).

The capacity of condensers.

restricted where

be path families

r2,

4.4.

elementary

A.

The (conformal)

Rn

and

C

capacity,

in

Rn

(in the

E = (A,C) is a non-empty cap E,

of

E

is

154

(4.5.1.)

cap E

Equivalently,

the

where R1

W(E)

with

is the

u(x)

general

condensers, capacity

0 < a < b < ~

of t h e see

is g i v e n

(4.5.2)

E

can be defined

| JRn

x ~ C

and

ACL

u(x)

two d e f i n i t i o n s , [Sr],

spherical

by

IVul n dm

non-negative

[H],

of the

of

inf u~W(E)

for all

equivalence

The

:

set of all

= 0

For the

M(F(C,~A;A)).

capacity

E

cap

:

functions

= 1 see

[MS i]

for all [G];

and

condenser

u:

Rn +

x ~ CA.

for m o r e

[MS 2].

E = (Bn(b),Bn(a)),

by

cap E

mn-I

=

'

where

: m n - i (S n-I ).

mn-i

E = (A,C)

is a c o n d e n s e r

both meet

a certain

We

such that

sphere

(4.5.3)

shall

also CA

sn-l(a,r),

cap

u s e the

fact

and

are

C

= E CA

and

that

if

connected, a E C,

then

E ~ n

where

an > 0

the

family

and

{tel:

4.8.

Now

condenser is a g a i n A

of p a t h s

let in

mapping.

Then

in

f: D + R n D

(meaning

a condenser,

Modulus

only

which

1 ~ t ~ ~}

is a n o r m a l

4.7.

depends

domain

and

on

n.

join

the

R n,

be a

In fact, line

[V2].

qm

mapping

A c D);

we

say

E = (A,C)

for

f.

capacity

inequalities.

is the m o d u l u s

segments

see

the

an

and

then

{tel:

a

= (f(A),f(C))

is a n o r m a l

Let

-i s t 5 0}

E = (A,C)

f(E)

of

condenser

f: G + R n

be a

if

qm

155

(4.7.1)

M(fF)

for all path families

~ KI(f)M(F)

F

in

G;

heme

fF = {foy : y E F}.

Further-

more cap f(E) ~ Ki(f) N(f~A)n-i M(f,C) n

(4.7.2)

for all condensers (4.7.3)

E : (A,C)

G

and

cap E ~ Ko(f) N(f,A) cap f(E)

for all normal condensers

E : (A,C)

Of these inequalitites, (4.7.2)

in

cap E

to Martio

[MI] and

(4.7.1) (4.7.3)

in

G.

is due to Poleckii

to Martio,

[P]~

Rickman and V~is~l~

[MRV i].

5.

5.1.

DISTORTION THEOREMS FOR

In these sections,

capacity

inequalities

tortion theorems are contained 5.2. N.

For

and

for quasiregular

Let

r ( (0,i) m(r)

MAPPINGS

we shall illustrate

(4.7.2)

in [M].

THEOREM.

qr

=

(4.7.3)

IV3].

f: B n ~ R n

be a

qr

denote

inf

If(x) - f(0) I

IxJ =r M(r)

:

sup

If(x) - f(0) I.

Ixl =r The~ there is an

r

o

> 0

such that

(5.2.1)

Alra ~ m(r)

~ A2r6

(5.2.2)

A3r~ ~ M(r)

~ A4r8

in proving

mappings.

See also

the use of the two several dis-

Some of these results

mapping with

i(0,f)

:

156

i/n-i

for all Ai,

r ((0,to)

~

and

Proof. that

8

= U(0,f,r),

neighborhood

constants

that

the

f(0)

and

which depend on

= 0.

0-component

with connected

f.

The

complement

Choose

of

R > 0

f-iBn(r),

whenever

such

is a n o r m a l

r ( (0,R]

(See

Let

n ~ = inf {IxI:

Fix

{ N ]i/n-i , 8 = k~ii ]

~ : (KoN)

are the best possible.

We m a y a s s u m e

U(r)

8.1).

where

are positive

i = 1,--.,4,

exponents

,

r ( (0,r o)

is a n o r m a l (3.i.i),

x ( ~U(R)}

and w r i t e

condenser

N(f,U(R))

in

and

r I = sup {IxI:

M = M(r) Bn

with

= M(f,U(m))

and f(E)

= N,

x E ~U(R)}.

m = m(r).

E = (U(R),U(m))

= (Bn(R),Bn(m)).

hence

from

(4.7.2),

KI KI = cap f(E) ~ -~- cap E ~ -N-- "

~n-i

By (4.5.2)

and 4.8 f o l l o w s mn- i n-i

This yields

the right

For the right er

inequality

E = (U(N),U(m)).

and 4.3,

inequality

Then

in

(4.7.3)

"

(5.2.1).

(5.2.2), f(E)

n-I

consider

the normal condens-

= (Bn(N),Bn(m)),

(4.5.3)

and

0 < on ~

cap E ~ N K 0 cap f(E)

N(f,U(M))

= N

give = NK 0

mn-1

(og

n-l"

consequently

(5.2.3)

where and

M ~ Cnm ,

cn N.

is a p o s i t i v e The r i g h t

and the right

f(E)

inequality

inequality

Next consider

constant

of

and

(5.2.2)

follows

o n l y on

n,

now f r o m

K0

(5.2.3)

(5.2.1).

the n o r m a l

= (Bn(R),Bn(M))

of

which depends

condenser

(4.7.3),

E = (U(R),U(M)).

(4.5.2)

and 4.3 y i e l d

Then

!57

~n-i

< cad E < NI<0 cap f(E)

mn-i

= NK 0

log

log

This proves inequality

the left i n e q u a l i t y of

(5.2.1)

follows

of (5.2.2),

from

To show that the r i g h t h a n d are best p o s s i b l e ,

consider

mapping

fN:

9 .-,x n)

to the p o i n t

and

Rn § Rn

0 S ~ < 27.

pute

KI,

and

= r x

and

= i.

h o l d s on the r i g h t

KI = N

let

N > 0

g = h 9 fN

where

and

h: R n § R n

is the r a d i a l stretching

h

is a

branch

qc set

direction, and

Ixl a-I

oNlxl n(~ and

mapping Bg,

g

and

Ig'(x)l

e = (N KO )I/n-I holds

Bg

BfN.

=

= q.

n-2

major

= olxl O-I

on the left

in

h(x) x

Consequently,

and

above

= Ixl~

in the radial direction J(x,g)

Ko(g) = m(r)

(5.2.2).

and

off the

Thus

M(r)

(5.2.2)

described

to the r a d i a l

directions.

(5.2.1)

and

o > N

olxl O-I

On the o t h e r hand

BfN =

(5.2.2).

by

At e a c h p o i n t

normal

To com-

set

Choose

defined

stretching

in a d i r e c t i o n

in all o t h e r

and

is the w i n d i n g m a p p i n g

has the m a j o r

Nlxl O-I

and

so e q u a l i t y

fN

p m 0

= N.

in (5.2.1)

be an integer.

let

sin %,x3,

I,N,I,--.,I. Thus ( N_N_~I/n-I and ~ : ~KI] =

in (5.2.1)

To s h o w th a t the left h a n d i n e q u a l i t i e s are best possible,

r

i(0,f)

(5.2.2)

the w i n d i n g

off the b r a n c h

Hence

and

.-., Xn),

has the m a j o r s t r e t c h i n g s

~(f~(x))

so e q u a l i t y

(p cos

p sin N%,x3,

= m(r)

(5.2.1)

N > O

sends e a c h p o i n t

M(r)

fN

of

for each i n t e g e r

n o t e that at each p o i n t

J ( x , f N) = N and

(5.2.3).

inequalities

(p cos Nr

Here

{x: x I = x 2 = 0},

1

which

and the c o r r e s p o n d i n g

=

_ on-iN = rO

and

This c o m p l e -

tes the proof.

5.3.

Next consider

[MS1]

that

n e x t two

f(x) + ~

sections

lim f(x) as

qr as

mappings x § =

f: R n § R n. if and o n l y

we s t u d y the r e l a t i o n

n + ~,

the d e g r e e

N(f)

It is shown in

if

between of

f

N(f)

< ~.

In the

the e x i s t e n c e

of

and the g r o w t h of

f

158

near

~.

For related results

m(r)

For

{lfCx)l:

inf

=

see [V3,w

r > 0,

let

= r}

Ixl=r M(r)

5.4.

THEOREM.

:

Let

sup { I f ( x ) l : Ixl:r f: R n § R n

Ix I : r } .

be a qr mapping with

N(f)

= N < ~.

Then

(5.4.1)

Alr8 ~ m(r) ~ A2ra

(5.4.2)

A3r8 ~ M(r)

for all sufficiently = (N/KI)I/n-I on

large value8 of

and

Ai,

r,

where

i : 1,''-,4,

~ = (KoN)

are constants

i/n-i

which depend

f. The ezponents Proof.

~

and

8

The fact that

qm extension, Rn

~ A4ra

N < ~

denoted again

is compact,

are best possible.

f,

f: ~ n § R n

implies to

Rn

[MS1] with

that f-l(~)

is a closed mapping;

f

has a

= {~}.

hence,

Since

by

(3.1.2) i(~,f)

Let with

iS(x)l

g = Solos -I _

1

=

/. xEf-l(~) where

for all

i(x,f)

= N.

S:R n § Rn

is a M~bius t r a n s f o r m a t i o n

x E R n \ {0}.

Then

g

is a

qm

map-

Ix; ping with the dilatations i(0,g)

= N.

sup {Ig(x)l: I/M'(I/r).

Letting Ixl = r}, Thus,

of

m'(r)

f,

N(g)

= inf {Ig(x)l:

we see that

applying

= N(f),

M(r)

g(0)

Ixl = r} = i/m'(i/r)

(5.2.1) and (5.2.2) to

g

= 0

and

and

M'(r)

and

m(r)

= =

we obtain

(5.4.1) and (5.4.2). In order to see that

(5.4.1) and

(5.4.2)

are sharp,

one can

159

take the m a p p i n g s (5.2.2)

fN

and

g

w h i c h give e q u a l i t y in (5.2.1) and

and form the m a p p i n g s

s-lofNoS

and

s-logoS

w h i c h will

give e q u a l i t y in (5.4.1) and (5.~.2).

5.5.

COROLLARY.

Let

(5.5.1)

or

f: R n § R n

be a qr m a p p i n g . If(x) l

log

lim sup x§

log

If

-

Ix I

if lim inf log ]f(x)l _ 0 x§ log Ix I

(5.5.2)

then

f

has no

Proof. [MS I]);

l i m i t at

If

f

and Thus

~.

has a limit at

~,

then

N(f)

-- N < ~

(see

(5.4.1) would imply that 1

lim sup log

If(x) I / log

and (5.4.2) that lim inf log contradicting

6.

If(x)l / log

Ixl >_ 8 = (KoN)I/n-I > - 0,

(5.5.1) and (5.5.2).

THE CONFORMAL MEASURE

6.1.

A,

Ix] _ e = kKi/

Let

D c Rn

A c BD,

be a domain.

w i t h respect to

The c o n f o r m a l measure

D

at the point

x ( D

@(x,A,D)

will be

defined bY

r

where

E c ~

A = 0

we set ~(x,A,D)

~(x,A,D) § 0 ~(x~A,D)

: inf M(P(E,A;D)) E is a eontinu~n w i t h r

and

E n ~D = 0.

If

= 0.

is n o n - n e g a t i v e , as

x ( E

d i a m A § 0.

and for Also,

D # Rn

and fixed

x ( D

it is easy to see that

is a c o n f o r m a l invariant for every

n ~ 2

and

monotone

of

160

w i t h respect to ~(x,.,D)

A

for fixed

x

and

D.

