Computational Methods And Function Theory

Lecture Notes in Mathematics Edited by A. Dold, B. Eckmann and E Takens

1435 St. Ruscheweyh E.B. Saff L.C. Salinas R.S. Varga (Eds.)

Computational Methods and FunctionTheory Proceedings of a Conference, held in Valparaiso, Chile, March 13-18, 1989 II I I I I

SpringerM~rlag Berlin Heidelberg New York London Paris Tokyo Hong Kong Barcelona

Editors

Stephan Ruscheweyh Mathematisches Institut, Universit~.t W~rzburg 8?00 WL~rzburg, FRG Edward B. Saff Institute for Constructive Mathematics Department of Mathematics, University of South Florida Tampa, Florida 33620, USA Luis C. Salinas Departamento de Matem&tica, Universidad Tecnica Federico Santa Maria Casilla 110-V,Valparaiso, Chile Richard S. Varga Institute for Computational Mathematics, Kent State University Kent, Ohio 44242, USA

Mathematics Subject Classification (1980): 30B?0, 30C10, 30C25, 30C30, 30C70, 30E05, 30E10, 65R20 ISBN 3-540-52768-0 Springer-Verlag Berlin Heidelberg NewYork ISBN 0-38?-52?68-0 Springer-Verlag New York Berlin Heidelberg

This work is subject to copyright.All rightsare reserved,whetherthe wholeor part of the material is concerned,specificallythe rights of translation,reprinting,re-useof illustrations,recitation, broadcasting,reproductionon microfilmsor in otherways,and storagein databanks. Duplication of thispublicationor partsthereofis only permittedunderthe provisionsof the GermanCopyright Law 0fSeptember 9, 1965, in its versionof June 24, 1985,and a copyrightfee must alwaysbe paid.Violationsfall underthe prosecutionact of the GermanCopyright Law. © Springer-VerlagBerlin Heidelberg 1990 Printed in Germany Printing and binding: DruckhausBeltz, Hemsbach/Bergstr. 2146/3140-543210- Printed on acid-freepaper

Preface This volume contains the proceedings of the international conference on 'Computational Methods and Function Theory', held at the Universidad T@cnica Federico Santa Mar/a, Valparaiso, Chile, March 13-18, 1989. That conference had two goals. The first one was to bring together mathematicimas representing two somewhat distant areas of research to strengthen the desirable scientific cooperation between their respective disciplines. The second goal was to have this conference in a country where mathematics as a field of research is developing and scientific contacts with foreign experts are very neccessary. It seems that the conference was successful in both regards. Besides, for many of the non-Chilean participants this was the first visit to South-America and these days left them with valuable personal impressions about the regional problems, an experience which may lead to active support and cooperation in the future. About 40 half- and one-hour lectures were presented during the conference. They are listed on the last pages of this volume. Of course, not all of them led to a contribution for these proceedings since many have been published elsewhere. However, the papers in this volume are fairly representative for the areas covered. To hold such a conference, in a place somewhat distant from the international mathematical centers, obviously requires strong support from funding agencies, and it is the organizer's pleasure to acknowledge those contributions at this point. The local organization was made possible through generous grants from the FundaciSn Andes, Chile, mad from our host, the Universidad T@cnica Federico Santa Mafia. In addition, foreign participants were supported by a special grant of the National Science Foundation (NSF), USA, and by other national agencies such as the Deutsche Forschungsgemeinschaft (DFG), FRG, the German Academic Exchange Service (DAAD), FRG, the British Council, UK, etc. We also wish to thank the Universidad T~cnica Federico Santa Maria for the hospitality on its marvellous campus overlooking the beautiful Bay of Valparaiso, and the many people who did help us with the organization. Especially, we wish to thank Ruth Ruscheweyh, who assisted the organizers during the conference and the hot phase of its preparation, and also was responsible for the typesetting (in ISTEX) of the papers in this volume. Finally, we should like to thank Springer-Verlag for accepting these proceedings for its Lecture Notes series.

For the editors: Stephan Ruscheweyh

Contents Preface . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

III

R.W. Barnard Open Problems and Conjectures in Complex Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . .

1

J.M. Borwein, P.B. Borwein A R e m a r k a b l e Cubic Mean Iteration

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27

A . C d r d o v a Y . , St. Ruscheweyh On the Maximal Range Problem for Slit Domains . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

33

R . Freund On Bernstein T y p e Inequalities and a Weighted Chebyshev Approximation P r o b l e m on Ellipses . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

45

D.M. Hough Conformal Mapping and Fourier-Jacobi Approximations . . . . . . . . . . . . . . . . . . . . . . . . .

57

J.A. H u m m e l Numerical Solutions of the Schiffer Equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

71

K.G. Ivanov, E.B. Saff Behavior of the Lagrange Interpolants in the Roots of Unity . . . . . . . . . . . . . . . . . . . . .

81

Lisa Jacobsen Orthogonal Polynomials, Chain Sequences, Three-term Recurrence Relations and Continued Fractions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

89

A. Marden, B. Rodin On T h u r s t o n ' s Formulation and Proof of Andreev's Theorem . . . . . . . . . . . . . . . . . . .

103

D. Mejfa, D. Minda Hyperbolic G e o m e t r y in Spherically k-convex Regions . . . . . . . . . . . . . . . . . . . . . . . . . . .

117

D. Minda The Bloch and Marden Constants . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

131

O . F . Orellana On Some Analytic and Computational Aspects of Two Dimensional Vortex Sheet Evolution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

143

N. Papamichael, N.S. Stylianopoulos On the Numerical Performance of a Domain Decomposition Method for Conformal Mapping . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . G . Schober Planar Harmonic Mappings

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155

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171

T.J. Suffridge Extremal Problems for Non-vanishing H p Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . .

177

VI W.J. Thron Some Results on Separate Convergence of Continued Fractions . . . . . . . . . . . . . . . . .

191

R.S. Varga~ A.J. Carpenter Asymptotics for the Zeros of the Partial Sums of e ~. II . . . . . . . . . . . . . . . . . . . . . . . . .

201

Lectures presented during the conference . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

209

Computational Methods and Function Theory Proceedings, Valparm'so 1989 St. Ruscheweyh, E.B. Saff, L. C. Salinas, R.S. Varga (eds.) Lecture Notes in Mathematics 1435, pp. 1-26 (~) Springer Berlin Heidelberg 1990

O p e n P r o b l e m s and Conjectures in C o m p l e x Analysis Roger W. Barnaxd Department of Mathematics, Texas Tech University Lubbock, Texas 79409-1042, USA

Introduction This article surveys some of the open problems and conjectures in complex analysis that the a u t h o r has been interested in and worked on over the last several years. They include problems on polynomials, geometric function theory, and special functions with a frequent mixture of the three. The problems that will be discussed and the author's collaborators associated with each problem are as follows: 1. Polynomials with nonnegative coefficients (with W. Dayawansa, K. Pearce, and D. Weinberg) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

2

2. The center divided difference of polynomials (with R. Evans and C. FitzGerald) . . . . . . . 4 3. Digital filters and zeros of interpolating polynomials (with W. Ford and H. Wang) . . . . . 5 4. Omitted values problems (with J. Lewis and K. Pearce) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

8

5. MSbius transformations of convex mapp!ings (with G. Schober) . . . . . . . . . . . . . . . . . . . . . . .

12

6. Robinson's 1/2 conjecture . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

13

7. Campbell's conjecture on a majorization - subordination result (with C. Kellogg) . . . . . 14 8. Krzyi conjecture for bounded nonvanishing functions (with S. Ruscheweyh) . . . . . . . . . . 15 9. A conjecture for bounded starlike functions (with J. Lewis and K. Pearce) . . . . . . . . . . . 16 10. A. Schild's 2/3 conjecture (with J. Lewis) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

18

11. Brannan's coefficient conjecture for certain power series . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

19

12. Polynomial approximations using a differential equations model (with L. Reichel) .... 20

2

R.W. Barnard 1. Polynomials with nonnegative coefficients

We first discuss a series of conjectures which have as one of their sources the work of mgler, Trimble and Varga in [66]. In [66] these authors considered two earlier papers by Beauzamy and Enflo [23] and Beauzamy [22], which are connected with polynomials and the classical Jensen inequality. To describe their results, let

p(z)= ~a3z j= j=O

ajz j, where aj =O, j > rn, j=0

be a complex polynomial ( 5 0), let d be a number in the interval (0, 1), and let k be a nonnegative integer. Then (cf [22], [23]) p is said to have concentration d of degree at most k if k

(1)

oo

~ la, I >_d E fail,

j=O

j=0

Beauzamy and Enflo showed that there exists a constant (~d,k, depending only on d and k, such that for any polynomial p satisfying (1), it is true that (2)

2zr f0

()

l°glp(ei°DIdO-l°g ~'~lajl >- Cd,k. j=0

In the case of k = 0 in (2) the inequality is equivalent to the Jensen inequality 1 f02~ log 2--~

[23],

Ip(e~°)ldO>_log ta01.

Rigler, etc., in [66] considered the extension of this inequality from the class of polynomials to the class of H °~ (of. Duren [36]) functions. For f E H ~ the functional 1

2~

(

J(f) := ~ f0 log tf(ei°)ldO - log ~ lajl

)

j=0

can be well-defined and is finite. They let (3)

Cd.k = i n f { J ( f ) : f C H ~ and f(z) = ~ ajzJ(~ 0) satisfies (1)}. j=0

For a (fixed) d 6 (0, 1) and a (fixed) nonnegative integer k, it was shown that there exists an unique positive integer n (dependent on d and k) such that

l ~ - - ~ ( n )
-

(n-I) J

"

Open Problems and Conjectures in Complex Analysis

p=

n-l) k

3

-1.

k (n-1)-d2n-1 E j=0 j With these definitions the following conjecture was made in [66]. C o n j e c t u r e 1. Let Cd,k be defined by (3). Then

1)

(4)

In [66] Conjecture 1 was verified for k = 0 and for the subclass of Hurwitz polynomials, i.e., those polynomials with real coefficients and having M1 their zeros in the left half-plane. In order to verify the conjecture for the entire class an interim step was suggested. This step was one of the motivations for the following problem which was solved recently by this author and others in [10]. Let p be a real polynomial with nonnegative coefficients. Can a conjugate pair of zeros be factored from p so that the resulting polynomial still has nonnegative coefficients? We gave an answer to one proposed choice for factoring out a pair of zeros. Fairly straightforward arguments show that if the degree of the polynomial is less than 6 then a conjugate pair of zeros of greatest real part can be factored out and the resulting polynomial will still have non-negative coefficients. However, the example p(z) = 140 + 20z + z 2 + 1000z 3 + 950z 4 + 5z s + 20z 6

shows that the statement is not true for arbitrary polynomials with non-negative coefficients. A large amount of computer data had suggested the following: C o n j e c t u r e 2. The nonnegativeness of the coe~cients of a real polynomial is preserved upon factoring out a conjugate pair of zeros of smallest positive argument in absolute value. Interestingly this last conjecture also arose quite independently in the work of Brian Conrey in analytic number theory in his work on one of Polya's conjectures. Conrey announced Conjecture 2 at the annum West Coast Number Theory Conference in December 1987. The conjecture was communicated to this author by the number theorist Ron Evans. Indeed Evans, using a large amount of computer evidence, has generated a closely related conjecture which we include. C o n j e c t u r e 3. If a polynomiM of degree 2n has zeros ei(t+ak) and e -~(t+~/,

k = 1,2,...n,

where the ak tie between 0 and re, then all the coettlcients are nondecreasing functions o f t for small t > 0 provided the coemcients are alI nonnegative for t = O. A special case of Conjecture 3 where the zeros on the upper semicircle are equally spaced would be of special interest. Although Conjecture 2 was verified in [10] the techniques do not appear applicable to Conjecture 3.

4

R . W . Barnard

2. T h e c e n t e r divided difference o f p o l y n o m i a l s Another series of polynomial problems was generated in classical number theory by the work of Evans and Stolarsky in [37]. Given a polynomial p and a real number A define ~ ( p ) , the center divided difference of p, by

{ p(z+~)-p(z-~) ~#o, ~(P)

=

p'(z),

2~

'

~ = o.

We did a study of the behavior of the 5~(p) as a function of A in [11]. A number of classical results of Walsh and Obrechkoff and of Kuipers [50] give some information about the zeros of g~(p) as a function of A. Let W[p] equal the width of the smallest vertical strip containing the zeros of p. It folIows from the classical work that

w[~(p)] < w~0] and that the diameter of the zero set of 6~(p) approaches oe as [A] approaches oe. The Gauss-Lucas theorem shows that

w[;'] < w[v]. It was shown in [11] that

(5)

w[~(p)] < w~']

and the conditions on p when equality holds in (5) are given. We were also able to prove that W[~(p)] = O(1/A) as [AI ~ oc. The numerical work done by the number theorists had suggested,

Conjecture 4. W[(~;~(p)] monotonically

decreases to zero as [A1 --~ c~.

In that direction it was shown in [11] that

(6)

w[~(p)] _<w[~(;)]

for all positive A and conditions for equality in (6) were found. In addition, if the zero set of p is symmetric about a vertical line then (7)

W[~(p)] = 0 for all A _> W[p'].

However, an example was given of a polynomial p~, that contradicts Conjecture 4 at least for some A. The polynomial p~ has its zero set symmetric about the imaginary axis and has the property that for small ~, W[61(p~)] = 0 and W [ ~ ( P , ) ] -- 0 for A _> x/1 + 2e = Wipe] while W[6~(p~)] > 0 for 1 < A < ~/1+2~.