However,

is not a d d i t i v e as a set f u n c t i o n on

a m e a s u r e on

8D

in the c o n v e n t i o n a l

BD

in general and thus is not

sense.

In certain proofs the conformal m e a s u r e can r e p l a c e the harmonic m e a s u r e w i t h the clear a d v a n t a g e that,

unlike h a r m o n i c measure,

c o n f o r m a l m e a s u r e is a c o n f o r m a l invariant in all d i m e n s i o n s

~ 2.

We shall i l l u s t r a t e the use of c o n f o r m a l m e a s u r e and the m o d u l u s inequality pings in r e m in

(4.7.1) in proving a two constant t h e o r e m for R n,

C

n ~ 2.

qr

map-

Recall that the c l a s s i c a l two c o n s t a n t theo-

is proved by the use of h a r m o n i c measure,

see [N,III2.1].

We shall need the following n o t a t i o n for our t w o - c o n s t a n t theorem.

6,2.

For

0 < m < r < 1

let

D = {x E Rn: m < lxl < i}. show that

~(r)

~(r)

1

~

= M(r(E,Sn-I,D))

and t h e r e f o r e

on

THEOREM.

subset

~-l(t)

extension

x ~ A,

Let

3D,

of

D

to

D U A

~(r)

E = {tel: m ~ t ~ r}.

It

is strictly increasing from

is strictly increasing

R n,

in

a nonconstant and

for all

from

m

0

to

A

qr

a non-empty mapping

with

proper a contin-

m ~ (0,1). x E D

If(x) l ~ m

and

for all

then

If~x~l ~

~-l(Ki(f)r

is d e f i n e d

Proof. x ~ D

where

be a domain

f: D § R n

If(x) I < 1

If

where

where

(0,|

6.3.

uous

: ~(rel,Sn-l,D),

By s y m m e t r i z a t i o n it is not hard to

is also not hard to show that to

: ~(r,m)

and

r a continuum

in S.1 and

\ A,D))

~

in 8.2.

By L i o u v i l l e ' s t h e o r e m for E > 0. < ~, E

We may assume that

qr

mappings,

if(x)J

> m

D ~ R n.

and that

since o t h e r w i s e there is n o t h i n g to prove. in

5

w h i c h meets

~

and

A

and

Let

such that

Choose

161

M(F(E,~DkA,D)

< r

+ e.

Denote

E' = f ( E ) \ B n ( m )

and let

r' = r ( E ' , ~ f ( D ) \ B n ( m ) , f ( D ) \ B n ( m ) ) . For each

y':

[a,h) + f ( D ) ~ B n ( m )

of

F',

with

y'(a)

E E'

and

lim y'(t) ~ ~ f ( D ) ~ B N ( m ) , choose a point z in E 0 f-l(y'(a)) t§ and let y: [a,c) § D be a maximal lift of y' from the initial point

y(a)

= z.

discrete mappings maximal and

[Rill.

Since the lift

f(A) c Bn(m),

and consequently subpath of

Such a lift exists by Rickman's

7'

it follows

~ ~ F(E,~D\A;D).

f(D) c B n,

7(t) + ~ D \ A

Note that

s M(fF) ~ KI(f)M(F) < KI(f)[r

F

that

was assumed to be

toy

In any case by virtue of 4.3 and

M(F')

where

y

is the family of lifts it follows,

Lemma for open

as

t § c;

may be a proper

(4.7.2)

~ KI(f)M(F(E,SD\A;D) + c],

y

of

by 4.3 and 6.2,

7'

for

7' E F'.

Since

that

~(If(x)l,m) ~ M ( F ( E ' , ~ B n , B n ~ B n ( m ) )

~ M(F').

Hence

~(If(x)l,m) and the result

~ KI(f)[r

+ e],

follows by 6.2 and letting

e + 0.

ACKNOWLEDGEMENTS

I with to thank the Mathematics Department at the University of Maryland for its hospitality during the special year in complex analysis.

REFERENCES

[G]

F.W. gehring,

[H]

J. Hesse, A p-extremal length and p-capacity equality, (to appear).

[M1]

0. Martio~ A capacity inequality for quasiregular mappings, Ann. Acad. Sci. Fenn. A.I. 474 (1970), 1-18.

[M2]

0. Martio~

[MR]

O. Martio and S. Rickman~

Extremal length definition8 for the conformal capacity of rings in space, Mich. Math. J. 9 (1962), 137-150.

On k-periodic mappings~ (to appear).

Measure properties of the branch set and its image of quasiregular mapping6, Ann. Acad. Sci. Fenn. A.I.

541 (1973),

1-16.

S. Rickman and J. V~is~l~, Definitions for quasiregular mappings~ Ibid. 448 (1969)~ 1-40.

[MRVI] O. Martio,

S. Rickman and J. V ~ i s ~ l ~ Topological and metric properties of quasiregular mappingsj Ibid. 488 (1971),

[MRV2] O. Martio~ 1-31. /MS1]

O. Martio and U. Srebro, Periodic quasimeromorphic mappings, J. d'Analyse Math. (to appear).

[MS2]

O. Martio and U. Srebro, Automorphic quasimeromorphic mappingsj Acta Math. (to appear).

[N]

R. Nevanlinna,

[P]

E.A. Poleckii,

Analy%ic

Funciions,

Springer Verlag,

1970.

The modulus method for non-homeomorphic quasiconformal mappingsj Mat. Sb. 83 (1970), 261-272 (in Russian).

[Rel]

J.G. Reshetnyak~ Space mappings with bounded distortion, Sibirsk. Mat. Z. 8 (1967), 629-658 (in Russian).

[Re2]

J.G. Reshetnyak,

On the condition of the boundedness of index for mappings with bounded distortion, Ibid. 9 (1968)~

368-374

(in Russian).

[Ril]

S. Rickman~ Path lifting for discrete open mappings, (to appear).

[Ri2]

S. Rickman,

[Sa]

J. Sarvas, Multiplicity and local index of quasiregular mappings (to appear).

[St]

U. Srebro, Conformal capacity and quasiregular mappings, Ann. Acad. Sci. Fenn. A.I. 529 ( 1 9 7 3 ) ~ 1-13.

(to appear).

163

[VI]

J, V~is~l~, Discrete open mappings on manifolds~ Ibid., A.I. 392 (1988), i-i0.

[V2]

J. V~is~l~, Lectures on n-dimensional quasiconformal mappings, Lecture notes in Math. 229 Springer Verlag, 1971.

[V3]

J. V~is~l~, Modulus and ~apacity inequalities for quasiregular mappings, Ann. Aead. Sei. Fenn. A.I. 509 (1972), 1-14.

[V4]

J. V~is~l~, XVI Seand. Cong. of Math.

(1972).

TECHNION HAIFA, ISRAEL

STARLIKE FUNCTIONS AS LIMITS OF POLYNOMIALS T,J,

SUFFRIDGE*

INTRODUCTION, This paper is a study of functions w h i c h are starlike of order (functions

satisfy the c o n d i t i o n of functions

a n a l y t i c in the unit disk w h i c h

f(z) = z + a2 z2 + .-. Re[zf'(z)/f(z)]

starlike of order

~

> e),

where

e ~ 1.

The class

is first c h a r a c t e r i z e d as the class

of limit functions of sequences of p o l y n o m i a l s having a simple r e s t r i c t i o n on the location of their zeros. from a study of these polynomials. te proof of the result of Brickman, functioDs

starlike of order

~

Our results then f o l l o w

These techniques yield an alternaet.al,

that the class of

lies in the convex hull of the

c o l l e c t i o n of r o t a t i o n s of the f u n c t i o n

z/(1 - z) 2(l-e)

Additional

information is o b t a i n e d c o n c e r n i n g the p r o b a b i l i t y m e a s u r e s [0,2~] ~.

for which

For each

IO~ z/(l - zeit) 2(I-~) d~(t)

e ~ i,

the p a r t i c u l a r case

~ = 0,

on

is starlike of order

a c o n v o l u t i o n - t y p e t h e o r e m is obtained.

Schoenberg conjecture Small).

U

For

this yields a proof of the Pdlya-

(recently proved by R u s c h e w e y h and Sheil-

Further results are obtained,

some of w h i c h bear on the

geometric effects of c o n v o l v i n g certain convex functions.

i.

A C H A R A C T E R I Z A T I O N OF FUNCTIONS STARLIKE OF ORDER C o n s i d e r the class

P

of p o l y n o m i a l s n

P(z)

~, n ic~ = H (l+ze J), j=l

where

(i)

*

2~/(n + 2) ~ ej+l - ~j'

i ~ j ~ n,

~n+l : el + 2~.

This work was supported in part by the National Science Foundation under grant number

GP-39053.

165

For such a polynomial

P,

zP'(z)/P(z)

n " [ zeleJ/(l j=l

=

Using the m a p p i n g properties

of

i@j + ze

). we see that for fixed

w/(l + w),

r < 1 min[

min Re[zP'(z)/P(z)]

]

n

is attained when equality holds m i n i m u m occurs for Let

(i.e.,

Then for

P E Pn'

p(zP(z)) ~ p(zQn(Z)). subsets of

Thus we conclude

u n i f o r m l y on compact

Thus,

the

= (i + z n+2) / (i - 2z c o s w ( n + 2) + z2).

P(f) = sup { r : Re [zf'(z)/f(z)]

uniformly on compact n § ~.

1 ~ j ~ n-l.

denote the radius of stamlikeness of functions

p(f)

Izl < 1

Qn(Z)

in (i) for

Izl < i,

that if

subsets of

Similarly, define Pn(8) n iej P(z) = ~ (1 + ze ) where

But so

> 0,

f

analytic in

Izl < I}).

lim ZQn(Z) = z/(l - z)2 n§ p(zQn(Z)) § 1 as

P

E P and zP (z) § f(z) nk nk nk Izl < 1 then f is starlike. to be the class of polynomials

j=l (2)

2e ~ aj+ 1 - mj,

Here we require

1 ~ j s n,

0 ~ e ~ w/n. min P~P (8)

Again

en+l = ~i + 2~.

it is clear that for

[ min Re[zP'(z)/P(z)] Izl-r

r < i,

]

n

is attained when equality m i n i m u m oecurs for

1

in (2) for 1 ~ j ~ n-l.

nH (i + ze i(2j-n-l)8) j=l n Qn(Z;8) = [ C(n,k,8) z k k=0

Qn(z;e)

prove by induction that

C(n,k,e)

holds

It is easy to

=

if

where

k = 0,n

= k sin(n - 4 + i)8 sin ~

j=1

We wish to show that for

1 < k < n-i

e = ~/(n+2-2~),

Thus the

166

(3)

lim n+~

ZQn(z;e)

For this value of

8,

n § ~.

while

we have

Further~

0 ~ C(n~k~8)

~ 1

8.

we see as before

that

for

Let

uniformly

~

such that starlike

be fixed.

for Sk

g

has degree

s

Define

g

as

> ~

for the

Izl < r}~

n + ~.

: z + a2z

The

2

+ "-"

is

> ~

be the

nk-i

: z + a2z2

+ "'"

and

= f(rs163

partial

ZSk(Z)

ZPnk

is starlike

be an increasing Then

and it is sufficient

nkth

and

Izl < i.

f(z)

g(z)

nk § ~

is the limit of polynomials

nk > k

g

is

to show that for of the required

sum of the power

is starlike

sequence

of order

series e.

Then

so that

< Re[z(ZSk(Z))'/ZSk(Z))]

=

1 + Re[zS~(z)/Sk(Z)]

which can be made arbitrarily

small

Izjl ~ 1

{zj}

for some

follows

been proved.

such that

0 < rs < i,

e + ~

ZSk(Z)

where

Assume

{rs

lim rs = i.

Let

f(z)

e =< 89

if and only if there exists a sequence

and let

arbitrar~ but fixed form.

Then

on compact subsets of

of order

§ 1

theorem has therefore

Proof of "only if". of order

(3)

p (f) ={r:Re[zf'(z)/f(z)]

Pnk E P nk(~/(nk + 2-2a)),

{Pnk}k~ I, f(z)

~

for

Hence

0~(ZQn(Z ; z/(n+2-2e))

~ ~ 1

starlike of order

89 ~ e ~ i.

Letting

"if"part of the following i.

F(2+k-2~) P(2-2~)F(k+l)

0 ~ C(n~k~8) ~

given choice of

THEOREM

-- ~ r(2-2~)F(k+l)F(2+k-2~)zk+l k=0

s i n ( n - j + i)8 : sin(n+2-2a-(l+j-2~))@ F(2+k-2~) = sin(l+j-2e)@ so that C(n,k,@) § F(2-2e)F(k+l)

= sin(~-(l+j-2~)@) as

: z/(l - z)2(l-~)

j.

Here

(near

=

-~)

~ - I -z/z. l 1 + Re[j_~l ] in

are the zeros of

Izl < 1 S k.

if

Thus we

conclude Izjl > i, 1 ~ j ~ nk-1. Now set ZPk(Z) : ZSk(Z) + 2n k z Sk(I/z). Then Pk(Z) has degree 2nk-l, and we wish to show ZPk(Z)

+ g

and

Pk E P2nk_l(~/(2nk+l-2a)).

Since

167

iznk r llon zJ l Sk(Z)

we conclude

Iz

Izl ~ i,

uniformly bounded.

z = e ir =

uniformly there so

Since {S k}

g(z) is

Izl < i.

Pk(Z) = 0

e -i(nk+ 89 ~ [eir

-i(nk- 89162

)

Pk(ei$ ) = 0

That is,

zS k + g

Izl ~ i.

ZPk(Z) = lim ZSk(Z) = g(z) k§

on compact subsets of

= e

when z = 0

It follows that

lim k~

0

0

2n k_ nk+l Sk(i/z) I ~ Izl ISk(Z)l,

is analytic on

For

=

+ e

l(n k

if and only if (ei~) + e i(2nk~) Sk(ei~)]

8 9

= 2 Re[e

if and only if

-i(nk- 89162 Sk(ei$)].

e -i(nk- 89162

)

is pure

imaginary. Since _i(nk_89 ~ d (arg[e d-~ and

]zj] > i,

(4)

Sk(eir

= -(nk- 89 + Re

nk-i _eir [ j=l l-eir

we conclude

-n k + ~ - 8 9~

(arg[e _i(nk_ 89162

]) < -nk/2,

where the left inequality comes from the fact that of order

~

equation.

elude that

Pk

Pk

Izl = i).

p(eir

zS k

is starlike

and the right inequality comes from the preceding -i(nk- 89 Since the argument of e Sk(e i~) is decreasing

and the total decrease in

lie on

,

has

~2 > 91'

2nk-i If

[0~2~]

is

zeros on 92

and

2~(nk-89 = (2nk-l)~ Izl = 1

~i

we con-

(i.e. all zeros of

are consecutive zeros of

then from the left inequality in (4) we have

168

(~2 - ~ l ) ( n k completes

that

is t r i v i a l function

2.

the above

for

~2

proof

a

= i, for

starlike

of o r d e r

= {i

+ yz n

[ii]

from

Pn(8)

a zn n

such

the

: IYI

This

2n k + 1 - 2c~

applies

in this 1

even

case

while

for

f(z)

~/(n

~ <

= z

0.

The r e s u l t

is the o n l y

+ 2 - 2~)

= w/n

and

= i}.

shown

that

that

strictly

then

of

lection which

show

degrees

Thus

which

polynomials

from

yields

if

of the

also form

~ P n=l n starlike

a2z2

the r e l a t i o n

polynomials

of this

f o r m and

THEOREM

Let

i8 c l o s e - t o - c o n v e x lie on

all

~/(n+l)

univalent

1 n + --z n

characterized

between

+ ...

the

the

as

subcol-

(a f u n c t i o n univalent).

of all

is d e n s e

(in the d i s k

It

8 ~ n/(n-l),

is not n e c e s s a r i l y

+ ...

~ P with n=l n normalized

functions

the c o l l e c t i o n

+

It w a s

sequences.

~

are

I.

described

univalent

in the c l a s s

Izl < i).

It is

close-to-convex

class

+ ~1 z n

if and only if all zeros of

Izl = l, and are separated in argument

E Pn_l(w/(n+l)).

from

of s u c h

functions

the p r e s e n t

= z + a2z2

Izl ~

and that

and we h a v e

functions

to n o t e

P(z)

lie

as l i m i t s

that

interesting

2.

n,

of p o l y n o m i a l s

e < 0

z +

= z + a2z 2 + .-.

in

k ~

P ~ Pn_l(e),

shown

(distinct

n

1 ~

the u n i v a l e n t

univalent

P(z)

P

- P(ze-ikn/(n+l)))/

is u n i v a l e n t

of o r d e r

it was

of n o r m a l i z e d

the

the c l a s s

(P(ze i k n / ( n + l ) )

are obtainable

sequences

[ii]

of

of a s e q u e n c e

[4] that

E Pn"

introduced

of p o l y n o m i a l s

- e-ikw/(n+l)),

is s t a r l i k e In

zeros

functions

to

author

here)

the limit

zP(z)

limits

all

increasing

univalent is e a s y

present

as u s e d

z(e i k ~ / ( n + l )

P'

- ~i ~

RELATION OF THEOREM i TO PREVIOUS WORK, In

S

or

the proof.

Note

Pn(w/n)

+ 89 - ~) ~ ~

Pn(w/(n+2)).

fnl

: i.

P'(z)

by

Then

P

are distinct,

2~/(n+l)

(i.e.

169

Proof.

We r e m a r k that

zP'(z)/g(z)

P

is c l o s e - t o - c o n v e x if and only if

has positive real part in

functions.

Izl < i,

where

g

is a starlike

This d e f i n i t i o n is due to Kaplan [2] and,

it implies univalence.

The f o l l o w i n g intuitive g e o m e t r i c a r g u m e n t

is based on the equivalent c o n d i t i o n is c l o s e - t o - c o n v e x

as he o b s e r v e ~

(proved by Kaplan

[2]) that

P

if and only if

e ~ IRe[zP"(z)/P'(z) + i] de > -~,

z = re ie ,

=

e2 ~ el,

when

r < i.

2

We first show that the c o n d i t i o n s given in the t h e o r e m are n e c e s s a r y for

P

To be c l o s e - t o - c o n v e x .

Since

P'(z) ~1 ,

the product of the zeros of

P'

is

of

P'

lie on

P

implies all zeros of

that if

P'

has a double zero on

= 1 + 2a2z + ---+ ~z

l~l = i,

then

It is easy to see P

is not univalent

in the i n t e r s e c t i o n of the unit disk with a n e i g h b o r h o o d of such a point.

It remains to show that the zeros must be separated

argument)

by

2~/(n+l).

when by

Izl = i, (n+l)/2

P'(z)

# 0.

9 (8 2 - 81 )

as

P' # 0,

(goes b a c k w a r d

at a cusp

are separated in a r g u m e n t by n+l -2~ + --~--y < -~ Conversely, Then as

e

Re(zP"(z)/P'(z)

+ i) = (n + 1)/2

Hence the tangent line rotates forward

along an arc where w)

8

varies from

81

to

82

and the tangent line r e v e r s e s d i r e c t i o n (a zero of y,

P').

Hence if two zeros

the tangent line turns an amount

if the zeros are separated by more than 81

to

e 2,

A arg zP'(z)

> -z.

i n e q u a l i t y t h e r e f o r e also holds on the circle of radius i,

and hence

P

is c l o s e - t o - c o n v e x .

the zeros are separated by exactly

2~/(n+l),

family of c l o s e - t o - c o n v e x functions

implies that

convex.

(z = e i8)

if y < 2~/(n+l).

varies f r o m

s u f f i c i e n t l y near

(in

It was shown in [i0] that for u n i v a l e n t

p o l y n o m i a l s of the given form,

,

so the univalence

Izl = I.

Izl = i,

n-i

2w/(n+l), This r,

r

If some of

c o m p a c t n e s s of the P

is c l o s e - t o -

170

T H E O R E M 3.

class

K

Let

g(z)

: z + A2 z2 + .--,

Then

Izl < i.

g

is in the

of functions which map the unit disk conformally onto

convem domains if and only if to-convex polynomials

g

is the limit of a sequence of close-

Inl

P(z) = z + a2z2 + ... + ~1 q zn,

: l,

having strictly increasing degrees. Proof.

It is w e l l - k n o w n that

only if

zg'(z)

only if

g'

is starlike

g

is in the class

(of order

0).

{Pnk }

is the limit of some sequence

Pnk_l(w/(nk+l)), if and o n l y if

nk § ~ P

as

k § ~.

Hence

By Theorem2,

is c l o s e - t o - c o n v e x .

K

if and

g E K where

if and P' E nk

P'nk E % _ l ( W / ~ + l ) ) ,

Hence the t h e o r e m follows.

nk

3. C O E F F I C I E N T REGIONS FOR THE CLASS If

P(z) = i + "'. + a n Z n E Pn(e),

since all zeros of P(zei8))

P

lie on

[zl = i.

a

n

= I.

then

We may r o t a t e

P

It is clear that

so that the c o e f f i c i e n t s satisfy the c o e f f i c i e n t s = C(n,k)

1 ~ k ~ n-i

and

C(n,k,8)

an_ k = a k.

(i.e. form

are p a r t i c u l a r l y

C(n,k,n/(n+l))

and

Pn(0)

9 -. + z n

w i t h all zeros on

~ i.

n = 3:

Let

C(n,k,n/n)

For example, = 0,

Pn(w/n)

= 1 + a2z 2+

Iz[ = I.

P (e), n

1 + az + z 2 E P 2 ( e ) ,

-2 cos e ~ a ~ 2 cos 8.

{Q2(z;8),Q2(-z;8)}

simple.

is the c o l l e c t i o n of all p o l y n o m i a l s

detail the c o e f f i c i e n t regions for We have

P(z) = znp(I/~)

Thus we have

In order to m o t i v a t e the next section,

n = 2:

P (e) n

For c e r t a i n values of

(the usual combinatorial),

{i + z n}

if

[anl = i

We m a k e this a d d i t i o n a l r e s t r i c t i o n on

in the r e m a i n d e r of this paper.

C(n,k,0)

0 ~ e ~ n/n,

w i t h o u t c h a n g i n g the s e p a r a t i o n of the zeros and hence we

may assume

8,

Pn(e),

Hence

(recall that

P E P3(8),

we now d e s c r i b e in some n = 2,3~4.

0 ~ 8 ~ g/2,

P2(8)

if

and only

is the c o n v e x hull of n Qn(Z~8) = [ C(n~k,e)zk). k=0

0 ~ 8 < n/3.

Then

w e have

171

P(z) = i + az + az 2 + z 3,

P(z)

Thus

A : i,

values for

= i + ~

e 2i~/3,

Q3(ze-2i~/3;8)

which we prefer to write

Az + ~

e -2in/3

respectively.

A

A z2 + z 3

yields

P(z)

of this polynomial

by

the closed triangle 8

8 § ~/3.

P3(9)

for the region of

8 ~ a ~ (2~/3)-8

-2~/3.

with vertices

increases with Thus

and

I,

The region e 2i~/3

,

e

and rotations

is contained

-2i~/3

and fills out the entire triangular is contained

Q3(zei2~/3;e),

the polynomial

+ ze-2ia),

2~/3

: Q3(z;%),

The b o u n d a r y curves

are found by considering

(i + 2ze ia cos e + z2e2ia)(l

in the form

in

it

;

region as

in the convex hull of

{Q3(z;e),Q3(zei2~/3;e),Q3(ze-i2~/3;e)}.

n : 4:

0 ~ 8 ~ ~/4,

B

is real.