Open ProbIems and Conjectures in Complex Analysis

5

Thus conjecture 4 needs to be modified to read C o n j e c t u r e 5. W[5~(p)] monotonically decreases to zero for A > W[p']. The original question that motivated the number theorist's interest in this problem was the determination of the zeros of 5~(pN) where N

pN(Z) =

I I (z - k). k=-N

Also occuring in their work were the iterates, ~(") of 5 defined inductively by

with The numerical work had suggested C o n j e c t u r e 6. A11 nonreal zeros o/" ~(~)(pu) are purely imaginary for a11 )~ and a11 ft.

Conjecture 6 has been verified in [11] for n = 1. Indeed, an interesting problem, with other ramifications in number theory, see Stolarsky [71], would be to characterize those polynomials for which 6~") has only real and pure imaginary roots.

3. D i g i t a l

filters

and zeros of i n t e r p o l a t i n g p o l y n o m i a l s

Some interesting problems arise when classical complex analysis techniques are applied to digital filter theory. Polynomials to be used in interpolation of digital signals are called interpolating polynomials. These polynomials may require modification to assure convergence of their reciprocals on the unit circle. Such modifications provide the opportunity to apply classical analysis theory as was done by the author, Ford, and Wang in [12]. A real function, g, defined for all values of the real independent variable time, t, is called a signal. A digital signal, 7, is a real sequence, {7,, : - o o < m < oo}, consisting of equally spaced values or samples, 7m = g(mAt), from the signal, g, with a time increment or sample interval, At. Thus, the independent variable for digital signals such as 3' is sample time, mAt, or simply sample number, m. The signal, g, is studied in terms of its classical Fourier transform, G, as a function of real frequency, w. The digital analog of the Fourier transform consists of the study of a sequence such as 7 in terms of its Z-transform, which is defined to be the power series, F, having 7m as the coefficient of z m. Frequency's digital analog comes from evaluation of Z-transforms such as F on the unit circle with the negative of the 0 in z = eiO referred to as frequency. If the coefficients in F are used without any actual evaluation o f / ' ( z ) or g is used without computation of G, such use is said to be in the time domain. But

6

R.W. Barnard

i f / ' ( z ) is used with evaluation for some z of unit modulus or G is used, such use is said to be in the frequency domain. Signals are based on even functions in a number of applications. This restricts digital signals to self-inversive cases meaning that F ( z ) = F ( z -1) for z ~ 0. Equivalently, 7 is a symmetric sequence meaning that 7m = %m for all m. A second signal, f , with Fourier transform, F , poses as a filter of the signal, g, if the convolution integral, g • f , of g and f is considered. Of course, the Fourier transform of g * f is the product of the Fourier transforms, G of g and F of f. The discrete analogy consists of the product of Z-transforms,/" and #, where the latter refers to the power series with the sample, #,~ = f ( m A t ) , taken from the filter, f, as the coefficient of z "~. Reduction of certain frequencies is a fundamental aim in the application of a filter, f, to a function, g. This can involve the definition of f by the requirement that F(w) be a constant, c, for ]w[ < COobut zero otherwise. If so, c can be chosen so that

(s)

f (t ) = sinc wot,

where sinc is defined by sin x

(9)

sine

X -X

These equations illustrate that the definition of a real signal is determined from the specifications of its Fourier transform. Similarly, digital signals are often defined by the specification of Z-transforms. T h e Fourier transform, F, of the f in (8) is referred to as a frequency window since it has compact support in frequency. Application of such a window to a signal, g, is known as a frequency windowing. These problems concern discrete time windowing. This consists of the scaled truncation of an infinite sequence such as V to obtain a finite sequence of the form {c,nvrn : - L < m < L} wherein the finite sequence, {cm : - L < m < L}, is referred to as a time window. Suppose a given digital signal, {bk : - o e < ~: < ec}, is such that bk is understood to correspond to the time, k N A t , with the sample interval, N A t , where N is a natural number such that N > 1. If this digital signal is to be compared with digital signals based on the smaller sample interval, At, the given digital signal must be interpolated to the smaller sample interval, At. For example, insertion of N - 1 zeros between every bk and bk+l, followed by multiplication of the Z-transform of the result by the interpolating series, ~r~, defined by

(10)

PN(z) = 1 +

(z m + z -'~) sinc --~-, rn=l

leads to (11)

A(z)=

anzn= n = - co

bjz jg \j=-

oo

PN(Z). /

Since the coefficient of z kN, akg, in A comes from products of bj and sinc(rmr/N) such that k N = j N + m, it follows that m - 0 (modN), sinc(rnr/N)=O for nonzero m, and a k y = bk. Thus, A is an interpolation of the given B with coefficients, bj.

Open Problems and Conjectures in Complex Analysis

7

A major goal is to study possible alternatives to the interpolation used in (10) in terms of truncation of the interpolating series in (11). In practice one truncates P to obtain the interpolating polynomial, PN,L defined by (12)

P N , L ( Z ) = Z L-1

1 + Z ( zm + z - m ) sine

,

ra= l

where N > 1. To assure stability and accuracy of evaluation it is important that alternative P ' s have no zeros on the unit circle. It is shown in [12] that all of the zeros of PN,L are of unit modulus when L < N and examples are given showing that when L > N + 1 almost any combination of zeros inside, on, and outside the unit circle can occur. A number of classical results are then combined to give sharp conditions on real sequences {cm : 1 < m < oc} so that the function P*N,L defined by -

-

-

-

(13)

P ~ , L ( Z ) = Z L-1

1 +

~ (z m + z - m ) c m sine m=l

has no zero of unit modulus. In particular, in order to define a useful test to determine if a specific sequence of numbers will work for the cm's in (13) the following theorem was proved in [12]. T h e o r e m 1. I f a real sequence, {bin : 0 <<_m < L, bo = 1} is such that

1 bl

bl 1

•" bl

bk-1 ...

bk bk-1 >_0

bk-1

"" "

bk

bk-1

for O < k < L, let cm=bm(1

bl

1

bl

bl

1

TML

21°gL)

define the coetticients in (13). Then P~V,L has no zero of unit modulus. A number of the standard "windows" that occur in the engineering literature are then shown to be just special cases of those defined in Theorem 1, including the very generalized Hamming window and the Harming window. (see Rabniner and Gold's book, Theory and Application of Digital Signal Proce~ing.) The distribution of zeros and the orthogonality property of the sine functions determine the interpolating properties in (11) and enables the classical results to be applied. Thus one can ask, can the sine functions be replaced by more general orthogonal functions, e.g., Jacobi polynomials, to create a more general setting in which m a n y more applications can be found? Discussions with several engineers have suggested this.

8

R.W. Barnard 4. O m i t t e d

values

problems

We now discuss a number of open problems in geometric function theory. Let A~ = {z: Izl < r}, with A1 = A. Let S denote the class of univalent functions f in A normMized by f(0) = 0 and if(0) = 1. The problem of omitted values was first posed by Goodman [38] in 1949, restated by MacGregor [57] in his survey article in 1972, then reposed in a more general setting by Brannan [5] in 1977. It also appears in Bernardi's survey article [24] and has appeared in several open problem sets since then including [27],[40] and [60]. For a function f in S, let A(f) denote the Lebesgue measure of the set A \ f ( A ) and let L(f, r) denote the Lebesgue measure of the set {A\/(AI)} N{w: ]wl = r} for some fixed r, 0 < r < 1. Two explicit problems posed by Goodman and by Brannan were to determine (14)

A = sup A(f), FeS

and (15)

L(r) = sup L(f, r). F~S

Goodman [38] showed that .227r < A < .507r. The lower bound which he obtained was generated by a domain of the type shown in Figure 1.

i

Figure 1

Later, Goodman and Reich [39] gave an improved upper bound of .38~r for A. Using variational methods developed by the author in [6] and some deep results of Alt and Caffarelli [4] in partial differential equations for free boundary problems, a geometric description for an extremal function for A was given by the author in [9] and by Lewis

Open Problems and Conjectures in Complex Analysis

9

in [54]. This can be described as follows: There is an f0 in S with A = A(fo) such that f0(A) is circularly symmetric with respect to the positive real axis, i.e., it has the property that for 0 < r <= 1,

~--~lfo(re'°)[ and ff--~[fo(re-ie)[ <_O, for 0 < 0 < 7r (cf. Hayman [44]). Moreover the boundary of f0(Zl) consists of the negative real axis up to - 1 , an arc 7 of the unit circle that is symmetric about - 1 and an arc A lying in A, except for its endpoints. The arc A is symmetric about the reals, connects the endpoints of 7 and has monotonically decreasing modulus in the closure of the upper half disc. These results follow by standard symmetrization methods. Much deeper methods are needed to show (as in [9] and in [54]) that f0 has a piecewise analytic extension to A with f~ continuous on foX(A) and [f~(foX(W))t - c < 1 for all w • AN{A\(--1,1)}. Using these properties of f0 it was shown by the author and Pearce in [19] that by "rounding the corners" in certain gearlike domains a close approximation to the extremal function could be obtained. This gives the best known lower bound of .247r < A. The upper bound is conceptually harder since it requires an estimate on the omitted area of each function in S. Indeed, it appears difficult to use the geometric description of f0 to calculate A directly. However, an indirect proof was used by the author and Lewis in [17] to obtain the best known upper bound of A < .317r. O p e n p r o b l e m . Show that fo is unique and determine A explicitly. For the class S* of functions in S whose images are starhke with respect to the origin, the problem of determining the corresponding A* = sup A(f) dES*

has been completely solved by Lewis in [54]. The extremal function fl E S* defined by A* = A(f~) ~ .235r is unique (up to rotation). The boundary of fl(A) has two radial rays projecting into A with their end points connected by an arc A1 that is symmetric about the reals and has [fi(()[ --- El for 811 ( • f~-x(A1). The problem of determining L(r) in (15) was solved by Jenkins in [47] where he proved that for a fixed r, 1/4 < r < 1,

L(r) = 2r arccos(Sv~ - 8r - 1). The extremM domain in this case is the circular symmetric domain (unique up to rotation) having as its boundary the negative reals up to - r and a single arc of {w: [wl -- r} symmetric about the point - r .

10

R.W. Barnard

T h e corresponding problem for starlike functions of determining L*(r) = supfes. L(f, r) was solved by Lewandowski in [53] and by J. Stankiewicz in [70]. The extremal domain in that case is the circularly symmetric domain (unique up to rotation) having as its b o u n d a r y two radial rays and the single arc of {w : lwl = r} connecting their endpoints. An explicit formula for the mapping function in this case was first given by Suffridge in [72]. For the class S ¢ of functions in S whose images are convex domains the corresponding problem of determining (16)

A¢(r) = sup A(f, r) IES c

and

(17)

L (r) = sup L(r, v). ]ESc

where A ( f , r ) denotes the Lebesgue measure of A~/f(A), presents some interesting difficulties. One particular difficulty is that the basic tool of circular symmetrization used in the solution to each of the previous determinations is no longer useful. The example of starting with the convex domain bounded by a square shows that convexity is not always preserved under circular symmetrization. However, Steiner symmetrization (cf. H a y m a n [44]) can still be used in certain cases such as sectors. Another difficulty is the introduction of distinctly different extremal domains for different ranges of r. Since every function in S c covers a disk of radius 1/2 (of. Duren [36]) r needs only to be considered in the interval (1/2, 1). Waniurski has obtained some partial results in [74]. He defined rl and r2 to be the unique solutions to certain transcendental equations where rl ,~ .594 and r2 ~ .673. If F,/2 is the map of A onto the half plane {w : R e w > - 1 / 2 } and F~ maps A onto the sector

whose vertex, v = -rr/4a, is located inside the disk, then

A~(r) = A(F./2,r) for 1/2 < r < rl, L¢(r) = L(F./2,r) for l/2 < r < r,, and LC(r) = L(F~, r) for rl < r < r2. This author had announced in his survey talk on open problems in complex analysis at the 1985 Symposium on the Occasion of the Proof of the Bieberbach Conjecture the following conjecture: C o n j e c t u r e 7. The extremal domains in determining AC(r) and LC(r) will be halfplanes, symmetric sectors and domains bounded by singles arcs of Iwl = r along with

tangent/ines to the endpoints of these arcs, the different domains depending on different ranges o f t in (1/2, 1).

Open Problems and Conjectures in Complex Analysis

11

This conjecture was also made independently by Waniurski at the end of his paper [74] in 1987. Another conjecture that was announced at the Symposium on the Proof of the Bieberbach Conjecture arose out of this author and Pearce's work on the omitted values problem. A significant part of characterizing the extremal domains for At(r) and LC(r) in (16) and (17) via the variational method developed in [6] would be the verification of the following: C o n j e c t u r e 8. If f E S ~ then (18)

lim27r

r~,

~ dO < sup f'(rei°) ~e,~

.

Using standard integral means notation this is equivalent to showing that the smallest c such that (19)

Adl [1/f'] < c,&4~ [z/f(z)]

holds is c = 1. Well known results (cf. Duren [36], pp. 214) on integral means show that the smallest c for all functions in S is two, while unpublished results of the author and Pearce show that the smallest c for the class of functions starlike of order 1/2 [cf G o o d m a n [40]] (a slightly larger class than S c) is c = 4/7r. It was also shown that equality holds in (18) for all domains bounded by regular polygons and it was conjectured that equality holds for those convex domains bounded by single arcs of {w : Iwl = r} and tangent lines at the endpoints of these arcs. Verification of Conjecture 8 would give an interesting geometric inequality. Let a convex curve F have length L and have its minimum distance from the origin be denoted by d. An application of the isoperimetric inequality along with the conjecture would imply

(20)

< - V27r Jo

lf'(ei

-< -2dTr'

We note that the normalization for the functions f in S c would force the first and last terms in inequality (20) to go to one as d goes to one. Determining explicit values for At(r) and LC(r) would involve computing the map that takes A onto the convex domains bounded by an arc of {w : ]w] = r} along with the two tangent lines at the endpoints of this arc. The function defining this map involves the quotient of two hypergeometric functions (cf. Nehari, [62]). In particular an extensive verification shows that the function g as shown in Figure 2

Figure 2

12

R. W. Barnard

is given by

z2F' (24-- 1' 2~+3'14 +o~;z) 9(z) =

~F~ 2a___l 3__-2o~ 1 - o E ; z '

4

'

)

A difficulty arises when determining the explicit preimage of the center of the circle so that g can be renormalized to the mapping function f in S taking Zl onto a domain whose boundary circle is centered at the origin.