The coefficient

contained

in the tetrahedron

(-i,0,i)

and

8

increases

(0,-i,-I) to

7/4.

special case of Theorem

4.

FURTHs Since

so that

has the form

sin 40 sin 48 sin 38 Bz 2 + sin 4e ~ z 3 4 = i + ~ Az + sin 8 sin 28 ~ + z ,

P(z)

where

P 6 P4(8),

PROPs P(z)

P(e ir

region

having vertices

(ReA, I m A , B ) (i,0,i),

is (0,i,-I),

and increases

to fill the whole region as

These assertions

will be verified below as a

6.

OF THs CLASS = zn P(i/z),

= einr162

pn(e),

we conclude where

R

e-in~/2p(eir

is real.

Consider

is real the

expression

(5)

P1 (z'~)

=

p(zeir _ p(ze-ir z(2i sin n~)

from which we see that

-

n [ C(n,k,e) s~--~---~zk-i k:l sln n~ ~ '

0 < ~ < e, --

172

Pl(Z,@)

:

n 7 C(n-l,k-l,8)Akzk-i k= 1

We have the following theorem. n [ C(n,k,e)Akzk where k=0 Ak : An-k (0 < k < n), and A 0 : i. Then P n k-i all zeros of Pl(Z,e) : [ C(n-l,k-l,8)AkZ k=l THEOREM 4.

P(z)

Let

Proof.

Assume

=

P E P (8). n

all zeros of

Pl(Z,~)

on

it follows

Izl = i,

pl(z,0 ) = in P'(z) 8 = 0.

For

z = e ir

IzI < i.

from Lucas'

if and only if

IzI < i.

lie in

Izl =< 1

0 < ~ < 8 < _w ~ = n

Since all zeros of

P

lie

theorem that all zeros of with equality possible only when

0 < ~ < 8 < w/n,

0 = P(ze i~) - P(ze -i~)

E Pn(e)

We will show that for

lie in

lie in

0 < e < w/n,

Pl(Z,@)

= einr162

= 0

if and only if

- e-in~/2R(r

But this is true if and only if (cos n~/2)[R(r which

implies

only for

R(r

~ ~ e

P~z,~)

- R(r : R(r

since

P ~

= 0.

Now assume Izl > i.

~,

It follows that for

Izl = I.

Since the zeros of

We wish to show that

We a c c o m p l i s h

vary the coefficients

of

of zeros of

Pl(Z,8)

to obtain

IzI > i

= A 1 + ... + z n-l,

Izl > i.

Recall that

if

has two zeros on

P(z)

exactly

28.

P

without

this implies

Pl(Z,8)

Izl = 1

Pl(Z,e)

Pl(Z,e)

in

changing

IAII > i.

Pl(Z,e)

has a zero on

Since the coefficients

the number of zeros of

Pl(Z,~)

has a

this by showing that we may

continuously

Pl(Z,8)

0 ~ ~ < e. <

this part of the theorem follows.

P ~ Pn(8).

in

+ R(r

The last equality can be true

~n(e).

has no zeros on

vary c o n t i n u o u s l y with

zero in

= 0 = (sinn~/2)[R(r

the number

Since

has a zero in

Izl = 1

if and only

which differ in argument by of

P

IzI

> 1

vary continuously, increases

if

there must at

173

some point

be a zero on

CASE i.

Assume

also has the factor P(z)

Izl

P

= i.

has a factor

(i + ! eiez) P

= (i + BlZ + ...

(I + pe!az),

since

P(z)

+ e-2iezn-2)(l

p > I.

= z n P(I/~)

+ (p+~)eiaz

Then

P

so

+ z2e 2ia)

a [~d sin ne

which

= IB1 + (p + ~) ei~l, P

can be made a r b i t r a r i l y

tending in

IAll

to

0

Izl > i,

large

does not a f f e c t the d e s i r e d

by letting

the n u m b e r

result

p + O.

of zeros

follows

for this

of

Since lying

Pl(Z,e)

case.

CASE

2. A s s u m e all zeros of P lie on Iz] = i, P(z) = iaj (I + ze ), a I < a 2 < ... < a < al+2~ : an+l, w h e r e for some j=l = = = n 9 < 28. A s s u m e the n o t a t i o n is c h o s e n so that j' aj+ 1 - a 3 n

an+ 1 - a n > 2e.

Choose

al~

a]2. - ~31. ~ 28,

a.3s

- a.3Z_ I < 2B,

aZ 0

by to

~s

either

~J2_l ...

" 'eJ2 ,''',a.]k a31

- ajl < 28,

, e.3k = ~n+l .

..-,~js For

~ 28~

and let

t

replace vary from

menormalize

that

'AI'

= slnSine9i S JIn e~

> sin Sinns8

I ~ ei(2j-n-l)el

j =l and the proof

P~(8) = {Pl(Z,e) n = {Q(z) = k=l[ Ak zk-I

all zeros

of

= i,

j =i

is complete.

We define P'(w/n)

- ~js

a.31 =

by r o t a t i n g 9 to o b t a i n A n = i. W h e n n P(z) now has the form ~ (i + ze 18j) where 8j+ 1 - 8j is j=l 0 or 28 and for at least one Jr 8j+ 1 - 8j = 0. Hence it

is clear

Finally~

SO that

jp =< s < Jp+l ~

= as + t(~ 1 + 2(p - 1)8 - es

i.

t = i,

successively

Q

lie in

: P

~ Pn(e)}

: An = i,

Izl ~ I}.

for

0 ~ e < W/n

An_ k = Ak,

Then for

and

1 =< k =< n-l,

0 ~ e ~ w/n

we have

and

174

the relation n

n

Pn(e) = { [ C(n,k,e)Akzk k=0

: k= I

C(n-l,k-l,e)Ak zk-I E P'(8), n

To simplify the notation,

or

{Ak}~= 0 ( P n ( e )

6 P'(8), n

if

n we will say {A k}k:0 ( P~(e) n A 0 = 1 and [ C(n-l,k-l,8)Akzk-i k= 1

A 0 : i}.

0 < 8 < ~/n. We wish to show that if then

{Ak}~= 0 ( P~(8 2)

0 < e I =< e 2 < ~/n

and that

P~(~/n)

and

n 0 ( p~(81) {Ak}k=

is the convex hull of

{ n[ ei2kj~/n zk_l : 0 ~ j ~ n-l}. This will verify the assertions k=l made in the previous section concerning coefficient regions for n = 3,~ where

and will imply that co(G)

pn(8)

co{Qn(zei2j~/n;e)

c

is the convex hull of

: 0 ~ j ~ n-l},

G.

0 =< 8 < ~/n. LEMMA I. Let qn~(k)(z;8) = Qn (zei2(k-l)W/n;e)' i =< k =< n, .^(k) n Then tqn (z;8)}k: I is a basis for the vector space (over the real n

~ Ajz 3 such that ~j : An_~. 0 < j <= n~ j=0 n 9 i2(k-l)j~/n}n and A 0 is real. Equivalently, { ~ C(n-l~j-l~8)z3-1e k=l j:l n is a basis for the vector space of polynomials [ C(n-l,j-l~8)Ajz j-I j-~l such that Aj = An_j, 1 < j < n-l, and A n is real, 0 < 8 < ~/n. numbers)

Proof.

of polynomials

We will prove the first statement.

Since the collection

n-1 n+l n-i n+l {l+zn,z+zn-l,i(z-zn-~, .'. ,z 2 + z 2 , i(z 2 - z 2 )} {l+zn~z+zn-l,i(z-zn-l), ... , i(s 2 l-z

is clearly a basis and contains prove

{Q~k)(z;8)}n k=l

elements,

n~ a kqn^(k)(z-)e, k=l

),z 2 }

odd) (n

even)

it is sufficient to

is linearly independent.

P(Z) =

Then

n

(n

-- 0.

Assume

175

pl(z;e ) =

n [ akei2(k-l)w/n k=l

Q(k) (z;e) E o n-1 '

i.e., n n [ C(n-l,j-l,e)( [ akei2j(k-l)~/n)z j-I - 0. j:l k:l Hence n n i2j(k-l)~/n " n akei2(k_l)~/n 1 - zn [ [ ake z 3-I = [ l_zei2(k-l)~/n j=l k=l k=l

0 =

Now set

z = e -2i(s

1 ~ ~ ~ n.

We obtain

as = 0

and the

proof is complete. LEMMA 2.

P~(~/n)

Proof.

Let

Q(z) =

Hence

Q E P~(wln).

= 0

-i ~ (Ale

Q(e i~ ) = 0

integer

%,

Q(z) =

and

Then

-i ~ r

..

+ A2 e

+.

implies

0 < ~ < n-l.

of real numbers

(6)

%e i ~r +

+

n

e I 7r

is real and

By Lemma I,

~ = 2~/n

for some n there is a sequence {ak}k= 1

such that

(since the coefficient of

z n-1

is

i)

9

Q(e lr

= 0

implies

0 ~ s ~ n-l, which implies

1 ~ k ~ n, then 0 < t < i,

~ e in-2 2 ~) _- _ei(n-l)~.

n akei2(k_l)~/n (nkl) n akei2(k_l)~/n 1 - zn [ Q _ (z;~/n) = [ l_zei2(k_l)~/n k=l k=l

observed above, fying

: 1 =< k =< n}.

if and only if

9

Hence

qn-l^(k) (z;~/n)

n k-i = A1 + A2z + "'" + A2 zn-3 + A1 zn-2 + zn-I [ AkZ k=l

Q(e ir

i ~ e

= co{ei2(k-l)~/n

Q(z) = zn-l.

~ = 2s

a~+ 1 = 0.

Hence for

n [ a~ = i. But as k=l for some s satis1 In (6), if a k = ~,

a k ~ 0,

1 ~ k ~ n, and

176

(~(l-t)

+ ta k)

k=l

e

has all its zeros in

Izl < i.

i2(k-l)~/n^(k) (z;~/n) qn_ 1

This implies

P'(~/n) m

=

co{ei2(k-l)~/n

^(k)(z;~/n) qn-i

Now suppose that aj > 0

for some

j.

n

: 1 < k < n} = = "

as < 0

for some

Replace

Q

by

given by (6) with

as

and all other

remain the same.

ak

replaced by

Al(t)

Qt

t.

Hence

Q

and

Then

aj

Qt

replaced by

and

Izl < i,

Q

aj + t

have the same

Izl = 1

and

Izl > i.

satisfies

IAl(t) I = IA1 + t(e 2i(j-l)~/n - e2i(s

for large

n [ a k = i, k=l where Qt is

Since (t > 0),

az-t

number of zeros in each of the sets But the new constant term

s

I > 2t Isin(j-s

has a zero in

Izl > 1

- I~I > 1

and

Q I P~(~/n)-

This completes the proof of the lemma. The following result,

which we state as a lemma,

of Szeg~ [13] and [5,p.47].

is a result

It is a consequence of Grate's theorem,

which has a beautiful generalization

to higher dimensions

[3].

It

will serve to motivate the theorems which follow. n

LEMMA 3. R(z) =

P(z) =

region

A.

n ~ k=O

c~rcular where

Let

~

n

[ C(n,k)Akzk , Q(z) = [ C(n,k)Bkzk, and k=0 k=0 C(n,k)AkBkZk a n d s u p p o s e all zeros o f P lie in a Then

i8 a s u i t a b l y

every

chosen

zero

y

point

in

of

R

has

A

and

8

Using this lemma we readily see that if R (Pn(0).

the

is a zero

P,Q ( P n ( 0 )

y = -aS, of

Q.

then

This suggests the following theorems which will be proved

in section 8. THEOREM 5.

form

If

{Ak},{B k} ~ Pn(e)

then

{AkB k} ~ Pn(e).

An obvious corollary to Theorem 5 is the following.

177

i. If

COROLLARY

{Ak},{B k} (P~(8)

then

{AkB k} E P~(e),

Corollary 1 is clearly equivalent to Theorem 5 when However Theorem 5 is trivial for for

1 ~ k ~ n-l)

8 = ~/n

(since

0 ~ 8 < ~/n.

C(n~k,~/n)

while Corollary i is not so obvious.

it is easy to see that Corollary 1 is true for

0 ~ e

= 0

Nonetheless

e = ~/n

by means of

Lemma 2. Observe that if where

B k = i,

n 0 {Ak}k=

0 ~ k ~ n~

P~(8).

THEOREM 6.

If

E P~(0),

{Bk} ~=0

E

P'(e)n

Szeg~'s theorem implies

This is a special case of the following theorem.

Further,

If

{Ak}~ P~(82).

then since

0 ~ 8 ~ ~/n~

0 ~ 81 ~ e 2 ~ ~/n

{A k} E Pn(82).

COROLLARY 2.

{Ak}k= 0n

and

{A k} E Pn(81),

then

Pn(8) r eo{Q~k) (z;8)} k=l n 9

0 ~ e I ~ 8 2 ~ ~/n

P~(e)

Eurther,

and A k (

P~(el),

then

c co{ei2(k-l)~/n ^(k) qn-1 (z'8)}~ ' : I"

Note that the last statement follows readily from the first part of the Corollary and Lemma 2.

5.

CONSEQUENCES OF THEOREMS 5 AND 6,

Let

Ck(~)

be defined by z/(l-z) 2(I-~)

=

~ Ck(e) zk+l k=0

i.e. F(2+k-2e) Ck(a) : F(2-2e)F(k+l) and let

S (e)

denote the class of normalized functions

origin with derivative order

~ < i.

Let

1

at the origin)

(0

at the

which are starlike of

178

co

(7)

f(z) =

and set

~ Ck(e)akzk+l , k:0

g(z) =

[

Ck(a)bkzk+l

(

S~(e)

k;O

f~g(z) =

[ Ck(m)akbkzk+l The operation e depends on k=0 but its meaning will be clear from the context. Then by Theorem i, there are sequences {Pnk},{Qnk } E Pnk(~/(nk+2-2~)) ZPnk § f Hence

and

ZQn k § g

uniformly on compact subsets of

Z(Pnk * Qnk) + f*g

Theorem 5)

such that

(Pnk e Qn k

IzI < I.

is the operation implied in

uniformly on compact subsets of

Izl < i, and Theorem 5

implies that

Pnk ~ Qn k E Pnk(~/(nk+2-2e)).

f*g E S (a).

We have thus obtained the following result.

THEOREM 7.

If

f

f~g(z) =

then

and

g,

given by(?),

[ Ck(~)akbkzk+l k=0

The special cases

~ = O

Therefore we obtain

are starlike

is starlike

and

~ = 1/2

nbnzn

are starlike then

of Theorem 7 were

if

~ anzn n=l

With

and

~ = i/2,

of order

1/2

~ bnzn n=l

then

Brickman,

[ anbnzn

Hallenbeck,

If

~ anbnzn n=l and

the

is convex.

[ bnzn

is starlike of order

showed that the extreme points of z/(l_zeiY) 2 (i-~)

[ a n zn

MacGregor,

e = 0

~ nanzn and n=l is starlike; equivalently,

are convex then

the result is:

With

If

~ nanbnzn n=l

n=l

~

of order

recently proved by Ruscheweyh and Shell-Small [7]. result is the Polya-Schoenberg conjecture:

of order

are starlike

1/2.

and Wilken [i] recently

S*(e)

are the functions

0 < y < 2~, and that each

f E S * (~)

has the

repre sentat ion (8)

where

f(z) =

~

I~

" 2(i-~) dw(t) z/(l-ze It)

is a probability measure on

[0,2~].

Using the fact that

179

each of the functions point of

z/(l-zeit) 2(I-~),

~(k). {{qn ~z ;~/(n+2-2e)) }n k=l }~ n=l

0 ~ t < 2~,

is a limit

together with Theorem 6,

we

obtain the following theorem. THEOREM 8.

If

[0,2~]

on

f ( S (~)

then there exists a probability measure

such that (8) holds.

set of probability measures on by (8) are starlike of order

Further,

[0,2~] ~.

If

let

U(~)

denote

the

for which the functions given ~ ~ 8 ~ ij

then

U(e) c U(8).

Theorem 7 can be expressed in terms of convolutions of measures in the class

U(e)

defined above.

If

f(z) : I ~

z/( i - zeit) 2(I-~) d~(t)

I~

z/(l zeit) 2(I-~) d~(t)

g(z)

:

and

then f,g(z)

=

12~ 12~ z/(l-zel(S+t)) 2(I-~) dw(s) dg(t) 0 0

=

IO ~ ~t+2~z/(1 zeiT) 2(I-~) d~(T-t) d~(t)

=

121T[ ; 21~z/(1-ze IT) "2 0

=

Here

~

= ~*v.

(l-a)dg(T-t)]dv(t)

0

I~

" 2(i-~) d~(T) . z/(l-ze IT)

has been extended to be periodic with period

2~.

We write

It is clear that Theorem 7 is equivalent to the following

theorem. THEOREM 9. U(e)

i8 closed under the operation

It is well-known it is starlike of order

([6],[9]) 1/2.

that if

*.

T[ akzk k=l

Equivalently,

is convex then

[ kakzk k=l

starlike

180

implies

[ akzk T is starlike of order i/2. This result is a k=l special case (a = O, 8 = 1/2) of the following easy consequence

of

Theorem 6 .

If

THEOREM 10.

Ck(e)akzk+ 1 E S * (~)

and

~ ~ 8 ~ 1

then

k=O Ck(8)akzk+l

E S~(8).

k=0

6.

SOMEGEOMETRIC PROPERTIES OF THE CONVOLUTION,

EXAMPLE.

Let

f(z)

=

[ kakzk ~ k=l

g(z)

=

be starlike

(of order

0)

and let

akzk = [O f(w)/w dw. k=l

Then

g

zg'(z) g

maps the unit disk onto a convex domain. : f(z),

[z I : r,

is normal to The curve

Also, g({Izl

the vector : r}),

maps an arc of the unit circle onto a straight line segment

only if

f

maps that arc onto a radial

normalized minus

=

=

For

~ = 0

(9)

slits along the real axis,

ze i8 1 [.----I~ 2i sin 8 l-ze ~

be The

i.e.

= l_ze_-~

sin k8 z k k=l ~ ,

0 < 8 < w.

we obtain

f~h(z)

ak ~._--;T:-~-zsin ~ • k8

=

From (9) we see that if

then

h(z)

z (l-zeiB)(l-ze -i8)

k:l

length

Let

if and

starlike function which maps the unit disk onto the plane

two radial h(z)

segment.

so

y

28

is constant when

7-28).

If,

_ g(ze-iB)].

is an amc of the unit circle whose

is greater than

arg f*h(z)

arc of length

F

k = 1 [g(zeiS) 2i sin 8

and ze i8

for example,

g(F) and g

is a line segment~ ze -i8

lie on

F

(an

maps the unit disk onto

181

the interior of

polygon and

a

B

is small,

then the Hadamard

product of

12 with

g

[

h(z)/z dz =

i sin k8 k [ sl~---~--~u z

k=l

is a m a p p i n g w h i c h smooths out the corners of the polygon

w i t h a slight amount of shrinking and stretching. The above example is a special case of a more general result which we obtain below lemmas concerning LEMMA 4.

Let

(Theorem 13).

be a given real number and let

{Qn(zei(0+2kS);8 )}k=0 n

of degree

less than or equal to

Since the d i m e n s i o n of the vector space is

sufficient

to prove that the given polynomials n independent. Assume 7. akQn(zel(~+2kS);8) k=0

Qn(zei(~+2k8);8) setting

has the

an

LEMMA 5. If

=

0,

Then

an_ 1

0 < 8 < ~/n

with :

0

, ...

and

,

:

P (Fn(8),

0

Since

0.

n

)}j:l'

we obtain in

9 then

for all real

has the r e p r e s e n t a t i o n 9

P(z)

where

ak

is real,

z = -e -(~+(n-2s

0 < k < n. ,

§

nke ) Qn(zei(~+2kS) ;8)

Further,

s = 1,2, .. .,p

an-p+ I 9 Proof.

n

n -l(~ 7 ak e = k=0

By Lemma 4,

then

it is

are linearly

s = 1,2,.-.,n+l, a0

n.

n+l,

{ (l+zei[~+ (2j-n-l+2k) 8 ]

factQrs

z = -e -i(~+(n-2s

succession

P

0 < e <w/n.

is a basis for the vector space of polynomials

Cover the complex numbers) Proof.

some preliminary

Pn(e).

the polynomials

~

We require

if

P(z)

= 0

for

0 = a n = an_ 1 : ...

~,

182

P(z) =

n[ ak e -i(~r k=0 n

+ i ~ bk e

nke)Qn(zei(~+2ke);e)

-i(~ @+ nke)

k=O

where

{a k}

R(r

where

R

=

and {b k} e- i n ~r

and

ir

Rn

n -~r [ bk e k=0 lemma is p r o v e d .

are real.

=

Setting

Hence

(zei(~+2ke)

The r e s t

(zei(~+2k8)

z : e ir

n [ a kRn(r k=0

are real. Qn

Qn

+ i

of the

we have

n [ b kRn(r k=0

n [ b kRn(r k=0

;8) ~ 0.

;8)

~ 0

and

Thus the first part of the

lemma i s

proved

by t h e

same d e v i c e

used in proving Lemma 4. THEOREM ii. A s s u m e {Ak},{B k} ~ Pn(e), 0 < 8 < ~/n. Further,. n assume P(z) = ~ C(n,k,e) A k z k has p zeros, z = -e lSj, w h e r e k=0 n ~j = r + 2(j-l)e, 1 < j < p, a n d Q(z) = [ C(n,k,8) Bkzk has k=0 q zeros, z = -e i0j, w h e r e ~2j = ~i + 2(j-l)e, i =< j =< q. I f n p + q > n,

then

P~Q(z) =

~ C(n,k,8)AkBkZk k=0

has

p+q-n

zerosj

i(r z = -e

,

Proof.

P(z) =

P

and

Q

1 < j < p+q-n.

have the representations

n-p i(~(r [ ake k=0

)

i(-r Qn(ze

i(~(~l+(n-l)e)-nMS) Q(z) = n~q bke k=0 Hence P,Q(z) : 2n-~q k[ ak_Rb s ei(~r162162 k=0

s

and the desired result follows.

;B)

i(-~l-(n-l)8+2kS) Qn (ze

;e).

183

Remark. for

Theorem ll holds trivially

8 = 0.

To see this,

suppose

zeros of multiplicity

p

{Pk }

{Pn(Sk)}

Pk

and and

{Qk } Qk

3.

k § ~,

q

and

Q have

We choose sequences

e k § 0,

Pk § P'

of Theorem ii.

we obtain the result for

and Ql(Z;e)=

0 < e < w/n, and that

Cj+l - Cj = 2e,

Qk § Q

Applying

and

Theorem

8 = 0.

P1

1 < j < p-l, and

n ; C(n-l,k-l,e)~z k-I k=l has

p

Q1

Zj = e iCj

seros

has

q

seros

~j+l - ~j = 2e, 1 < j < q-l, where p+q >. n [ C(n-l,k-l,8)AkBkZk-i has p+q-n+l zeros z. = e 18j k=l ]

zj = e n-l.