5. M S b i u s

transformations

of convex

mappings

Another problem on convex mappings originated from a question of J. Clunie and T. Sheil-Small. If f E S and w ~ f(A), then the function

(21)

] = f/(1 - f/w)

belongs again to S. The transformation f --~ ] is important in the study of geometric function theory. It is useful in the proofs of both elementary and not so elementary properties of S. If F is a subset of S, let = {f:

f C F,w e C ' \ f ( A ) } .

Here C* = c u { o e }. Since we admit w = oe, it is clear that F C F C S. If F is compact in the topology of locally uniform convergence, then so is _/'. If F is rotationally invariant, that is, f=(z) = ei=f(e-i~z) belongs to F whenever f does, then /~ is also rotationally invariant. It is an interesting question to ask which properties of F are inherited by/~'. Since S = S, this question is trivial for S. In [20] and [21] the author and Schober considered the class S c of convex mappings. Simple examples show that ~c is strictly larger than S c. Since the coefficients of functions in S ¢ are uniformly bounded (by one), J. Clunie and T. Sheil-Smalt had asked whether the coefficients of functions in S c have a uniform bound. The affirmative solution of this problem was given by R.R. Hall [42]. O p e n Q u e s t i o n . Find the best uniform bound as well as the individual coefficient bounds for ~c. In [20] the variational procedure developed in [17] is applied to a class of extremal problems for ~c. If ~ : Sc -* R is a continuous functional that satisfies certain admissibility criteria, it was shown that the problem max )~ has a relatively elementary extremal function ]. More specifically, it was shown that either is a half-plane mapping f(z) = a/(1 - eiaz) or is generated through (21) by a parallel strip mapping f E S e.

Open Problems and Conjectures in Complex Analysis

13

The class of functionals considered in [20] contain the second-coefficient functional A(]) = Re a2 and the functionals A(f) = Re #(log f ( z ) / z ) where # is entire and z is fixed. The latter functionals include the problems of maximum and minimum modulus (#(w) = +w). In general, the extremal strip domains f ( A ) need not be symmetric about the origin. This adds a nontrivial and interesting character to the problems. A sharp estimate for the second coefficient of functions in :~c is given explicitly in the following result. Surprisingly, the answer is not an obvious one.

Theorem 2. If f ( z ) = z + a2z 2 + . . . belongs to Sc, then 2 ]as[ < - - s i n x 0 - c o s x 0 ~ 1.3270 X0

where x0 ~ 2.0816 is the unique solutlon of the equation cot x -

1 x

1 x 2

in the interval (0, re). Equality occurs t'or the/'unctions e-i:f(ei~z), o~ e R, where ](z) = f(z)/[1 - f ( z ) / f O ) l and f is the verticaJ strip mapping de~ned by

1 f(z) -- 2i sinz~

(22)

1 + eiX°z log 1 + e-i':oz"

We make the following:

Conjecture 9. The extremM functions for maximizing [as [ over ~c are the vertical strip mappings detlned by (22) where a different xo is needed for each n. In [21] the Koebe disk, radius of convexity, and sharp estimates for the coefficient functional [ta3 + a~[, for t in a certain interval, were found for functions in the class ~c Also, in [3], R.M. Ali found sharp upper and lower bounds for lI(z)t for f in ~c.

6. R o b i n s o n ' s

1/2 conjecture

A conjecture that has been open for more than 40 years is Robinson's 1/2 conjecture. Let .4 denote the class of analytic functions on A. For a subclass X (possibly a singleton) of .4 let r s ( X ) denote the minimum radius of univalence over all functions f in X. For a function f in S define the operator O : S --+ .4 by

O f = (z f)'~2. In 1947, in [67], R. Robinson considered the problem of determining rs[O(S)] which will be denoted by r0. He observed that for each f in S, [zf]' 7~ 0 for A1/2 and noted that for the Koebe function, k, k(z) = z(1 - z) -2,

rs(k) : rs.(k) : 1/2

R. W. Barnard

14 which implies r0 < 1/2. Robinson made

Conjecture 10. If f C S then (z f)'~2 is univalent in Ar /'or 0 < r < 1/2, i.e. r0 = 1/2. He was able to show that .38 < rs.[6~(S)] < r0. Little or no progress was made directly on the study of the operator O following Robinson's work until Livingston in [56] proposed a shift for the setting of the problem from the full class S to subclasses of S. He showed that {9 preserved many of the well-known subclasses of S. e.g., S* and S c. Livingston's work renewed interest in the study of (9. Numerous papers by various authors followed (see [13]) connecting the operator {9 to various subclasses of 6'. It was shown by the author and Kellogg in [13] that most of these results follow directly from the Ruscheweyh-Sheil-SmalI theory on Hadamard convolutions. However, for the entire class S, if appears that the easily obtained lower bound of approximately .41 is the most that can be obtained from the convolution methods. Thus Conjecture 10 is still open. Although Bernardi had suggested that rs.[(9(S)] = 1/2 might even be true, in [7], it was shown that rs. [6)(S)] < .445, while Pearce proved in [64] that .435 < rs. [6~(S)]. In [8] the author proved that .490 < r0 _< .50 using the Grunsky inequalities. The closeness, but non sharpness, of this result has intrigued a number of people in the field. Robinson's conjecture and the progress on this problem appeared in A. W. Goodman's book, Univalent Functions [40], and in [27].

7. C a m p b e l l ' s conjecture on a majorization- s u b o r d i n a t i o n result A conjecture relating majorization and subordination was made by Campbell in [34]. Let f, F, and w be analytic in A~. f is said to be majorized by F, denoted by f << F, in Ar if If(z)] =< ]F(z)l in At. f is said to be subordinate to F, denoted by f -< F, in Ar if f(z) = F(w(z)) where tw(z)l < tzl in A~. Majorization-subordination theory began with Biernacki who showed in 1936 that if f'(O) > 0 and f -< F ( F E S) in A, then f << F in A1/4. In the succeeding years Goluzin, Tao Shah, Lewandowski and MacGregor examined various related problems (for greater detail see [33]). In 1951 Goluzin showed that if f'(0) > 0 and f < F ( F e S) then f ' << F ' in A0.12. He conjectured that majorization would always occur for Izl < 3 - v ~ and this was proved by Tao Shah in 1958. In a series of papers [32,33,34], D. Campbell extended a number of the results to the class Us of all normalized locally univalent (f'(z) # 0) analytic functions in A with order < o~ where Ul = S c, the class of convex functions in S. In particular in [34] he showed that if f'(0) > 0 and f -< F ( F G bt~) then f ' << F ' in Izl < ~ + 1 - (a 2 + 2oL)1/2 for 1.65 __< a < co where o~ = 2 yields 3 - x/~. Note that a = 1 yields 2 - v'~, the radius of convexity for S. Campbell's proof breaks down for 1 < a < 1.65 because of two different bounds being used for the Schwarz function with different ranges of a. Nevertheless, he made the following:

Open Problems and Conjectures in Complex Analysis

15

C o n j e c t u r e 11. If f'(O) >_ 0 and f -~ F (F E 14~,) then f ' ~ F' for Izl < a + 1 (a 2 + 2~) 1/2. In [14] the author and Kellogg combined Ruscheweyh's subordination result [68], variational methods, and some tedious computations to verify the conjecture for a = 1, i.e., it is shown that if f'(0) __>0 and f ~ F ( F E S ~) in AI then f ' (( F ' for lzl < 2 - v/3.

8. K r z y i ' s

conjecture

for

bounded nonvanishing functions

Another conjecture that has been investigated by a large number of function theorists is Krzy~'s conjecture. Let B denote the class of functions defined by f ( z ) = ao + alz + • .. + a~z ~ + . . . for which 0 < [f(z)I < 1 for z E A. In 1968 in [49] J. Krzy~ posed the fundamental problem of determining for n > 1 An = sup [a~ I. feb That A1 = 2/e dates back to 1932 (see Levin [51]) and appears explicitly in Hummel, etc. [46] and Horowitz [45]. That A2 = 2/e appears in [46] and A3 = 2/e in [65]. For a fairly complete history of this problem see [46] or Brown [31]. These results suggest what has become known as the Krzy~ Conjecture, C o n j e c t u r e 12. An = 2/e, for all n > 1, , with equality only for the functions [,+1] 1 +2z~+... Kn(z) = exp tz n _ lJ = e e and its rotations e~K,~(e~Vz). A~ is to equal the apocryphal Pondiczery constant, named by Boas in [25]. A sharp uniform bound less than one is expected. However, the bound 2/e ~ .7357.-., is somewhat surprising in view of the fact that the best uniform estimate known to date is 1 4 (1) la,~l _~ 1 - ~ + sin = 0.9998772... given by D. Horovitz in 1978 in [45]. The open problem of Krzy~'s Conjecture is stated in A. Goodman's book "Univalent Functions" [40, page 83]. De Branges' recent solution to the Bieberbach Conjecture gave hope to solving many of these type problems. However, not withstanding the amount of effort by several function theorists to solve the corresponding coefficient problem, Conjecture 12 still remains open. A related conjecture made by Ruscheweyh upon verification would give a much improved uniform estimate for An. Consider f ( z ) = exp[-Ap(z)] for ), > 0 and p E P where P = {p: p(z) = 1 + p l z + ' " , a e p ( z ) > 0,1zl < 1}. Then consider the following: For 0 < r < 1, choose x = x(r) such that

16

R. W. Barnard •

J1 \{ix 1 -2r r 2 ] 2r ~] Jo ~(ix x-r~

(J0, J1 are Bessel Functions) -

~

and define

F(~)

(23)

= ~(r)~

_=<~/~? 2~ ~-~ ,,0 (,~(~) 1--=--g )

Ruscheweyh conjectured that for any positive integer n, Ak 2 0, I~k[ = 1, k = 1, 2,---, n and n

p(z) = E

(1

Conjecture 13. 1

r2~"

i

'

--2rrJo e-ReP(~ ~)Re{p(re'~)}d~ < - F(r'~)'

(24)

with F defined in (23). We have shown by using the Legendre polynomial expansion for Bessel functions that (25)

{ 1 + m e i~'} 1+~2~ /'. 2r '~ , 1 /2~-xRe f ~ 2-~Yo e l'l-r"e"~J'Re 1 - rne '~' dc2=e-'i---Zrrz~"J° V~I_--=-~).

Equation (25) shows that the estimate (24) would be sharp for fixed r for/5 defined by

/5(z) = z(<)=±~

+

z n z n

Upon verification of Conjecture 13 it can be shown that

_2

(26)

F(~o)

la~l _< n r~-1(1 - r2) '

0

< r < 1.

Choosing r 2 = (n - 1)/(n + 1) in (26) it would follow that (27)

,a~,< lim 2

F[(k-l"~k/2] [\k+l] j \k+l]

=eF (!) ~.869

k+l

by numerical calculations.

9. A c o n j e c t u r e for b o u n d e d starlike f u n c t i o n s A conjecture that was made by this author in [6] in 1975 was recently disproved with computer methods by Pearce leaving the problem now as one that probably can only

Open Problems and Conjectures in Complex Analysis

17

be done numerically. The conjecture involved coefficient estimates for bounded starlike functions in S. Define, for a fixed M > 1,

SM = { f e S : f(z) = z + a2z 2 + a3z 3 + - - - , If(z)l _ M, z E z2}. The fact that la2l is maximized in SM by the function mapping onto Pick's domain of the disk A M minus a single radial slit has been known since 1917 [see Goodman, vol. I, p.38]. In the early sixties Tammi [73] used Schiffer's variational methods to determine the explicit extremal domains for maximizing the first few coefficients in SM. In particular he proved that the extremal domains for maximizing ta31 in SM are as shown in Figure 3 for the different values of M. I

I

I I

I I

l<M<e

e_<M
3

There is a difficulty in modifying Schiffer's variational methods to allow for preservation of both boundedness and starlikeness at the same time. Also the fact that the forked slit domains occurring for M > 3 are no longer starlike suggested the need for a local variational technique that preserved these properties. This was developed by combining the Julia Variational formula with the Loewner Theory in [6] and in [17]. Let

S~t = { f 6 SM: f ( A ) is starlike with respect to the origin}. It was shown in [6] that the extremal domain maximizing la3t in S~ is the disc AM minus at most two symmetric radial slits. Define DM a s A M minus two symmetric radial slits where 20 is the angle between the 2 slits. Let Aa(M, O) be the third coefficient for the function in S~/mapping A onto the domain DM. From the properties of the extremal domains in the class SM, along with initial computations and the observation that A3(3, 0) = A3(3, 7r/2) = 8/9 led this author to the following: C o n j e c t u r e 14. For ali f 6 S ~ (28)

[a3[ < A3(M, rr/2), 1 < M < 3,

18

R. W. Barnard la3l < Aa(M,O), 3 < M < ee.

(29)

It follows from T a m m i ' s results that (28) holds for 1 < M < e and it was shown by the author and Lewis in [16] that (29) holds for 5 < M < oe. Verifying Conjecture 14 for e < M < 5 remained an open problem. This conjecture was announced at the 1978 Brockport Conference and appeared in the open problem set in the proceedings [60] for t h a t conference. It was announced by J. Lewis at the 1980 C a n t e r b u r y Conference and a p p e a r e d in the open problem set in its proceedings, [27]. It was also announced at the 1985 Symposium on the Proof of the Bieberbach Conjecture. Motivated by the observation that the domain DM is indeed a "gearlike" domain and now having the computer software available, Pearce was able to compute Aa(3, 0) and discovered that A3(3, 0), as a function of 0 from 0 to 7r/2, was convex downward, i.e., it took its minimum at the endpoints. Thus Conjecture 14 was false. Indeed further computations shows that there exists a O(M), 0 < O(M) < ~r/2, such that, for some M0 > 0, m a x [Aa(M, 0), A3(M, 7r/2)] < A3(M, O(M)) for 2.83 < M < M0 < 5.