P

Assume

Pn(8),

satisfy, ing

and is also true

and that

respectively. where

n Pl(Z;8) = [ C(n-l,k-l,e)~z k-I k=l belong to

e = w/n

P,Q E Pn(0)

satisfy the hypotheses

ii and letting COROLLARY

from

and

for

satisfying Then

8j+ 1 - 8j : 28,

satisfying

Proof.

For

and the remark.

1 <= j <__ p+q-n.

0 ~ e < ~/n, For

e = ~/n,

the fact that all zeros of z = e ins

this result follows from Theorem ii the result follows from Lemma 2 and

P1

for some integer

and s

Q1

on

[z I = 1

satisfy

The proof then proceeds much like

the proof of Theorem ii. THEOREM 12.

fCz) =

Let

open arc of length satisfies

{Pnk }

zPnk § f

uniformly

zeros

s

),

on

Re{zf'(z)/f(z)}

sequence

s

y

e

where

~ Ck(a)akzk+l k=0 Izl=

= a

of polynomials

J

on

Then F

f

and let

r

be an

is analytic on

r

and

if and only if there is a

such that

on compact subsets of

(I ~ j ~ s )

s

i.

~ S*(m)

satisfying

Pnk E Pnk(~/(nk+2-2e)), Iz] < I,

and

~j+l - ej = 28,

Pnk

has

(i ~ j

k § y/2~.

We will give the proof of Theorem 12 after stating and proving

184

- ~(mod

[(n+2)/2]Ar

R.

If

aAr is

2~),

where

Ar = 2 ~ / ( n + 2 - 2 ~ ) , so the a v e r a g e

A#

then

is the :separation of the zeros of

[(n+2)/2]Ar

r a t e of c h a n g e

- ~ = a(2~/(n+2-2a))

of the a r g u m e n t

=

o v e r this i n t e r v a l

e. Now assume

a sequence

K

P

as d e s c r i b e d in T h e o r e m 12 is nk s u b a r ~ of F and e > 0. We m a y (by

given.

Let

be a c l o s e d

Theorem

1 and the d i s c u s s i o n

above)

choose

0 < r0 < 1

such that for

r0 < r < 1 e < Re{zf'(z)/f(z)}

Hence

zf'(z)/f(z)

is a n a l y t i c

on

< e+e

F

(z ~ rK).

and

Re{zf'(z)/f(z)}

= e

there. Conversely,

suppose

Re{zf'(z)/f(z)}

= a

on

F

and

zP

§ f, nk

Pnk

(Pnk(~/(nk+2-2~)),

(i ~ j ~ s )

on

F.

a n d that It f o l l o w s

Pnk that

has if

s

zeros

e > 0

and

e iej k

is s u f f i c i e n t -

ly l a r g e t h e n

and we m a y r e p l a c e that s a t i s f i e s {Pnk}

7.

2~ ~j - ~ j - i - n k + 2 - 2 ~

<

{Pnk }

from

by a s e q u e n c e

the conditions

as in T h e o r e m

In this s e c t i o n ,

then follow

L E M M A 6.

described in s e c t i o n

For

'

{Pnk(~/(nk+2-2a))}

12.

That is, we m a y c h o o s e

12.

A PARTICULAR SUBCLASS OF

lar s u b c l a s s

of T h e o r e m

2E nk+2

Pn(8).

we w i l l p r o v e T h e o r e m s in the f o l l o w i n g 8.

0 < m < n

we

have

5 and 6 for the p a r t i c u -

lemma.

The g e n e r a l

case w i l l

185

the main result of this

section,

example

given at the beginning

THEOREM

13

f(z)

Suppose

9

which was illustrated

of this section.

:

Ck(~)akzk+l

,

g(z)

~[ Ck(e)Bkzk+l ( k:0

:

k:0 S (~) Xg

Ff,Fg

and that there are arc8

respectively

such

that

f

and

: a,

Re{zg'(z)/g(z)}

tively,

yf+yg

> 2~.

f*g(z)

on which

:

Proof. 12.

f

This theorem

and

ZQmk § g

many elements

in common.

Theorem

12

a > 0,

univalent).

Ff

on

is analytic

is an easy consequence is in doubt so that

and

Fg

of Theorems

Ii and

is whether we can choose

{n k}

12.

and

{m k}

have infinitely

i).

Note that if

ZPnk § f

i (zP + z2P~ ) n+l nk k

P 6 P (2~I(n+2-2~)), n

9n+2. zP(z) = e I 2 9R(r

yf+yg-2~

and satisfies

then also ' f

it was shown in [ii] that these polynomials For

respec-

This will be clear from the proof of

n+2 zP n+l nk (for

: a

Then there i8 an arc of length

(and the proof of Theorem

Proof of Theorem

and

: a.

The only point which

ZPnk

yf

of length

are analytic and satisfy

~ Ck(~)akbkzk+l k--O

Re{z(f*g)'(z)/f*g(z)}

Izl: I

on

g

Re{zf'(z)/f(z)} where

in the

are

we have i~[

zP(z)+

z2P'(z) = e

~R(r162

(z = e ir

so that 9

n+2 ~.. _ 1 n•

, ,~ = [ ~ R ( , ) [zP(z) + z2P'~z;j

Since the zeros of the above (and hence

expression

R as

and r

from one zero of

R'

interlace,

varies P

n+2

i + n--~R'(r

the change in argument

form one zero of

to the next)

(z = el~, "

is

R

to the next

i

of

186

Qn_m(Zei(me+2mt(7/n-e));e

) Qm(ze-i((n-m)e+2(n-m)t(n/n-e));e

)

(i0) m[ a e i(q-mt)(7-ne) q:0 q

Qn(zei(2mt(7/n-e)+2qe);8)'

0 < t < 1 = = '

where m-q

q ~

sin( (n-j +i) e +t (7-ne)) a

q

: C(m,q,e)

sin((s

s

j:l m

sin(n-p+l)e p:l (the products Remark. zeros

are understood

Note that the polynomial

into two collections

consecutive

zeros

28.

varies

As

t

ly from (n

to be

Qn(Z;e)

zeros

from

symmetric

0

zeros about

to

n-m

i,

when the upper limit is

defined and

in each collection

(n

around the circle

of

1

by (10) separates m

the

zeros respectively

being

separated

the polynomial

0). n with

in argument

changes

continuous-

^(m)(z;e ) qn

symmetric

about

-i)

to

-e -i2mT/n)

with

n-m

zeros traveling

in one direction

and the other

by

m

zeros traveling

in (i0),

we wish to show

in the other direction. Proof.

Equating

coefficients

of

zk

k [ C(n-m, k-s e)C(m, Z, e) e i(km-ns (ll)

= C(n,k,8)

m q=O

where if

(e +2t (7/n-e))

s

nl'

Z > m

~172 C(m,q,e)ei(q-mt)(~-ne)eik(2qe+2mt(~/n-e))

73

72, and or

73

are the products

in the lemma and

C(m,s

= 0

s < 0.

We have 1 m-q 71 = (2i) m-q j=l

(e

i((n-j+l)e+t(7-ne))

e-i((n-j+l)e+t(7-ne)))

1 e i (m-q) ((2n-m+q+l) 8/2+t( ~-ne ))(m~q( l_ei (2j-re+q-l)eei((m-q-l-2n~ -2t(~-ne ) ) (2i)m-q j=l

187

el(m-q) ((2n-m+q+l)e/2+t(~-ne)) ~q(_ei((m-q-l-2n)e-2t(~-ne) );e). (2i)m-q Similarly, ~2 -

i eiq((q-l)e/2+t(~-ne)) (2i) q

Multiplying both sides of (ii) by

Qq(_e-i((q-l)e+2t(~-ne)))

e-i~(e+2t(~/n-8))'

we need to

show k C(n_m,k_~,e)C(m,s

-i~(ne+2t(~-ne))

4=0 m C(n,k,e) sin(n-p+l)e

m~ (_l)q q=O

i

e im(2n-2k-m+l)e/2 ~ (-1)4

(2i) m

s

[ C(m-q,s s=O

p=l C(q,s,e)e-iS(m-2n-2)eeis163

C(m,q,8)e -i2s

Thus it is sufficient to show that for fixed

4,

0 ~ s ~ m,

0 < k-s < n-m, sin(n-k)e

... sin(n-k-m+s

sin ke -.. sin(k-s

m (12)

= (_i)~

1 [ (_i) q [ C(m-s (2i) m q=0 s=0

eiq(2k-2n-s

e

c(~,s,e) e-is(m~2n-2)e

im(2n-2k-m+l)8/2eis

where we have used the fact that

CCm'q~e)C(m-q'Z-s'e)C(q~s'9) C(m,s163

Using the exponential

=

C(m-s

form of the sine function,

we can write

the left hand side of (12) as 1 ei(m_s163 (2i) m

m-Z (I e i(2j-m+s j=l

e

i(m-s

9

188

(_l)s163163

(i - e

i(2p-s

e

i(2k-s

p:l (_l)Zeim(2n-2k-m+l)8

e-is

eims

~

~(-ei(2k-2n's

(2i) m Qs163

Hence (12) follows. LEMMA 7.

Assume n

P(z) =

belongs

~

Pn(e),

to

~

[ C(n,k,e)Akzk : Qm(zelS;e) Qn_m(ZeiY;%) k=O where

m8 + (n-m)y : 2 ~

for

0 < m < n,

come

s

0 < e < T/n,

Then

for

(and h e n c e

sufficiently

small

positive

n

e,

Pc(z) :

[ C(n,k,O+e)Ak zk E Fn(e+s k=0

Proof. for some rotation. of

0

We first observe that we may assume

t,

n - m {zj(e)}j= 1

Let

such that

terms in (i0)).

Pe(Zj(e))

be continouus functions in a neighborhood = 0,

z.(0) ] We wish to show that

This will mean that for arg zj(e) > 2(8+~).

d arg

e

> 2.

0,

arg Zj+l(e) -

If we apply this same result to the polynomial

Re(V) = Pc(z),

m

Pc(z)

and

n-m

and use the

it will follow that consecutive zeros of

are separated in argument by more that Hence

zj(0) =

Je=0

positive and near

obtained from (10) by interchanging

and positive.

and

are the common zeros of the

z~(e)|]

de

de

P (z)

1 <__ j _<_ n-m,

(i.e.

d arg Z~+l(e)

fact that

is given by (i0)

since the other possibilities may be obtained by

-e -i[(n-2j+l)9+2mt(~/n-8)]

Re(z)

P

~ Pn(e+c).

2(e+e)

for

e small

189

We have 0

-

dPr dc m

ei(q-mt)(~-ne)

a

q:0

q n

where

Qn(z;e+e)

_dd QnCzjCe)eiCC2mt)(~In-e)+2qe);e+e dc

)

(i + zei(2s

:

Z:I Therefore i ~

Qn(Zj(e)ei(2mt(~/n-e)+2qS);e+e)l"e=

:

n H ~=I s

0

~ d arg zj (e) I [ l-e2i(g-n-l+J +q) e][(n+l- 2j-2q) + de ] e=0 -q

= (_ 2 i ) n _ l e - i ( ~

- n(j +q) )8

n~ ~=i

[ sin( s

+q) 8 ][ n+l-2j-2q +

d arg

z~(e) I

de

],

s

and we have m

n E [sin (A-j-q)][n+l-2j-2q g=l

aq(-l) q q=0

+

d arg zj(e) I de e=0 ] = 0.

s

It then follows that m

n+l-2j

+

d

arg

de

z~(r

I

r

:

m

;. a q ( - l ) q=O

and so

n

~. q aq(-l) q H sin(s q:0 ~=I

n

q II s i n ( ~ - j - q ) O ~,=i

s

190

d arg Z~+l(o)

_ d arg zj

de

(o)

de m

n

qaq(-i) q :2+2

m

ZCj~q+l

q[0 aq(-l)q

s

q=0

~ s

m

n

_

[ %(-1) q ~ sin(~-j-q)e

sin(s

q=0

s s

s

We must

show that the quantity

1 ~ j ~ n-m-I

and

in brackets

0 ~ q ~ m

in the first denominator n H sin(s Z:I

sin(s

so that

is positive.