10. A.

Schild's

2/3

conjecture

Another long standing conjecture that was proved false was the 2/3 conjecture. Let rl = rl(f) be the radius of convexity of f , i.e. r a ( f ) = sup{r : f ( A r ) is a convex domain}. P u t d* = min{If(z)l : Iz[ = rl} and d = inf 1/31 for which f(z) 7~/3. In 1953 in [69], A. Sehild conjectured that d*/d >_2/3 for all functions f G S*. Here equality holds for the Koebe function k(z) = z(1 + z) -2. Schild noted that d*/d _> r I ~ 2 - - V ~ and proved the conjecture for p symmetric functions, p _> 7. He also showed for a certain class of circularly symmetric functions that d*/d > .49. Lewandowski in [52], proved the conjecture true for certain subclass of S*. In [58], McCarty and Tepper obtained the best known lower bound of .38 for all starlike functions. The conjecture was shown false by the author and Lewis in [15] by giving two explicit counter examples. The first example is given simply by the two slit m a p defined by Z

f(z) = (1 - z ) o ( l + z) 2-~ ' where a is sufficiently near 0. It was noted that if d is computed as a function of a, then a'(d) --+ +co as a --+ 0 so that a minimal value of .656 for d*/d was obtained for this function at a ~ .03. The second example is a more complicated function that m a p s A onto the circularly symmetric domain shown in Figure 4.

Open Problems and Conjectures in Complex Analysis

19

i!iiiiiii iiiii ii!ii iii i!~: !i!iii!

! :

Figure 4 An explicit formula for this function g~, determined by Suffridge in [72], is given by

1 } z

~l----z-~

]

+

1-zJ

2

+2log

[(1 + 2az + z2) 'I2 + 1 + z]

'

where a = 2b2 - 1 and d = [(1 + b)l+b(1 -- b)'-b]- " with ¢ = ~r(1 - b). A close approximation to the minimum of d*/d for this function is 0.644 given by a ~ 0.89. Also "~b~ .03zr for this minimum value. The author's work suggests: C o n j e c t u r e 15. inf d*/d = min {d*/d for g~} ~ .644...

.f 6 S *

11. Brannan's

a

coefficient

conjecture

for certain

power

series

An innocent looking, but not so trivial, conjecture was made by Brannan in 1973 in [26] on the coefficients of a specific power series. The problem originated in the Brannan, Clunie, Kirwan paper [28] (later completed by a Aharanov and Friedland in [1]) solving the coefficient problem for functions of bounded boundary rotation. Consider the coefficients in the expansion

(1 + xz)o _ ~ A~O,~(x)z,, (1 - z ) ~

Ixl = 1,~ > o,~ > o.

n_-0

Brannan posed the problem as to when (30)

A(~'~)(x) < A(~'Z)(1).

R. W. Barnard

20 He gave a short elegant p r o o f t h a t (30) held if/3 t h a t for/~ = 1, 0 < a < 1, (30) did not hold for the (30) held for o d d coefficients in a sufficiently small t h a t for 0 < c~ = / 3 < 1, IAf~'~)(x)l _< A~7'~)(1). By

= 1 and a > 1. However, he showed even coefficients a n d t h a t for x = e i°, n e i g h b o r h o o d of 8 = 0. He Mso n o t e d using the e x p a n s i o n

A(~,Z)(x) = (/3)(/3 + 1 ) - - . (/3 + n - 1) n! 2Fl(-n, ~, 1 -/3; - x ) a n d t h e p r o p e r t i e s of 2F1, the h y p e r g e o m e t r i c function, this a u t h o r has s h o w n t h a t (i) (30) holds for a _ 1 and IA(~'~)(x)l < A(~'a)(1) for Ixl = 1, x # 1 a n d n=1,2,3,.... (~,I)

(ii) ]A~:~_~(x)] <_ A2~+l(1),n = 1 , 2 , 3 , . . . for 0 < a < a + e, 1 - 6 < a < 1 for e, 6 sufficiently small a n d positive, and (iii) ]A(~'a)(x)] <_ A(~'~)(1), 0 < a < / 3 < 1. In [61], D. Monk has shown t h a t (30) holds for a > 1,/3 > 1. Milcetich, in [59], has recently shown t h a t (30) holds for n = 5,/3 = 1 and 2 < c~ < n but does not hold for n o n integer o/s less t h a n n - 1,/3 near zero, for o d d n > 3. T h e basic

Conjecture

16. [A~:~(x)I < A ~ : ~ ( 1 )

is still open.

12. P o l y n o m i a l approximations using a differential equation model A n o t h e r conjecture on special functions arose out of the a u t h o r ' s a n d L. R e i c h e l ' s work on p o l y n o m i M a p p r o x i m a t i o n s using a differential e q u a t i o n model. G i v e n equidis t a n t d a t a (x~, y~) with x~ = 1 - (2i - 1)/M, the p r o b l e m is to best fit a p o l y n o m i a l of given degree N - 1 to M d a t a points. A c o m p a r i s o n is used b e t w e e n the discrete n o r m If" liD, defined b y 1 M

IIIlt2D = -~ ~ If( x, )l 2 t

a n d the continuous n o r m , I1" He, defined by 1 tlfllo =: ~ mea~cIf(x)l • G r a m p o l y n o m i a l s {~j} are used where they are o r t h o n o r m a l in the discrete n o r m with an e x p a n s i o n for p given by

J so t h a t [[p[[~ = ~ a j.2 T h e s e are defined recursively by

Open Problems and Conjectures in Complex Analysis

(31)

~ N ( X ) -----2X~N-I~N-1

21

-- (O~N-1/C~N-2)~N-2(X),

where

U ( N 2 : 1 / 4 ~ 112 aN = ~ - ~,M2 _ N2 ]

•

The asymptotics as M and N --* oo are studied by letting r = N/v/-M and x =

1 - (/M. T h e n the recurrence relation in (31) can be used to obtain qON -- 2TN-, + ~ N - 2 = [T2 - ( 1 / 4 r 2) -- 2¢] ~ON/M + O(1). This in turn can be used to obtain the differential equation model: (32) where t = v - 1 / ~

~"(t) = [t2 - ( 1 / 4 t ~) - 2~] ~(t), and the initial condition as t ~ 0 is defined by CN [1 - e/M] =

2~/-2-~v/t+ O(1/M),

i.e., T(t) ~ v~(t --~ 0). A normalization is made by ~(t) ~ ~/~f2-vt-M where ~ is an odd positive integer if and only if x is a grid point. The solution to (32) is given by

~(t) = tl/~e-t2/21Fl ( ~ 2 ~,1;t2) , where 1F1 is Kummer's confluent hypergeometric function (see Gradshteyn and Ryzhik [411). To find error estimates for least square approximates by these polynomials an application of Brass's result in [29] can be used that gives error estimates for least square norms in terms of the uniform sup norm. But in order to apply this result all the ~0N(1 -- ~/M)'s must have their sup norms occur at the right end point of the interval [ - 1 , 1]. An extensive computer analysis suggested that this does occur. W h a t is needed then is to verify C o n j e c t u r e 17. For all ~ > 0 and reM t we have

Indeed, by converting to the Whittaker functions M~,v(x) see [41], for a more convenient range of variables the conjecture is equivalent to showing that

~'I~,o(X) <_Mo,o(X) for all ~ >__0 and x _> 0. We have verified Conjecture 17 for the regions dotted in Figure 5.

R. W. Barnard

22

:::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::: ::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::: /~0 ::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::: ~ ~::::::::::::::::::: ~i

~i ~i 1 ~i

~!~! ~i

~i!!!i!i!!!iii!iii::

i~~:::::::::::"~"::: ,..:t:i!!i!!~iiii;iiiiii ,--:::::::~::::::::::::::::::: ...:::iiiiiiiiiiiii~iiiiiiii~!~iiiiii ~iiiiiiiiiiiiiiiii i...:::ii !i i i i i i i i i ~i i?!?i~i !i ~i i i

?

ii~i Ti iTi :~ i:~ !

X0

I{ I I I

X

Figure 5

References

[1] D. Aharonov, S. Friedland, On an inequality connected with the coe~icient conjecture for functions of bounded boundary rotation, Ann. Acad. Sci. Fenn. Ser. A.I.

524 (1972). [2] H.S. A1-Amiri, On the radius of univalence of certain analytic functions, Colloq. Math. 28 (1973) 133-139.

[3] R.M. All, Properties of Convex Mappings, Ph.D. Thesis, Texas Tech University.

[4] H. Alt, L. Caffarelli, Existence and regularity for a minimum problem with free boundary, J. Reine Angew. Math. 325 (1981) 105-144.

[5] J.M. Anderson, K.E. Barth, D.A. Brannan, Research problems in complex analysis, Bull. London Math Soc. 9 (1977) 129-162.

[6] R.W. Barnard, A variational technique for bounded starlike functions, Canadiaa Math. J. 27 (1975) 337-347.

[7] R.W. Barnard, On the radius of starlikeness of (z f)' for f univalent, Proc. AMS. 53 (1975) 385-390.

[8] R.W. Barnard, On Robinson's 1/2 conjecture, Proc. AMS. 72 (1978) 135-139. [9] R.W. Barnard, The omitted area problem for univalent functions, Contemporary Math. (1985) 53-60.

[10] R.W. Barnard, W. Dayawansa, K. Pearce, D. Weinberg, Polynomials with nonnegative coe~cients, (preprint).

Open Problems and Conjectures in Complex Analysis

23

[11] R.W. Barnard, R. Evans, C. FitzGerald, The center divided difference of polynomials (preprint). [12] R.W. Barnard, W.T. Ford, H. Wang, On the zeros of interpolating polynomials, SIAM J. of Math. Anal. 17 (1986) 734-744. [13] R.W. Barnard, C. Kellogg, Applications of convolution operators to problems in univalent function theory, Mich. Math. J. 27 (1980) 81-94.

[14] R.W. Barnard, C. Kellogg, On Campbell's

conjecture on the radius of majorization of functions subordinate to convex functions, Rocky Mountain J. 14 (1984) 331339.

[15] R.W. Barnard, J.L. Lewis, A counterexample to the two-thirds conjecture, Proc. AMS 41 (1973) 525-529. [16] R.W. Barnard, J.L. Lewis, Coej~cient bounds for some classes of starlike functions. Pacific J. Math. 56 (1975) 325-331. [17] R.W. Barnard, J.L. Lewis, Subordination theorems for some classes of starlike functions, Pacific J. Math. 56 (1975) 333-366. [18] R.W. Barnard, Lewis, J.L. On the omitted area problem, Mich. Math. J. 34 (1987) 13-22. [19] R.W. Barnard, K. Pearce, Rounding corners of gearlike domains and the omitted area problem, J. Comput. Appl. Math. 14 (1986) 217-226. [20] R.W. Barnard, G. Schober, Mdbius transformations for convex mappings, Complex Variables, Theory and Applications 3 (1984) 45-54. [21] R.W. Barnard, G. Schober, Mdbius transformations for convex mapping8 II, Complex Variables, Theory and Applications 7 (1986) 205-214. [22] B. Beauzamy, Jensen's inequality for polynomials with concentration at low degrees, Numer. Math. 49 (1986) 221-225. [23] B. Beauzamy, P. Enflo, Estimations de produits de polyndmes, J. Number Theory (1985) 21 390-412. [24] S.D. Bernardi, A survey of the development of the theory of schticht functions, Duke Math. J. 19 (1952) 263-287. [25] R.P. Boas, Entire Functions Academic Press, New York, (1954).

[26] D.A. Brannan,

On coefficient problems for certain power series, Symposium on Complex Analysis, Canterbury, (1973) (ed. Clunie, Haymasa). London Math. Soc. Lecture Note Series 12.

[27] D.A. Brannan, J.G. Clunie, J.G. (eds.). Aspects of contemporary complex analysis. (Durham, 1979), Academic Press, London (1980).

24

R. W. Barnard

[28] D.A. Brannan, J.G. Clunie, W.E. Kirwan, On the coefficient problem for functions of bounded boundary rotation, Ann. Acad. Sci. Fenn. Set. A.I. Math. 3, (1973). [29] H. Brass, Error estimates for least squares approximation by polynomials, J. Approx. Theory. 41, (1984) 345-349. [30] J. Brown, A coefficient problem for nonvanishing H p functions, Rocky Mountain J. of Math. 18 (1988) 707-718. [31] J. Brown, A proof of the grzyi conjecture for n = 4, (preprint). [32] D. Campbell, Majorization-Subordination theorems for locally univalent functions, Bull. AMS 78, (1972) 535-538. [33] D. Campbell, Majorization-Subordination theorems for locally univalent functions II, Can. J. Math. 25, (1973) 420-425. [34] D. Campbell, Majorization-Subordination theorems for locally univalent functions III, Trans. Amer. Math. Soc. 198 (1974) 297-306. [35] S. Chandra, P. Singh, Certain subclasses of the class of functions regular and univalent in the unit disc, Arch. Math. 26 (1975) 60-63. [36] P. Duren, Univalent Functions. Springer-Verlag, 259, New York, (1980). [37] R.J. Evans, K.B. Stolarsky, A family of polynomais with concyclic zeros, II, Proc. AMS. 92, (1984) 393-396. [38] A.W. Goodman, Note on regions omitted by univalent functions, Bull. Amer. Math. Soc. 55 (1949) 363-369. [39] A.W. Goodman, E. Reich, On the regions omitted by univalent functions II, Canad. J. Math. 7 (1955) 83-88. [40] A.W. Goodman, Univalent Functions. Mariner (1982). [41] Gradshteyn, Ryzhik. Table of integral, series and products, Academic Press, 1980. [42] R.R. Hall, On a conjecture of Clunie and Sheil-Small, Bull, London Math. Soc. 12 (1980) 25-28. [43] E. Gray, A. Schild, A new proof of a conjecture of Schild, Proc. AMS 16 (1965) 76-77, MR30 #2136. [44] W.K. Hayman, Multivalent functions, Cambridge Tracts in Math. and Math. Phys., Cambridge University Press, Cambridge, 1958. [45] C. Horowitz, Coefficients of nonvanishing functions in H ~, Israel J. Math. 30, (1978) 285-291. [46] J.A. Hummel, S. Scheinberg, L. Zalcman, A coefficient problem for bounded nonvanishing functions, J. Analyse Math. 31 (1977) 169-190.

Open Problems and Conjectures in Complex Analysis

25

[47] J. Jenkins, On values omitted by univalent functions, Amer. J. Math. 2 (1953) 406-408. [48] J. Jenkins, On circularly symmetric functions, Proc. AMS. 6 (1955) 620-624. [49] J. Krzy~, Coefficient problem for bounded nonvanishing functions, Ann. Polon. Math. 70 (1968) 314. [50] L. Kuipers, Note on the location of zeros of polynomials III, Simon Stevin. 31 (1957) 61-72. [51] V. Levin, Aufgabe 163, Jber. Dt. Math. Verein. 43 (1933) p. 113, LSsung, ibid, 44 (1934) 80-83 (solutions by W. Fenchel, E. aeissner). [52] Z. Lewandowski, NouveUes remarques sur les thdor~mes de Schild relatifs d une classe de fonctions univalentes (Ddmonstration d'une hypoth~se de Schild), Ann. Univ. Mariae Curie-Sklodowska Sect. A. 10 (1956). [53] Z. Lewandowski, On circular symmetrization of starshaped domains, Ann. Univ. Mariae Curie-Sklodowska, Sect A. 17 (1963) 35-38. [54] J.L. Lewis, On the minimum area problem, Indiana Univ. Math. J. 34 (1985) 631-661. [55] R.J. Libera, A.E. Livingston, On the univalence of some classes of regular functions, Proc. AMS 30 (1971) 327-336. [56] A.E. Livingston, On the radius of univalence of certain analytic functions, Proc. AMS 17 (1966) 352-357. [57] T.H. MacGregor, Geometric problems in complex analysis, Amer. Math. Monthly. 79 (1972) 447-468. [58] C. McCarty, D. Tepper, A note on the 2/3 conjecture for starlike functions, Proc. AMS 34 (1972) 417-421. [59] J.G. Mileetich, On a coefficient conjecture of Brannan, (preprint). [60] S. Miller, (ed.). Complex analysis, (Brockport, NY, 1976), Lecture Notes in Pure and Appl. Math. 36, Dekker, New York, (1978). [61] D. Moak, An application of hypergeometric function to a problem in function theory, International J. of Math and Math Sci. 7 (1984). [62] Z. Nehari, Conformal Mapping McGraw Hill, (1952). [63] K.S. Padmanabhan, On the radius of univalence of certain classes of analytic functions, J. London Math. Soc. (2) 1 (1969) 225-231. [64] K. Pearce, A note on a problem of Robinson, Proc. AMS. 89 (1983) 623-627. [65] D.V. Prokhorov, J. Szynal, Coefficient estimates for bounded nonvanishing functions, Bull. Aca. Polon. Sci. Ser. 29 (1981) 223-230.

26

R. W. Barnard

[66] A.K. Rigler, S.Y. Trimble, R.S. Varga, Sharp lower bound8 for a generalized Jensen Inequality, Rocky Mountain 3. of Math. 19 (1989). [67] R. Robinson, Univalent majorants, Trans. Amer. Math. Soc. 61 (1947) 1-35. [68] St. Ruscheweyh, A subordination theorem for aS-like functions, J. London Math. Soc. (2) 13 (1973) 275-280. [69] A. Schild, On a problem in conformal mapping of schlicht functions, Proc. AMS 4 (1953) 43-51 MR 14 #861. [70] J. Stankiewiez, On a family of starlike functions, Ann. Univ. Mariae CurieSklodowska, Sect A. 22-24 (1968-70) 175-181. [71] K.B. Stolarsky, Zeros of exponential polynomials and reductions, Topics in Classical Number Theory. Collog. Math. Sec., J£nos Bolyai, 34, Elsevier, (1985). [72] T. Suffridge, A coefficient problem for a class of univalent functions, Mich. Math. J. 16 (1969) 33-42. [73] O. Tammi, On the maximalization of the coefficient a3 of bounded ~chlicht functions, Ann. Acad. Sci. Fenn. Ser. AI. 9 (1953). [74] J. Waniurski, On values omitted by convex univalent mappings, Complex Variables, Theory and Appl. 8 (1987) 173-180. Received: August 30, 1989

Computational Methods and Function Theory Proceedings, Valparafso 1989 St. Ruseheweyh, E.B. Saff, L. C. Salinas, R.S. Varga (eds.) Lecture Notes in Mathematics 1435, pp. 27-31 (~) Springer Berlin Heidelberg 1990

A Remarkable Cubic Mean Iteration J . M . B o r w e i n and P . B . B o r w e i n

Mathematics, Statistics and Computing Science Department Dalhousie University, Halifax, N.S. B3H 3J5, Canada

1.

Introduction

Consider the two term iteration defined by (1.1)

a,~ + 2b,~ a,,+l . - - - ,

a0 := a,

3

and

(1.2)

b0 := b.

3

Then since (1.3)

3

z

%+1 -

b.+l

(as - b.) 3 =

27

'

it follows that, for a,b E (0, co), and for n >_ 1,

la.+l

-

b,~+ll _<

ta. - b.I -

-

27

and

(i.4)

F ( a , b ) := nlira as = nlim bn --~oo --+oo

is well defined, and that on compact subsets of (0, oo) the convergence is cubic. It is also easy to see that F(1, z) is analytic in some complex neighbourhood of 1. All of this is a straightforward exercise. What is less predictable is that we can identify the limit function explicitly, and that it is a non-algebraic hypergeometric fimction. Thus,

J.M. Borwein and P.B. Borwein

28

it is one of a very few such examples; and it is certainly the simplest cubic example we know. The most familiar quadratic example is the arithmetic-geometric mean iteration of Gauss and Legendre. Namely the iteration

an + bn

where, f o r O < x <

an+l

:=

2

bn+l

:= ~ ,

'

ao

:---- a ,

b0

:= x,

la::l lim an = lim b~ =

1

~F, ( 1 , } ; 1 ; 1 - x2) "

For a discussion of this and a few other examples see [2] and [31.

2. T h e

main

theorem

The point of this note is to provide a self-contained proof of the closed form of the limit of (1.1) and (1.2). This is the content of the next theorem. T h e o r e m 1.

Then

the

Let 0 < x < 1. Let a,~ + 2bn an+l

.--

b=+l

:=

Proof. The (2.1)

3

common limit, F ( 1 , x ) , is 1 = F(1,x)

ao

3

Z==o(n!)3 33,~

( 1 - x a) =2F1

b0

:= 1

X.

-'1;1-x '3'

3

limit function F (a, b) must satisfy r (a0, b0) = F (el, bl) . . . .

and since the iteration is positively homogeneous so is F. In particular (2.2)

F(a°'b°)=F(al'bl)=F(a°+2b° b ° ( a' 2~° + a ° b ° +3 b ~ ) ) 3

or

F(1,x) = F ( l + 2xx( l +' ~x + x 2 ) )33 (2.3)

"

A Remarkable Cubic Mean Iteration If we set mes

H(x) :--- ~

29

- x)/F(1, (1 - x)~) then the functional equation (2.3) becoH(x) = ~

(2.4)

H(t(x)),

where

(2.5) Furthermore

t(z) := 1 - 9x*(1 + x* + ~'~) (1 + 2x*) 3

x" '

~)~.

(1 :=

-

x/~H(x) is analytic at 0. The point of the proof is to show that x) 2Ft (3, 2-.1;x) 3,

G(x) := ~ 1 -

(2.6)

also satisfies the functional equation (2.4). From this it is easy to deduce that G(x) = H(x); as follows from the functional equation for H/G, and the value at x = 1. The (hypergeometric) differential equation satisfied by G is

(2.7)

a(x)

a"(x)

{-8x~ + 8x- 9~

•- G(~) - k 3-~x~--~7~ ]

Now it is a calculation (for details see [2]) that

(2.8)

a ' ( x ) :=

~,/-,3 ,a(t(x)) V ~'Lx)

also satisfies (2.7) exactly when (2.9)

a(x) = (t'(x))2a(t(x))-

t(x)

2 \t,(x) /

It is now another calculation, albeit a fairly tedious one, that a and t defined by (2.7) and (2.5) satisfy (2.9). We have now deduced that G*(z) and G(x) both satisfy (2.7). Furthermore, since the roots of the indicial equation of (2.7) are (1/2, 1/2) there is a fundamental logarithmic solution. Since both G* and G are asymptotic to vz~ at 0, they are in fact equal. Thus (2.8) shows that G satisfies (2.4). This finishes the proof. • As a consequence we derive the following particularly beautiful cubic hypergeometric transformation.

Corollary 1. For x E (0, 1)

~F1 l ! 2_. 1\3'3'

'1-x3

)

3

- -l +-2 x

2F1

;5;

\l+2x]

Proo£ This is just a rewriting of the functional equation (2.3).

]" •

J.M. Borwein and P.B. Borwein

30

The above verification entirely obscures our discovery of Theorem 1. This arose from an examination of some quadratic modular equations of Ramanujan [1, Chapter 21]. Notably, Ramanujan observed that,

(2.10)

(1 - u3)(1 - v 3) = (1 - uv) 3

is a quadratic modular equation, for 2F1 (1, ~, 2 . 1, . .). We then observed, with the aid of considerable symbolic computation, that if

(2.11)

qm~+m~+~2

L(q) := ~

and

(2.12)

R(q) . - 3L(q3)

1

2L(q)

2

then

u := u(q) := R(q)

v := v(q):= R(q 2)

and

solve (2.10) parametrically. From (2.12) it is natural to examine the cubic modular equation for R. This leads to the following result. T h e o r e m 2. Let

L(q) := ~ qm:+.~.+~2 --00

and

M(q) :--

3L(q 3) - L(q) 2

Then, L and M parameterize the mean iteration of (1.1) and (1.2) in the sense that if a := L(q) and b :-- M(q), then L(q3 ) . - a + 2b

3 and M(q3) = i b(a2 4- 3ba --k b2)

and the limit function F (of Theorem 1) satis6es M(q) ~ 1 F 1, L(q) ] - L(q) "

The derivation of this, which requires some modular function theory, will be discussed elsewhere [3].

A Remarkable Cubic Mean Iteration

31

References [1] B.C. Berndt, Ramanujan's Notebooks Part II, Springer-Verlag, New York, 1989. [2] J.M. Borwein and P.B. Borwein, Pi and the A GM - - A Study in Analytic Number Theory and Computational Complexity, Wiley N.Y., New York, 1987. [3] J.M. Borwein and P.B. Borwein, A Cubic Counterpart of Jacobi'3 Identity and the AGM, Trans. A.M.S., to appear. Received: October 8, 1989

Computational Methods and Function Theory Proceedings, Valpara/so 1989 St. Ruscheweyh, E,B. Saff, L. C. Salinas, R.S. Varga (eds.) Lecture Notes in Mathematics 1435, pp. 33-44 (~) Springer Berlin Heidelberg 1990

O n t h e M a x i m a l R a n g e P r o b l e m for Slit D o m a i n s 1 Antonio C6rdova Ydvenes and Stephan Ruscheweyh Mathematisches Institut, UniversitSt Wfirzburg D-8700 W/irzburg, FRG

A b s t r a c t . Let/2 C C be a domain, 0 E/2. For the family P~(/2) of complex polynomials p of degree < n satisfying p(0) = 0, p(D) C/2 (D the unit disk) we define the maximal range/2n as

/2~ :=

U

p(D).

pc~.(n)

We are interested in the explicit characterization of/2n for some specific domains as well as the corresponding extremal polynomials p E P,~(/2), i.e. the ones with p(D) M(0/2n \ 012) # 9. In this paper we solve completely the maximal range problem for the slit domains /2(a, b) = C \ ((-c¢, -a] u [b, oo)),

a, b > 0.

These results yield, for instance, new inequalities relating IlpII, IRepl, IImpl for typically real polynomials.

1. I n t r o d u c t i o n Given a domain $2 in the complex plane C, with 0 E 12, we define the family P,~(I2) of complex polynomials p of degree at most n which map the unit disk D into $2, with the normalized value p(0) = 0, i.e. 79,(0) := {p E 79,: p(0) = 0, p(D) C g2}. The maximal range for this family is defined as the set

n~:= U p(D). 1Research supported by the Fondo Nacional de Desarrollo Cientffico y Tecnol6gico (FONDECYT, Grant 237/89), by the Universidad F. Santa Maria (Grant 89.12.06), and by the German Academic Exchange Service (DAAD).