Here we have

1 ~ j+q ~ n-l.

The product

is j+q : ( H sin(s s

n H

sin(~-j-q-l)e)

s

s

=

j+q (-I) j+q ( ~ sin s Z=l

where the last factor

is to be replaced

the first denominator

in the equation

[arg Zj+l(e)

- arg zj(e)] m ~

(-i) j

aq(

q=O

at

0

m a

(

q=0 q and as before Since

a q => 0

occur above, we must

j+q : n-l.

the derivative

Thus

of

sin s

is

> 0

j+q-i R sin ~e) ~=I

is to be replaced for the values

the two denominators

show that

if

sin s

is

n-j-q R sin s A=l

the last factor and

for

1

n-j-q-i E s

j+q n-j-q-i ~ sin~e)( ~ sin s 1 Z:I

Similarly, the second denominator

(_l)J -I

by

(

by of

are of opposite

1 q

when and

sign.

j+q = i. s

which

Therefore,

lgl

m

m

n

~ aqap(-l)P+q(q-p)[ q=0 p=0

n

~ sin(~-j-q)e 4=1 s

~ sin(~-j-p-l)e] s=l s#j+p+l

>

0,

that is, m

q~l

n

g aqap(-l)P+q(q-p) q=0 p=0

n

[ H sin(~-j-q)e ~=i s

H sin(~-j-p-l)e s=l s#j+p+l

n

sin(s

H sin(~-j-p)8] s=l s#j+p

s s m

qzl j'+q-i [ aqap(q-p)( ~ sin s ~=0 p=O s

n-j-q-i ~ sin s s

n-j -p-i 9( H sin sS)[-sin(n-j-q)8 s=l

The quantity sin n8 sin(q-p)8

in brackets > 0,

sin(j+p)8

is

P(z)

first observe P

and

values a

q

and

Q

ii

k-m

and

Q.

course,

=

n [ C(n,k,%)AkBkZk, k=0

different

m

We will use the coefficients

b

(10) for

that

zeros respectively zeros

independent

Q

and replace

P,Q(z) each

k = m

or none at all.

and

t

k = m = n/2, These

zeros and the other

by

k

m ~ k ~ n/2. of

separation

in argument

t

instead of

q

We assume

having minimum

or

m

and

has two collections

(i.e., they are separated

we could have

one such eollection called

P*Q(z)

in this representation.

then implies

consecutive

=

n n [ C(n,k,8)Akzk and Q(z) = [ C(n,k,8)Bk zk, we k=0 k=0 some ways of forming the convolution. We will assume

respectively

Theorem and

P

- cos(n+q-p)8]

> 0.

and the lemma now follows.

i n the representation s

+ sin(n-j-p)esin(j+q)8]

are given by (i0) with possibly

for

j+p-i ~ sin sS) s=l

1 ~[cos(n+p-q)8

In order to study the convolution where

=

by

n-m-k

between 28).

Of

in which case we have

zeros of

P*Q

will be

zeros will be called

dependent

192

zeros.

The independent

zeros of

P*Q

arise from the fact that

m

[ aq(-l) q e-i(q ne+mt(~-ne)) q=O

P*Q(z) =

so that if all

terms are

m+l

O~

is such that all the terms are for

z = e i(~+28)

Then

0

Q(zei(2mt(~/n-e)+2qe))

P,Q(z)

is zero.

Assume

z = e i~

but that the last term is not zero

e-i(n/2)~P,Q(eir

is real and has the

m

value

[ aq(-l) qs (r q:0

< ~ < ~+2e, P,Q(e i~) # 0.

(~/n-8)+2q8 )

where

Similarly,

P*Q(e ir

e i(@-28).

~ 0

if

If

consider the dependent

k

zeros are

Thus,

in order

we need only

with

n

C(n,k,e)B k = is the elementary

variables.

Then

P*Q(z)

=

and we may form this convolution by forming

n ~ C(n,k,e)Akzk , k=0

n-i [ C(n-l,k,e)(Ak+~iZAk+l)zk, k=0

n-s ~~ C ( n - ~ , , k , 8 )S(s [ p=0 k=0 P )( { l Z , . . . , ~ z ) A k + p Z k sequence of polynomials

by

9 .-, Pz(z;~iz,...,~s

....

-i~Sps

28.

then

~k-(n) = [~jl~j 2...~jk

symmetric function of degree

Ps163

but the first term is

of this type,

n Q(z) = H (l+{-z),j j=l

where

k=0~k ~ -(n)(~l , 9 .,~n)Akzk ,

e i@

where

zeros.

Observe also that if s(n), k ~ l , ~ 2 , ' . . , ~ n ),

4-28 < r < ~,

zeros by more than

to prove Theorem 5 for polynomials

(13)

real.

This implies that the independent

separated from the dependent

successively

is

then all terms in the sum have the same sign and

is chosen so that all terms are zero at not zero at

S

...

... ,

Denote this

P(z) = P0(z), Pl(Z;~iz),

P2(z;~iz,~2z),

Then

z) :

,

,~s

+ ei~SpE(ze-iS;~l

z,

~s +

= 2 cos~

n-s

8

193

Ps163

-

+ (~Z+I -i)

It is clear that function of

2i sin(n-s Ps163

~iz,-..,~s

is linear and symmetric as a hence the convolution

is independent of the order of 9

p~(ze,-ie;$1z,...,~zz)

61'''''~n"

P*Q = Pn(~iz,...,~nz)

Further,

if

k

Q(z) = Qn_k(zeiY) ~ (l+~jz),

j:l

then

P*Q(z)

is a rotation of

Pk(Z; ~i z, 9 "", ~k z) . We are now ready for the following lemma. LEMMA 8.

Theorem 5 holds for polynomials of the form

Qm(zeiS;e) Qn_m(ZeiY;e) ~ Pn(8).

Proof.

As in the above discussion,

are given by (10) with representation of where

Q

aq, m, t by formula

I~ll = I~k+iI = 1

and

we will assume

replaced by (i0).

Assume

~j = ~j_l e-2is,

P

and

bq, k,s

in the n Q(z) : ~ (i +~jz),

j:l

for

2 ~ j ~ k

k+2 ~ j ~ n. Consider Pn_2(z;~iz,''',~k_iZ,~k+iZ,''',$n_l z) n-2 =

k:0

s~n-2)(~l,--',~k_l,~k+l,''',~n_l)Akzk

+ 2(cos O)z

Q

n-2 ~(n_2)(~l, ", k [ bk "" ~k-l'~k+l'''''~n-l)Ak+l z k=0

+ z2 n~2 ~(n-2) k A(z) + 2(cose) zB(z) + z2C(z) k=0 bk Ak+2Z =

where C(z) : zn-2 $1~2...~k_l~k+l'''~n_ I A(I/~).

and

194

Then (14)

P*Q(z)

Assume

P,Q(z)

: A(z) + (~k+~n)ZB(z)

= 0 = P~Q(ze 2ie)

for some

+ ~k~nZ2C(z).

z.

Then we have

A(z) + (~k + ~ n ')zB(z) + ~k~n z2C(z) = O, (15) A(z) + (~i e2ie +~k+le2ie)zB(z)

Assume for the moment that the equation

+ $1~k+l e4i8 z2C(z)

IB(z)I < IA(z)I

A + (~+~)zB + {nz2C = 0. ~Z

=

-

IBI < IAI,

assumption

arg ~

~ = ~k

I+(B/A)~z .

ering the location of impossible to have The results imply that for

decreases as

when

~ = ~n

~j,

in Marden ~ = 0,

and

when

C"

arg n ~ = ~n

i ~ j ~ n,

~ = ~ie2i8

and consider

We must have

B/C + ~z Since

( : IC(z) I)

= 0.

increases and by when

n = ~k"

Consid-

we see that it is clearly n = ~k+l e 2i8

[5, pp.38-43]

IA(z) I > IB(z)I

in conjunction with (13) when

z

is a dependent zero.

Taking P(z) = Qn m (zei(me+2mt(~/n-e));e)

Qm (ze-i(~-m)o+2(n-m)t(=/n-e));8)

and ^

.

Q(z) = qn_k~ze with

m, k, s, t

false then

i(kS+2ks(~/n-%))

fixed and

8

8 ) Qk(ze-i((n-k)8+2(n-k)s(~/n-e));8 )

variable, we see that if Lemma 8 is

IA(z) I = IB(z) I = Ic(z)I

for some

z

and

P*Q(z) = 0, z a dependent zero of P*Q. Since ~(n-2)/2 ($1$2"--~k_l~k+l..-~n_l) 89 B(z) is real when becomes

A(z)(l +z~keir

+ Z~neir

for some

e

such that

Izl : i,

(14)

~, and one of these

195

factors is

0.

Thus the value of

z

is independent of either

or ~n' and we may assume it is independent of n [ s(n-l)(~l'~ '''''$n l)Ak zk-I 0, Izl = i. 2 = k=l k-i n [ C(n-l,k-l,e) Akzk-I k=l we see that

Pl(Z,e)

tion of n-m-i

This means

Since

P(zel8) - P(ze-le) 2i z sin ne

:

has one zero in

m-i zeros with consecutive

~n"

Izl < i,

zeros separated by

zeros with consecutive

{k

: Pl(Z,e)

one collection of 28,

and one collec-

zeros separated by

28.

Let-

ting Q(z) =

n-i ]I (l+[jz) : Q ( z ) / ( l + ~ n Z ) : 9:1

n-i [ C~(n-l,j,~)B.z j j :0 ]

and 9

m

Pl(Z,e) = Qn_m_l(zel@;8) where

Iqjl = i,

1 ~ j ~ m-i

the remark preceding Lemma 8)

E (l+njz), j=l

and Iqml > i, we have (in view of n (n-l) [ Sk_ 1 (~l,~2,---,~n_l)Akzk-i is a k=l

rotation of ^

Qm(Z;qlZ,q2 z,...,qm z) =

n-l-m m ^ 9 [ [ C(n-m-l,j 8) s(m)(qlZ,''',qmZ)Bj+pZ3 j:0 p:0 ' P

n-l-m m-i ~(m-l), zj : j[0: p=0[C(n-m-l,j,e) ~p ~nlZ,..',~m_lZ)Bj+ p

+ nmZ

= Since

l~jl = 1

ID(z) I = IE(z) I when

n-l-m m ^ 9 j!0 p!l C(n-m-l'J'8)S~ ll)(nlz'''''Nm-lz)Bj+pz3

D(z) + qmZE(z) .

(i < j < n-l)

and

I z I = i. Therefore~

lqjl : 1 (i < j < m-l), n ~(n-l),~ .. r )Az k-I A ~ 1 ~%1'" '~n-i "I< = 0 k=l

196

for some

z,

Izl : i,

implies

D(z) : E(z) : 0.

location of the zero is independent of

~(z)

form

Qn_m_l(zei~)Qm(zeiY)

Nm"

Thus we may change

for some

y

Q

are of degree

conclude that the zero an independent zero.

8.

PROOF OF THs

n-l, z

nm

nm

of

to has

SO that

a zero which does not depend on one of the zeros and

This means the

P.

Since

we may use an induction argument to

of

P*Q

for which

IA(z)I

= IB(z)l

is

This concludes the proof of Lemma 8.

5 AND 6,

We require the following lemma, extreme points in the class ( n-i {Qnm)(z;e)}m=0 9

P (8) n

which implies that the only are the polynomials Pn(e) c

This gives an alternate proof that

n-i co{Qn(m) (z;e) }m=0" L~MMA 9.