34

A. Cdrdova Ydvenes, St. Ruscheweyh

We call a polynomial p E P,~(J~) extremal for J?~ if

p(D n (0x?n \ 0x?) ¢ 0, and the points ~ E 0D with p(~) E 0D are called the points of contact of p. Our main interest is the description of this set f2, and the associated extremal polynomials. In our previous work on this problem (see [1], [2] and [3]) we gave the explicit characterization of D,~ and the corresponding extremal polynomials for some typical domains such as interior and exterior of disks, halfplanes and strips. We obtained in this way new sharp estimates relating ]IPIID, IlRepllD, IlImpllD, min~eD [p(z)[, etc. for p E P~(f2). The following general characterization for the extremal polynomials, which turned out to be constructive in many cases, has been derived in [3]. For earlier, somewhat

weaker versions, see [11, [2]. T h e o r e m 1.1 Every extremat polynomial p E 7),~(X?) with the normalization p(1) E Of2~ \ OD has the following properties: 1. p' has M1 of its zeros on OD. Let e i~°,, j = 1,... ,n - 1 denote these zeros, where 0 < ~/~ < ... < ¢~_~ < 2re.

2. There exist a t / e a s t n points of contact e iOJ, j = 1 , . . . , n (muItiplicities counted) such that (1) 3. If ~ is simply connected, then p is univMent in D. Moreover, for every co E OJ2~ \ Of2 there exists an extremM polynomial p E 7~( ~ ) with p(1) =

In this paper we apply these results to solve completely the maximal range problem for the slit domains

(2)

:= C \ ( ( - ~ , - a ] U [b, +c~))

a, b > 0.

As an application we obtain a set of new inequalities for polynomials: T h e o r e m 1.2 I f P E P~(J2), J2 = C \ [1,oo) then for z E D n+l zr IImP(z)l _< - - - ~ cot n +--~'

-

COS22n~2 rr < Re P(z) < c°t2 2n + 2 - sin ~ sin 2--~+2 ip(z)l _< cot~

A11 bounds are best possible.

~r n+2"

On the Maximal Range Problem t'or Slit Domains

35

T h e o r e m 1.3 Let ~? = C \ ( ( - o % - 1 ] t3 [1,oo)), n odd. Then for P E T',~(~) we have in D 1 ,n r,~ ,,ir~er~z)l < sin '~ ' n+l

IP(z)l <

n+l

These bounds are sharp for P as given in (24) at z = z e - - ~ , i, respectively. Note that this last result holds for typically real polynomials with the normalization Ie(x)l _< 1, - 1 < x < 1.

2. Slit d o m a i n s One of the problems which initially inspired us to study the general maximal range problem was to determine the best constants c(n) such that

(3)

[IPlID ~ c(n)llpll[-1,11

for typically real polynomials p in D with p(0) = O. Here ti" liD, I1" Itt-1,11 denote the sup-norm of the modulus on the corresponding sets. This question is related to the work of Rahman and Ruscheweyh [5] and the conjecture was that certain polynomials first studied by Suffridge [4] could be extremal. Clearly, the maximal range problem for the slit domain J2 := C \ ( ( - e c , -1] U [1, +c~)) is related to (3) although the condition "typically real" is not referred to when working with maximal ranges in our sense. Fortunately, for n odd, the extremal polynomials turn out to be typically real (the expected ones) thus permitting a solution of (3) in this case. The cases n even remain open for "typically real" (not for the maximal range problem, though). We study the slightly more general case of the domains J2(a, b) as defined in (2). Of course, we can normalize the situation letting b = 1 and writing ~ ( a ) for ~2(a, 1). We shall show that all extremal polynomials in the sense of Theorem 1.1 for ~2(a) can be described in terms of the above mentioned polynomials of Suffridge, which are given as follows: P ( z ; j ) = ~ Ak,iz k,

j = 1,...,n,

k=l

where

kj~ Ak,j -=

n-k+lSinn+t n

k,j = l,...,n.

sin J---~ n+l

We list a few properties of these polynomials (see [4]). See Figures 1,3 for typical graphs of P(0D; j).

A. C6rdova Ydvenes, St. Ruscheweyh

36 L e m m a 2.1 For P(z,j) the following holds:

1. Let a = Az_ Then, for I0[ < 7r n+l

"

P(ei°;j) = {

+ i (1 -- (--1)Je i(n+l)O) sin 0 n+l , 2 . ( c o s 0 - cos ~) 2n(cos e - cos a) 2

O¢+oL,

(n + 1 ) c o s a + i ( n + t) 2 4n sin 2 a 4n sin a '

~=+a.

2. P(z;j) is typically teat and univalent in O. 3. P'(z;j) has all zeros on OO. i(2k+J)~ 4. P(e ~+1 ;j) c R f o r k = l , . . . , n + l , k ~ n + l - j , k # n + l . •

5. R e e - . + 1

P'(e

2k

7r

.+1 ; j ) = O ,

k=l,...,n+l.

Figure 1

2a. The

one-slit

case

We start with the one-slit case O := O(oo). Since O is starlike with respect to the origin and symmetric with respect to the real axis the same properties hold true for J~n, whose boundary therefore consists of a connected portion of the slit and a Jordan arc connecting the upper and the lower shore of the slit. Let w be any point on this arc. According to Theorem 1.1 the corresponding extremal polynomial P satisfies

On the Maximal Range Problem for Slit Domains

37

P E T'=(~2), w E P(0D), P is univalent in D, and P ' has all the zeros on OO, interlacing with points of contact. The velocity of the argument of the tangent vector at P(e i°) is -__+_! except at the zeros of P ' where the argument jumps back by 7r. Hence between two 2 ' successive points of contact with the lower (upper) shore 0 has to move by exactly ~-gT" 2, Near the vertex at 1 there are two possibilities, schematically shown in Fig. 2. In either case, increasing 0 by 2~ from the last point of contact with the lower shore will lead us to the next point of contact ( with Re (p'(¢) = 0 (P'(O = O or horizontal tangent). At the other end we find that between the last point of contact with the upper shore and the first one with the lower shore the argument of the tangent vector has to turn by 3Tr which means for 0 a change of n6~ +l " Without loss of generality we may assume that the preimage of the last point where P has a horizontal tangent before it reaches the lower shore is z = 1. Then we readily obtain the following necessary conditions for P, using ~k = e~ik/(n+l): (4)

(5)

ImP((k)=0,

k=l,...,n-1,

Re 6 P ' ( 6 ) = 0 ,

k=l,...

,n

+1,

n+l

(6)

~-]P(~k)=0. k=l

The condition (6) follows from the mean value property for polynomials and the assumption P ( 0 ) = 0. We wish to show that the set of polynomials satisfying (4)-(6) is a one parameter family. However, in the sequel we shall need a slightly more general result.

Figure 2 P r o p o s i t i o n 2.1 Let P e "P, satisfy (5), (6) and (7)

ImP(4k)=0,

k=l,...,n+l,

k¢r,s,

with r,s 6 { 1 , . . . , n + 1}, s > r. Then, f o r a certain .~ E R,

(8)

=

_ .(~+r)~

Furthermore, ~. Rep(~k) > O, &. Imp(~T) _< O.

_

A. Cdrdova Ydvenes, St. Ruscheweyh

38

Proo£ Let Q(z) = 1 - z TM. By Lagrange interpolation we obtain P(z) =

Q(z) ~

¢JP(~i)

j+l

and, after differentiation,

~kP'(~k) = ~., ~ k - ~j +

(9)

,~+1 j = l ~jp,(~j___~) 2 j#k

=~ = e(O

(10)

P(~k)

cot X : / ~

+i

+

jCk Taking real parts and using (5), (6) we get n+l

(11)

(n + 1)ReP(#k) = ~ I m P ( ~ j ) c o t (j - k)~ , j=l n+l j#k

k = 1 , . . . , n + 1.

Now let k = r, s. Then using (8) we obtain r -

3)~

(n + 1)ReP((~)

=

ImP(G)cot--

(n + 1 ) R e P ( G )

=

(~ - ~)~ ImP(~r)cot - n+l

(12)

n+l

'

Taking imaginary parts in (12) one gets

(13)

I m P ( G ) = -ImP(<~),

which together with (12) shows that P(G) = P(~). For k ¢ r, s (11) yields

P((k) = Re P((k)

=

Im P((~) cot (rn-+___~k)r+ Im P ( G ) cot (S__n+lk)~"

=

ImP((~)cot

n+-I

cot n--+-I J"

Using this and (12) and (13) one deduces that all values P(~k) , k = 1 , . . . , n + 1 are uniquely determined by ImP(~r) and depend linearly on this parameter. Lemma 2.1 shows that the function on the right hand side of (8) satisfies (5), (6), (7) which implies the representation (8). The final conclusions follow easily. •

On the Maximal Range Problem [or Slit Domains

39

We return to the one slit case and conditions (4)-(6) for the extremal polynomials. Proposition 2.1 yields (with r = n , s = n + 1)

P(z) = •P(e , .+1; ..2__ 1) and by our construction we have Im P((n+l) < O, hence A < 0. Since rotations are not important for our problem we m a y use the identity P(z;j) = P ( - z ; n + 1 - j) to claim that every extremal polynomial (up to rotations) is of the form

P(z) = AP(z;n) ,

A > O.

But P(z; n) has contact with the tip of its corresponding slit at z = 1 and this determines A. We have proved: T h e o r e m 2.1 For 12 = C \ [1, oo) we have 012.\012=

{ P(e~e; n) P(1;n)

: [8-rr[<

~

3re }

.

Theorem 1.2 is a corollary to this; the bounds for Im P aald Re P in Theorem 1.2 nr are attained at 0 = ~-~ and 0 = 7r, hn --- 1~ r respectively, for P = P(e~°; n). For estimating ]P(z)[ we used the fact that the coefficients of P(-z; n) are negative. Figure 1 shows a typical one-slit case (n -- 3).

2b. The

double-slit

case

The case of double slits is more complicated. As before, 12,~(a) is starlike and symmetric with respect to the real axis. We may assume that the left side has contact with 12n(a) since otherwise we are essentially back at the one slit case. This means that we must take a < cot 2 ~ (Theorem 1.2). On the other hand, if a is too small, namely a < cot -2 ~ then 012,~(a) looses contact with the right hand slit and we have again a one slit situation. Hence we may assume that cot -2

~r 2n+2

< a < cot 2 - -7r 2n+2"

Then 012,~(a) consists of two connected segments of the two slits and two symmetric Jordan curves connecting the upper (lower) shores of the two slits. If w is a point on the upper Jordan arc then there exists an extremai polynomial P which has contact with both slits and connecting Jordan arc between the slits. The upper arc cannot contain a vertex ( a point P ( ( ) , with P ' ( ( ) = 0) but there may be one vertex on the lower one. These observations follow from Theorem 1.1. Again assuming that P((,~+I) is the last point with horizontal tangent before we find the first point of contact with the lower shore of the right hand slit we see that for the extremal polynomial (14)

ImP((k)=O,

k=l,...,n-1,

k#r,

A. C6rdova Ydvenes, St. Ruscheweyh

40

Figure 3 where r E { 1 , . . . , n - 1}. P(•) is the point in the upper half plane where P has horizontal tangent. Conditions (5) and (7) remain valid. We now proceed as in the proof of Proposition 2.1, and arrive at the representation (11). Choosing k = r,n,n + 1 in (11) we obtain with (14)

(15)

( n - rr"j+ImP(C ~ +l)cot\(n + 1 - r r~J'

(n + 1)ReP(C,)

=

ImP(C,)cot

( n + 1)ReP(~,)

=

ImP(¢~)cot (\ nr - +n z1r ~ ] + ImP(~n+l)cot ( ~

(n + 1)Re P((,~+I) =

ImP(~r)cot(r-n-l~r)

~r )

+ Im P(~,~) cot ( ~ - - i - )

and in the remaining eases ( n + 1)P(~k) = (16)

=

(n + 1)Re P(~k) ImP((r)cot \ n + 1 ] + I m P ( ( , ) c o t ~,n+ 1 ]

+ ImP(~+,)cot ( n + l - k ) n+l

7r

.

Finally, taking imaginary parts in (7) we get (17)

,

Im P((,) = - I m P((~) - Im P(~,+,).

On the Maximal Range Problem for Slit Domains

41

Inserting (17) into (15), (16) we deduce that all values P((k) depend linearly of the two values Im P((~), Im P((,,+~). Furthermore Im P((~) and Im P ( ( , + I ) are non-positive if we are not in the one slit case which we have excluded. Hence we may write

~,3 > o,

P(z) = ~P,(z) + 3p~(z) ,

where PA, /'2 satisfy (14), (5), (6) and ImPs(G) = 0, Px(C~+~) < 0, ImPz(G) < 0, Pz((n+l) = 0. But then we are back at the assumptions of Proposition 2.1 for P1 (with s = n + 1) and P2 (with s = n) and we conclude i(,~+1 +r)

p~(z)

=

a~p(~- o+---r--~;~ + 1 - r ) ,

P2(z)

= a=rke

.+~ z ; n - r ) ,

al < O,

Aa
Again, using the relation - P ( - z ; j) = P(z; n 4- 1 - j ) we can rewrite this as

Pl(z)

= A1P(e-'~z;r)

,

P2(z) = A2P(~z;r + 1),

~1 > 0, A2 > O,

-n+r

with ( = -e-'-+~ *. Since rotations in the argument are of no significance for our problem we finally arrive at (i8)

P(z) = aP(z; r) 4-/~P(e .+, z; r 4- 1 ) ,

a, fl >__0,

as a general representation for extremal polynomials related to boundary points of f2,~(a) in the upper half plane. The final step is to show that for every a E (cot -2 2~-~,cot 2 ~-7~) there is exactly one system (a, fl, r) which fulfills the remaining conditions for an extremal polynomial + r~ (r+2)~ as pointed out in Theorem 1.1. Using (18) and Lemma 2.1, we get for 0 # n+l, - ,+1 (mod 2rr)

p(eiO)

=

n+l

=

__i (1 - ( - 1 ) % i("+O°) [aft(0)4- #f'+l(' 0 + 7)] • 2n

2-'---~ [aft(0) 4- ~fr+l(O 4- 7)]

(19)

where ^t = ,%-f,~fj(8) = 1/(cos 8 - cos ~,~+1)"Let us now fix r and assume 04 fl # 0. Then

af'~(O) 4- fif;+l( 4 - ? ) # 0 ,

8 = -i~-n+1,J

0,...,n, j # r

and it never changes sign when passing through one of these points. Hence the imaginary part of P(e ~e) cannot change sign in these points as well. But since P(0) = 0, the graph of P(e~°), O E [0, 2~r) has to cross the real axis at least twice, and this must occur at the zeros of g(8) = aft(O) 4- flf'+l(O 4- 7).

A. Cdrdova Ydvenes, St. Ruscheweyh

42

However, the terms of g have different sign only in the i n t e r v a l s / 1 = ( - 7 , 0), I2 = (~r - 7, rr). It is easily checked that h(O)=

frl(8) __ sin8 (cos(8 +_ 7_.)- _cos((___r_+ 1 ) 7 ) ) 2 y'+, (8 + 7) sin(8 + 7) cos 8 -- cos r 7

is monotonically increasing from - o o to 0 in b o t h / 1 a n d / 2 , such t h a t the equation

h(8)=---z

(20)

has exactly two solutions: 81 E / 1 and 82 E/2. But if P is an extremal polynomial of the t y p e described in T h e o r e m 1.1 then P(e ~°~), P(e ~°~) must be the vertices of the two slits otherwise there would be an are between (and including) the images of two zeros of P' without point of contact ( compare Figure la). Since P(e ~°~) > 0 , p(e~0~) < 0 we can identify the corresponding slits. To have really candidates for our normalized situation ~2(a) we therefore have to replace P by P / P ( e ~°~). Then, in the new notation, we have

p(eie2) --_ aft(02) +/~fr+l(82 + 7) ~f~(0,) + ~f~+1(81 + 7)' where we obviously can replace/3 by 1 - a and assume 0 < a < 1. In fact, all considerations r e m a i n valid, in a limiting sense, for a = 1. T h u s all what remains to be done is to show t h a t for each a E (cot -~ ~ ' k cot 2 ~ k ) there is at most one pair (a,r) with 0
(21)

c~f,(O,) + (1

-

c~)f,+,(O, + 7)'

(that there is at least one follows from T h e o r e m 1.1). If a = 1 then 81 = 0, 82 = 7r and hence (21) becomes --a

=

--

tan 2

rTc

--.

--an

,

r =

l,...,n--

I,

2n+2

a decreasing sequence in r, with a,` = c°t2 7;-;Z,~ al = cot -2 2g--+2'~- I/a,,. T h u s we only need to show t h a t the right h a n d side of (21) is monotone in a for r fixed. This, however is a simple exercise in calculus, differentiating with respect to a and using (20) (with /3 = 1 - a), whose details we can omit. We arrived at the following theorem: T h e o r e m 2.2 Let a C [a~,a~+1), r e { 1 , . . . , n a £ (0,1] with h(e,) = h ( e ~ ) -

ce - 1 O~

,

- 1}. Then there is exactly one

e, C (-~+~,01,

6~ ~ ( - ~ , , ~ ] ,

for which (21) is satisfied. Then

(22)

P(z)

:=

c~P(z; r) + (1 - c~)P(e i J-+, - e; r + 1) e Pn(~?(a)), c~P(eie~ ;r) + (1 - c~)P(ei.-#v+e';r + 1)

On the Maximal Range ProMem for Slit Domains

43

and (23)

r--2 < ~ < n+," r+2_l Of2~(a) \ Of2(a) = {P(eie), P(e{e) : ~-~Tr J'"

If a ~ an or a ~ al then the maximal range problem f2n(a) is essentially a one slit problem and can be solved by Theorem 2.1. Note that cases a = a~ are simple since then a = 1 , 81 = 0 and the polynomials (22) are typically real. As the special case of probably greatest interest we discuss the one with a = 1 (symmetric slit, for an example see Figure 3): C o r o l l a r y 2.3 Let f2 = C \ ( ( - o c , - 1 ] U [1,oo)). Then for n odd, we have (24)

P(z) =

p (z;

e Pn(f2),

and (25)

0f2, \ 0f2 = {P(ei°), P(ei°) : yr

21r

n+,

2~r < 8 < ~r + h-~}"

For n even, the extremM polynomial and the range f2,~ al'e described by (22), (23), with a=l,a=½, r = t 2 ~, and h(el) = - 1 , 0, e In the case n odd it follows from the properties of the extremal polynomial that IReP(ei°)[ is maximal for 0 - z ~1" The same extremal polynomial is odd and its coefficients are alternating in sign. Hence it takes its maximum modulus at z = i. The value at this point can be obtained using Lemma 2.1. This proves 1 (26)

t R e P ( z ) I ___ sin n+l

(27)

n+l IF(z)[ _< - - , 2

and Theorem 1.3. We note that Corollary 2.3 holds, in particular, for typically real polynomials in 7~n(f2). Hence it contains the solution of the problem (4) for n odd. We obtain the sharp bound (28)

c(n)-

n+l 2

Clearly, Theorem 1.2 gives also a bound for c(n), n even. However this bound is not sharp since our extremal polynomial is not typically real in this case.

References [1] C6rdova A. and Ruscheweyh St., On Maximal Range~ of Polynomial Spaces in the Unit Di~k, Constructive Approximation 5 (1989), 309-327.

A. CSrdova Ygvenes, St. Ruschewevh

44

[2] C6rdova A. and Ruscheweyh St., On Maximal Polynomials Ranges on Circular Domains, Complex Variables 10 (1988), 295-309. [3] C6rdova A. and Ruscheweyh St., On the Univalence of Extremal Polynomials for the Maximal Range Problem, to appear. [4] Suffridge, T.J., On Univalent Polynomials , J. London Math. Soe. 44 (1969), 496-504. [5] Rahman, Q.I. and Ruscheweyh St., Markov's Inequality for Typically Real Polynomials, J. Anal. Appl. (to appear). Received: August 31, 1989

Computational Methods and Function Theory Proceedings, Valparaiso 1989 St. Ruscheweyh, E.B. Sail', L. C. Salinas, R.S. Varga (eels.) Lecture Notes in Mathematics 1435, pp. 45-55 @ Springer Berlin Heidelberg 1990

On Bernstein Type Inequalities and a Weighted Chebyshev Approximation Problem o n Ellipses 1 Roland Freund Institut ffir Angewandte Mathematik und Statistik, Universit//t Wiirzburg, Am Hubland, D - 8700 Wiirzburg, F R G and RIACS, Mail Stop 230-5, NASA Ames Research Center, Moffett Field, CA 94035, USA.

A b s t r a c t . We are concerned with a classical inequality due to Bernstein which estimates the norm of polynomials on any given ellipse in terms of their norm on any smaller ellipse with the same loci. For the uniform and a certain weighted uniform norm, and for the case that the two ellipses are not "too close", we derive sharp estimates of this type and determine the corresponding extremal polynomials. These Bernstein type inequalities are closely connected with certain constrained Chebyshev approximation problems on ellipses. We also present some new results for a weighted approximation problem of this type.

1. Introduction Let H~ denote the set of all complex polynomials of degree at most n. For r _> 1, let

(1)

E,'={zeC

I Iz-ll+lz+ll
!r }

be the ellipse with loci at :t:1 and semi-axes (r :t: 1/r)/2. Moreover, we use the notation I1" IIE~ for the uniform norm IIfIle~ = max,¢e~ If(z)l on gr. It is well known (see e.g. [8, Problem III. 271, p. 137]) that, for any n E N and

R>r>_l,

(2)

R~

Ilplle. < 7

Ilplle~ for all

p

•/-/..

XThis work was supported by Cooperative Agreement NCC 2-387 between the National Aeronautics and Space Administration (NASA) and the Universities Space Research Association (USRA).

R. Freund

46

We remark that for the case r = 1, CI = [-1, 1], this inequality goes back to Bernstein (see [2] and the references therein). It is also well known (see [12, p. 368]) that the estimate (2) is not sharp, i.e. equality in (2) holds only for the trivial polynomial p -- 0. In this note, we are mainly concerned with the following two problems: find the best possible constants C,~(r, R) and Cn+l/2(r,R) such that [[Plier -< G ( r , R ) [IpIG

(3)

for all

p E H~

and

IIwplG <_G+l/2(r,n)

(4)

Ilwplle~ for all

p E H,~,

respectively. Here, and in the sequel, w denotes the weight function w(z) = ~/z + 1, and it is always assumed that the square root is chosen such that w maps the z - p l a n e onto {Rew > 0} t3 {i~/l~? _> 0}. We notice that the usual proof (e.g. [8, p. 320]) for (2) immediately carries over to the weighted case (4) and leads to the upper bound C~+I/2(r,R) < Rn+l/2/r '~+~/2. For the classical case r = 1, Frappier and Rahman [2] conjectured that

C.(I,R)-- I(R'~ + R n-2)

(5)

and C~+,/:(1,R)= ~ ( R ~+1/2 + R~-a/2).

The first identity in (5) was proved in [9] for n = 1 and in [4] for n = 2, R second relation in (5) is known to be true for n = 1 and R _> 1.49 [4]. It these are the only cases for which the best possible constants in (3) and (4) In this paper, sharp estimates (3) and (4) will be obtained for the case n and R not "too close" to r. More precisely, we will prove the following

>__v~- The seems that axe known. E N, r > 1,

T h e o r e m 1. Let n E N and r > 1. 73r 4 - 1 R>_r - r4-1

(6)

resp.

R>r

33r - 1 - r-1

then (7)

C~(r,R) = nn + 1/n~ r n + 1/r n

resp.

R~+ll 2 + 1/R~+1/2 Cn+l/2(r, R) = r~+1/2 + 1/r~+1/2

with equality holding in (3) resp. (4) only for the polynomials 26 (8) p(z) -- ~/(Tn(z) + i R a _ 1 / R , ) , ~/C C, 6 e [-1,1]

resp.

p(z) =- 7V,(z), ~/E C.

Moreover, if n = 1, the first identity in (7) holds true for all R > r. b) R ~ + 1/R ~ Rn+l/2 + 1/R'~+I/2 C,(r,R) > and C~+I/2(r,R) > r ~ + 1/r ~ rn+ll 2 + 1/r'~+l/~ for all R > r.

On Bernstein Type Inequalities ...

47

In (8) and in the following, the notation Tk is used for the kth Chebyshev polynomial which by means of the Joukowsky map is given by

T~(z) =_ (v ~ + ~ ) ,

(9)

1

!),

z - 50, + v

and, moreover,

Vk(z) =--

(10)

Tk+~(z) + Tk(z) V/2(z + 1)

Notice that (10) defines indeed a polynomial of degree k, and that Vk is up to a scalar factor just the kth Jacobi polynomial p~-1/2,1/~) associated with the weight function (1 - z)-1/2(1 + z) '/2 on [--1, 11 (cf. Szeg5 [13, p. 601). R e m a r k 1. The estimates (6) are very crude in the following sense. Let Rn(r) resp. R,~+~/2(r) denote the smallest numbers such that the first resp. the second identity in (7) is satisfied for all R >_ R~(r) resp. R >_ R~+l/2(r). Numerical tests reveal that R~(r) and R,~+l/2(r) are much smaller than the upper bounds in (6). Moreover, these experiments suggest that Rn(r), R~+~/2(r) ,~ r for large n. However, we were not able to prove these numerical observations. Although the weighted norms in (4) might appear somewhat artificial, note that (4) arises naturally if, using the Joukowsky map (of. (9)), one rewrites the estimates (3) and (4) for the disks lvl _< R, tv[ _< r, and the class of self-reciprocal polynomials

I vms(ll v)

zm := { s • / 7 . (cf.

[2,4,5]).

-

-

s(v) }

More precisely,

(11) resp.

v

~+1-2

1

! w(~(v +

1

~

))p( (v +

1

))=_ s(v),

p • 17n,

s • G'2n+l,

defines a one-to-one mapping between Hn and 2~2, resp. Z72n+l. With (11), it is easily verified that (3), (4) are equivalent to (12)

max [s(v)[ < Dm(r,R) max [s(v)[ Ivl_
for all

s E Zm,

where (3) and (4) correspond to the case m = 2n and m = 2n + 1, respectively. Moreover, the best possible constants in (3), (4), and (12) are connected by Din(r, R) = (R/r)m/2C,~/2(r, R). Rewriting Theorem 1 for (12), then yields the following C o r o l l a r y . Let m >_ 2 be an integer and r > 1. a)zr f (73r 4 - 1)/(~ 4 - 1) if

R __ ~" <( (33~ - 1)/(r

then

-

1)

if

m

is

even

m

is odd '

R. Freund

48

Dm(

,n) -

Rm +1

with equality holding in (12) only for the polynomials 45 s ( v ) - 7 ( v 2~ + z"R ~ _ I , Rn /

v "~+1),

7•C,

s(v)=_7(v m + l ) ,

7•C,

if

,~• [-1,1],

rn=2n

is even,

and m

if

is odd.

Moreover, D~(r, R) = (R 2 + 1)/(r 2 + 1) for atI R > r. b)

Rm+l Dm(r,R) > _ - rm+l

for all

R>r.

Our proof of Theorem 1 is based on the obvious representations (13)

Cn(r, R) -- max - -

1

E.(r, c)

and

Cn+l/2(r, R)

-----

1 max c~oen En+l/2(r, c)

of the sharp constants in (3) and (4) in terms of the optimal values of the family of constrained Chebyshev approximation problems

(14)

(E.(r, c):=)

min

p6 nr~:p(c)=l

I[Pller

and (15)

(En+l/2(r,c) :=)

min IIw~pll~ where p~lln:p(c)=l

we(z) =- w(z)/w(c).