P E Pn(8),

Assume

where

0 < e < n/n,

and that

i~. P(z) :

K

Qkj(ze

J;8),

8j + (kj-1)e < 28+Sj+l-(kj+l-l)8

where

j=l (i < j < 9,-1) and P

into factors

1 < p <= s

< 28+2~+81-(kl-l)e

which have zeros separated by

number of factors). and

8s + (ks

Then

for

~

real,

lel

2e

(i.e.,

factor

using the minimum

sufficiently

small,

we have 9

~

.

Pc(z) : P(z) + ie(kpzeiBIQ~_l(zeZSl;e)Qk

(ze18p;e) p

iSpQ~ kI -

(zeiSl; 8) ze

" (zelSp;e))

Qkl

s " H__2Qkj(ze lsj ;8) E j

j~P Proof.

Let e

so that

-i(kjl2)r

9 Qkj (elf;e) = Rj(r

l~j~,

Pn(e).

197

-i n r e

!T.kj 8j

2 p (eir e

• j:IH Rj(r

since of

9

[kjSj

Ps

9. H Rj(r j:2 j~p

RI(r

+ e(kPR{(r162162162

is a multiple of

0 ~ k ~ n.

functions such that where

Let

j:2H _R](r j~p

27.

Hence the coefficients

r

> r

We will show that < 0

for

the zeros of

1 ~ q ~ kl-l. Rp(r

Differentiating 0 : dRe(r de

~l,p

•

> "'" > Ck I' and

{Ak(S)}

A0(g) = i,

r162162162

-i ~ r 2 pe(elr

Re(C) = e

RI(r

r

[ H Rj(r j :I

satisfy the required coefficient relation

An_k(g),

r

kl + ie(kp(--~ RI(r162

s

2

k - kl(-~2 Rp(r162

9Rp(r

=

e

Ak-~-~=

be continuous Re(r

: 0,

r162

1 ~ q ~ kl,

are the zeros of

! Cq+l(0) - r

= 0

and

~q+l~"(0) -

The same result will clearly hold for

and the proof will therefore be complete. Re(r

and setting

: • [ [ ( H Rj(r162 s j~s

e = 0,

we find

s)

J

j~l,p,s

(kpR{(r162162162

H Rj(r j#l,p

8j)

so that

H Rj(r162162162

j~l

I) H R.(r j~l

]

I)

198

and r '( 0) = -kp,

i =< q ~ k I.

Differentiating once more and setting :

[R~(r

+ 2 ~l

l)

H

j~l,s

H Rj(r162 j~l

e = 0

+ [R[(r

R~(r162162

I)

yields H Rj(r j~l

+ 2(kpR~(*q+Bl)Rp(r

+ kpRI(~q+81) Rp(~q+BP) - klRi($q+Bl)Rp(~q+SP))(j~l,pll R.(r162

+ 2 kpRi(r162

sIl,p j~l,p,sH Rj(r162162

Hence

R"(r ~"(0) = k 2 I ~ -q p R{(r

I)

-

R~(r 2 kpk I Rp(r

Using the fact that ~+Sj+(2s-kj-l)8 k S ks R-(~+6~), = 2 H cos ( ) s=l and Cq = ~/2 - 81 + (-2q+kl+l)e, we have k 8n-81- (kn-2j- k]+2q) 8 = 2 9 cot(s-q)8 - kpk I ~ c o t = P s=l j=l s#q k1

r

=

q

so that

k 2

p)

199

0

~p-81- (kp-~+2q) T~ Cq+l(0)

- -q~"(O) : k 2p (cot (-q@) -cot(kl-q)8)-k p k I (cot ~p-~l+(k~§ 2

- cot

k sink 8 = -kp [sin qP8 sin(kPl-q)0

Since both quantities Proof of Theorem P*Q(z) true for

k I sin kp8 ~9-~y(kn-kl+2q)8 sin 92 in brackets 5.

Assume that

near

z/n

{8 E [0,w/n]

is therefore

is such that all zeros of

Izl = 1

when

P,Q (Pn(8)

: P,Q ~ Pn(@)

both open and closed,

=

C(n,k,8),

under the conditions

and the theorem follows.

> 0.

and

P

Rp(Z) = ze iBp Q~ (zeiBp;e) p

Let

P(z) + is[kpRq(Z)

~(e)

be a continuous

(P + ie[kpRq- kpRp]) property of

P,

, Q(e i~(s))

Q, and

~

8

fixed.

e-i(n/2)(~+2@)P*Q(e i(~+2@))

P,Q E Pn(8), If

@

and

Q

P*Q(e i@) = 0,

iBj E Qkj(ze j=l j~P

- kqRp(Z)]

E Pn(e)"

function

such that

= 0.

We

is

and

are both as in Lemma 8,

there is nothing to prove. Therefore, assume that s E (zelSJ;8) where s > 3 and that P,Q, and j=l Qkj = the above minimum is achieved. Let

By Lemma 9,

1 ~ k ~ n-i

P,Q ( P (8). n v ~ 6 [8,w/n]}

P*Q ( P n ( @ ) ,

0 < @ < 8 < w/n,

wish to show that the minimum of

d/d@(e -i(n/2)@P*Q(ei@))

(this is

We will show that this implies

To begin the proof, let

positive

the lemma follows.

since the coefficients

are small in this case). The set

0

~p-~l+(kp+kl-2q) '3]" 2

sin

are positive,

are distinct and lie on @

)

P(z) = ~

are such that

;0).

~(0) = ~

and

Then due to the minimizing

we have

ie i~ (PeQ)'(e i~) ~'(0) + i[kpRq - kqRp] , Q(e i~) = 0

200

and [-i~P*Q(e i(~+2r Fixing

p

and letting

Rq*Q(e i(~+2r gin.

+ iei(~+2r

Since

q # p

we may fix

Rq*Q(e i(~+2r

= 0.

we see that the vector

1 < q < 4.

q,

is contant.

~(e)

of the above type,

q

If

~'(0) = 0

this is impossible.

p

vary to obtain the

P

for every choice of

p

and

Taking a sufficient number of variations

we see that

(i.e., any two zeros of

and let

lies on the same straight line through

the origin for then

vary,

'(0) + i[%Rq-kqRp],Q(ei(@+2r

always lies on the same straight line through the oris > 3,

result that

'(ei(~+2r

P

is an interior point of

are separated by more than

Hence for some choice of

p

and

Pn(e)

2e); q,

but

~'(0) # 0.

Since s

s

Rq(Z) : zP'(z),

~ Rq~Q(z) = z(e~Q)'(z), q=l

q=l it follows that

Rq~Q(e i(@+2r

on the same line, or

i ~ q ~ 4.

ei(~+2r162

IP,Q(z)I

covered

as

lie

This implies either

P,Q(e i(~+2r

is real.

= 0

Since

maps the unit disk onto the half plane n

times, a point

z

on

is real (and therefore= n/2)

z(P~Q)'(z) /P*Q(z) of

ei(~+2r162

/ P~Q(e i(~+2r

z(P*Q)'(z) /P~Q(z) {Re w < n/2}

and

z

varies on

Izl = i.

Izl = 1

for which

is a local maximum

The set of

r ~ (0,8)

such

that the extremal problem under consideration has a positive solution is clearly open and non-empty. set of local maxima of bounded away from Similarly, 9P*Q(e i(~+2r

However,

Ip*Q(z)I,

it is also closed since the

Izl = i,

for

P,Q ~ Pn(8)

is

0. we may show that the maximum of

is negative when

d/d~( e-i(n/2)~ P*Q(ei~)) < 0. This completes the proof.

P,Q E Pn(8),

e -i(n/2)(~+2r P*Q(e i~) = 0,

and

201

Proof of Theorem proof of

Theorem

that Lemma

9.

5

6.

The proof of Theorem

except that

7 applies,

6 is identical

r

is allowed to vary in n Q(z) : ~ C(n,k,8+s)z k. k=0

and that

to the

(0,e+r

DISCUSSION AND OPEN QUESTIONS The proofs of Theorems

Possibly

there is a generalization

would lead to a simpler applies

5 and 6 are admittedly

proof.

of Grate's

Also,

for

theorem

e = 0,

to a much larger class of polynomials

Theorems

5 and 6 can be generalized

For instance,

rather

than

tedious.

[5,p.45]

SzegS's

result

Pn(0).

Perhaps

to a larger class than

for which polynomials

P

which

Pn(e).

is it true that if

W

and

9

are probability

measures

such that

P(ze It) d~(t)

and

9

P(ze It) d~(t)

have all their zeros on

P(ze ~(t+s))

d~(s)

d~(t)

It follows from Lucas'

has its zeros

theorem,

then for

1 < s < n-I

Izl < I.

We have shown that if

Izl > i,

Izl __> l? n ~ C(n,k)Akzk k=O

n ~ C(n-s163 k=~

zk-s

n ~ C(n,k,e)Akzk k=0

6 Pn(8)

all zeros of

In view of the representation likely that functions

in

that if

n [ C(n-s163163 has all its zeros in k=s this result true also for Z = 2,--., n-l?

it seems

then

Izl > 1

given by L e m m a

starlike

of order

a

lie in then

for

5 for

E Pn(0~

s = i.

Is

P ~ Pn(8)

have a represen-

tation

I~ f(z)

where

A

is a complex

not a probability length

y

such that

= A

constant

measure).

on the unit circle d~

ze i(l-e)t (l_zeit)2(l_a)

and

W

Further, there

d~(t),

is a positive if

Re ~

measure = e

on an arc of

should be representation

has no mass on an arc of length

y.

Such a

(but

as above

202

representation is b o u n d e d . bounded

does

This

convex

exist

follows

functions

for

e = 0,

readily given

at

from an

in [12].

least

when

integral

f

z

f ( w ) / w dw 0 r e p r e s e n t a t i o n for

REFERENCES 1.

L. Brickman, D.J. Hallenbeck, T.H. MacGregor and D.R. Wilken, Convex hulls and extreme points of families of starlike and convex mappings, Trans. Amer. Math. Soc. 185(1973), 413-428.

2.

W. Kaplan, Close-to-convex 1(1952), 189-185.

3.

L. HSrmander, 55-64.

4.

S. Mansour, On extreme points in two classes of functions univalent sequential limits, Thesis, University of Kentucky, 1972.

5.

M. Marden, The Geometry of the Zeros of a Polynomial in a Complex Variable, Amer. Math. Soc., Math Surveys no. 3, 1949.

6.

A. Marx, Untersuchungen ~ber 8chlichte 107(1932/33), 40-67.

7.

S. Ruscheweyh and T. Sheil-Small, Hadamard products functions and the P61ya-Schoenberg conjecture, Helvet. 48(1973), 119-135.

8.

G. P61ya and I.J. Schoenberg, Remarks on de la Vall~e Poussin means and convex conformal maps of the circle, Pacific a. Math. 8(1958), 295-334.

9.

E. Strohh~cker, Beitr~ge zur Theorie Math. Z. 37(1933), 356-380.

8chlicht functions,

On a theorem of Grace,

Math.

Mich.

Scand.

Math.

J.

2(1954),

Abbildungen,

der schlichten

with

Math. Ann. of 8chlicht Comment. Math.

Funktionen,

10.

T. Suffridge, On univalent 44(1969), 496-504.

Ii.

T. Suffridge, Extreme points in a class of polynomials having univalent sequential limits, Trans. Amer. Math. Soc. 163 (1972), 225-237.

12.

T. Suffridg e, Convolutions 15(1966), 795-804.

13.

G. Szeg~, Bemerkungen zu einem Satz yon J.H. Grace ~ber die Wurzeln algebraischer Gleichungen, Math. Z., 13(1922), 28-55.

polynomials,

J. London Math.

of convex functions,

J. Math.

Soc.

Mech.