Here, c • C \ $~ in (14) and (15), and, in (13), Ogn denotes the boundary of gR. The class of complex approximation problems (14) was investigated recently by Fischer and Freund [3]. In particular, the part of Theorem 1 which is concerned with the Bernstein type inequality (3) will follow from results given in [3]. In the present note, we will also derive some new results for the weighted variant (15). The outline of the paper is as follows. In Section 2, we establish some auxiliary results. In Section 3, the complex weighted approximation problem (15) is studied. Finally, remaining proofs are given in Section 4.

2. P r e l i m i n a r i e s In this section, we introduce some further notations and list two lemmas. It will be convenient to define, in analogy to the Chebyshev polynomials (9), the functions (16)

Tk+l/2(z) --

l(vk+,/2

-~.

1___~), 1 1), + vk+,/2 z = -~(v + v

k = O, 1,...

On Bernstein Type Inequalities ...

49

Here the square root v/~ is chosen, correspondingly to w(z) = x/z + 1, such that v/~ maps the v - p l a n e onto {Re~ > 0} U {iqlq > 1 or - 1 < r / < 0} (cf. [4]). With (10) and (16), one readily verifies that then

Tk+~/2(z) = w(z)Vk(z),

(17)

z E C,

holds. For the boundary points z E Og~ of the ellipse (1), we will use the parametrization (18)

z=z,(~)=l(r+!)cosg~+~(r-1)sing%

-~r<~<_~r.

With (16) and (18), it follows that

(19)

T k + l / 2 ( Z r ( ~ ) ) = ak

cos(k + 1)¢p + ibksin(k + 1)~, Z,

-~r < ~ <_ %

where (20)

1 (rk+ll 2 ~ ) ak := ~, + rk+l/2

and

1 (rk+ll ~ bk := ~,

1 rk+,/2).

Next, assume that n E N and r > 1. Using (17) and (19), we deduce that

IiwV, ll& = a,.

(21)

All corresponding extremal points zt E $~, defined by lw(zt)V,(z,)l = IIwTV~IIe~, are given by (22)

z, := z,(~,),

~"-

2br 2n+l'

l=-n,-n+l,...,n-l,n.

Moreover, we note that, in view of (17), (19), and (22), as

V,(z,) = ( - 1 ) ~ w(z,)'

(23)

t = -n,...,n.

The following property of the numbers ¢p~ will be used in the next section. L e m m a 1. Let j E Z. Then:

(24)

= { t=-~

+ 1

2z + 1

0

otherwise

Proof. With q := (2j + 1)/(2n + 1), we have e~'(j+1/2) = (eWe)t. If q E 2Z + 1, then e q'i = - 1 , and (24) is obviously true in this case. For q • 2Z + 1, (24) follows from n

,=-n

a n d ( - e q ~ ) 2~+1 = 1.

1

I

1 + e qri

•

R. Freund

50

Finally, we will apply the following result due to Rogosin'ski and Szeg5 [11] in Section 4. L e m m a 2. Let Ao, At,.. •, A~ be real numbers which satisfy A~ >_ O, An-t - 2A~ >_O, and Ak-1 -- 2Ak + Ak+l _> 0 for k = 1,2 . . . . , n - 1. Then:

A0 }-~Akcos(k~)

(25)

t(~) := T +

> o foraU

~eR.

k=l

3. Results for the weighted approximation problem (15) In this section, we are concerned with the constrained Chebyshev approximation problem (15). In the sequel, it is assumed that n C N, r > 1, and c E C \ gr. Standard results from approximation theory (see e.g. [6]) then guarantee that there always exists a unique optimal polynomial for (15). For the case r = 1 of the unit interval gt = [-1, 1] and c E R \ [-1, 1], Bernstein [1] proved that the resealed polynomial (10)

v,&)

(26)

v~(z; c) - v,~(e)

is the extremal function for (15). For purely imaginary c and, again, r = 1, Freund and Ruscheweyh [4] showed that the optimal polynomial is a suitable linear combination of v,~, v~-l, and v~-2. To the best of our knowledge, these two cases seem to be the only ones for which the solution of (15) is explicitly known. For the rest of the paper, we assume that r > 1. It turns out that, somewhat surprisingly, (26) is also best possible for the general class (15) with complex c as long as c is not "too close" to gr. For the following, it will be convenient, to represent c ¢ g~ in the form

(27)

c=cr(¢)---~(R+~)cos¢+

(R-

)sine,

R>r,

-Tr
In analogy to (19) and (20), it follows that

(2s)

dk := Tk+a/2(c) = Ak cos(k + ~)~b + iBk sin(k +

)¢,

where (29)

1

Ak := ~(R

k+i/2

1 + Rk---T~i~)

and

I

Bk := ~(n

k+l/2

1

Rk+i/2).

Based on Rivlin and Shapiro's characterization [10] of the optimal solution of general linear Chebyshev approximation problems, we next derive a simple criterion for the polynomial (26) to be best possible in (15). Note that the extremal points of v,~(z; c) are just the zl, l = - n , . . . , n, stated in (22). By applying the theory [10] to (15), (26), and by using (23), we obtain the following

On Bernstein T y p e Inequalities ...

51

C r i t e r i o n . v~( z; c) is the unique optimal polynomiaJ for (15) iff there exist nonnegative reM numbers a-n, a - n + 1 , . . . , a~ (not all zero) such that (30)

~

al(--1)tw(zl)q(zl) = 0

f o r all

q e II~

with

q(c) = O.

Clearly, it suffices to check (30) for the polynomials

q(z) - Vk(z) - Vk(c),

k --- 1, 2 , . . . , n.

With (17) and (28), this leads to the following equivalent formulation of (30): (30')

~

al(-1)'(doTk+l/2(z,)

-

dkT1/~(z,))

=

O,

k

=

1, 2 , . . . , n.

It turns out that there are simple formulae for all real solutions crz of (30'). The trick is to use the ansatz (31)

n at = ~'~(#j cos(j~l) + ujsin(j~l)), j=0

~t -

2br 2n + 1 '

l = -n,..

''

n,

where #j, uj E R, j = 0 , . . . , n. Note that such a representation (31) is possible for any collection of c~_~,..., c~n E R. Now we insert (31) into (30') and rewrite Tk+l/2(zt) and T~/~(zl) in the form (19). Then, a routine calculation, making repeatedly use of Lemma 1, shows that (30') reduces to the equations (323)

tLkan-k -- iukb~-k - (d,-k/do)(#nao - iu~bo) = O,

k = 1, 2 , . . . , n - 1,

and

(32b)

2#0a~ - ( d , / d o ) ( p , ao - iunbo) = O.

By determining all real solutions #k, uk of the linear system (323,b) and with (31), one easily verifies that all real numbers satisfying (30') are given by al = wa~, l = - n , . . . , n with r E N arbitrary and a~ defined in (33). Hence, in view of the Criterion, we have proved the following T h e o r e m 2. v~(z; c) is the unique optima1 polynomial in (15) iff the numbers

(Re(<_k, O)cos(k ,)+ m(eo_kdo)sin(k ,),)

1 IdnJ2 +

(33)

cr~ .-- 2 a,~

k=i

a~-k

b~-k

l = -n,-n+

1,...,n-

1,n,

are either all nonnegative or all nonpositive. Here ak, bk, dk, and ~t are deigned in (20), (28), and (22). The numbers (33) are positive whenever R / r is sufficiently large. In particular, in the next section we will prove the following

R. ~'~eund

52 T h e o r e m 3. Let c = cR(¢) with R > r > 1 (c£ (27)). Then: a)

(34)

En+l/2(r, C)

rn+l/~ + 1/r'~+'/2

~/(R n+l/2 Al-1/R~+1/2) 2 - 4 sin2(n +

1/2)¢"

b) If R >_ r(33r - 1 ) / ( r - 1), then v~(z; c) is the unique optimal polynomial for (15) and

equality holds in (34). For the case that c in (15) is reM, we have the following sharper result. T h e o r e m 4. Let r > 1 and c E R. If (i) c > r + 1/r - 1/2 or (ii) c ~_ - r , then vn(z; c) is the unique optimal polynomial for (15) and

c) =

r~+ll 2 + 1/r~+l/2 Rn+ll 2 + 1/Rn+l/2

in c se (5

r'~+l/2 + 1/r ~+112 R~+1/2 _ 1/R,~+1/2

in case (ii)

R e m a r k 2. In contrast to the case r -- 1, for r > 1, the polynomial v~(z; c) is not best possible in (15) for all c E R \ Er. Indeed, numerical tests show that among the corresponding numbers (33), in general, positive and negative a~ occur if c is very close to £~. Finally, we note that Theorem 3 is analogous to the following result for the unweighted approximation problem (14).

a)

T h e o r e m A. (Fischer, Freund [3]). Let c = cn(¢) with R > r > 1 (cf. (27)). Then:

r ~ + 1/r ~ E~(r,c) < R ~ + 1 / R .

(35) b) I f R >_ r(73r 4 - 1)/(r 4

-

1), then

pn(z; c) -- (Rn - 1/Rn)T~(z) + 2i sin(n¢) ( R '~ 1/R~)Tn(c) + 2i sin(n¢) is the unique optimal polynomial for (14) and equality holds in (35). R e m a r k 3. For n = t, (14) was solved completely by Opfer and Schober [7]. From their result, one can deduce (see [3]) that, for the case n = 1, the statement in part b) of T h e o r e m A is true for all R > r >_ 1. Clearly, in view of (13), Theorem 1 is an immediate consequence of Theorem 3, Theorem A, and Remark 3. The proofs of Theorem 3 and 4 will be given in the next section.

On Bernstein Type InequMities ... 4. Proofs

53 of Theorem

3 and

4

P r o o f of Theorem 3. With (17) and (21), it follows that

w(z)V~(z)l _ a, E,,+a/z <_ w(c)Vn(c) e. tTn+a/~(c)i By (20) and (28) (both with k = n), the right-hand side is just the upper bound in (34). We now turn to part b). Using (28) and (29), it follows that I Re(dkd~)J

<_ AkA,~ + BkBn <_ R n+k+1,

I Im(dkd~)l

< A k B . + BkAn <_ R n+k+l,

(36) and

(37)

Idol 2 > A~ - 1 > 4Re~+l(1 - 2/R2~+1).

Let a~ be given by (33). With (36), (37), and (20), we obtain the lower bound (38)

n-1

I(R2~Rv~(1 a[ > ~ --~-z r - ~

2 4R~+,v~ y~ ( R ) k r 4k+1 R 2'~+1)r 4k+2 1" k=O ' r

Now assume that R >_ r(33r - 1)/(r - 1). With

1

2 1 R 2"+~ >-2

and

r 4k+l

r4 k + 2 - 1

_<

F

ra-1

, k=0,1,

. " "

we deduce from (38) that 1 (R 2~ ~ Rv/r r2 °r > g'T' (," + 1 ) ( R - r) ( R - r - 3 2 r - 1 ) -> 0.

In view of Theorem 2, this concludes the proof of Theorem 3.

Proof of Theorem 4. First we consider the case (i), i.e. assume that (39)

1 1 1 1 c = ~(R + ~ ) _> r + -r - -'2

Then ¢ = 0 in (28), and the representation (33) reduces to a? = An 2 an +

A~-k cos(k~2t)), k=l

l = -n,...,n.

an-k

It follows that a~ = A,t(qpt) where t is the trigonometric polynomial (25) with (40)

An )~0:=-an

and

~k.-

An-k an-k

,

k=l,...,n.

R. Freund

54

Therefore, Theorem 2 together with Lemma 2 implies that vn is best possible in (15) provided that the numbers (40) satisfy the assumptions of Lemma 2. Hence, it remains to verify that the estimates (41)

A-A>2A° 31 ao

and

Ak+l

2A--3-k+Ak-1 _>0,

ak+l

ak

k=l,...,n-1,

ak-1

hold. It is easily seen that the first condition in (41) is equivalent to (39). A more lengthy, but straightforward, computation shows that (39) also guarantees that the remaining inequalities in (41) are satisfied. We omit the details. For the case (ii), c < - r , one proceeds similarly. Now ¢ = 7r in (28), and from (33) we obtain 1 Bn @, Bn-k a~=Bn(~7+~ cosk(~,+,~)), l = - n , . . . , n . k=l an-k

By applying Lemma 2, this time with (42)

B,.,. ~0:=-and an

~k.-

Bn-k

,

k=l,...,n,

an-k

and Theorem 2, we conclude that v~ is the optimal polynomial for (15) if the assumptions of Lemma 2 are satisfied. A lengthy computation shows that the condition c < - r indeed implies that the numbers (42) fulfill the required inequalities. Again, details are omitted here. • A c k n o w l e d g e m e n t . The author would like to thank Dr. Bernd Fischer for performing some numerical experiments which were very helpful for developing the results of Section 3.

References

[1]

S. Bernstein, Sur une classe de polynomes d'dcart minimum, C. R. Acad. Sci. Paris 190 (1930), 237-240.

[2]

C. Frappier, Q.I. Rahman, On an inequality of S. Bernstein, Can. J. Math. 34 (1982), 932-944.

[3]

B. Fischer, R. Freund, On the constrained Chebyshev approximation problem on ellipses, J. Approx. Theory (to appear).

[4]

R. Freund, St. Ruscheweyh, On a class of Chebyshev approximation problems which arise in connection with a conjugate gradient type method, Numer. Math. 48 (1986), 525-542.

[5]

N.K. Govil, V.K. Jain, G. Labelle, Inequalities for polynomials satisfying p(z) = znp(1/z)., Proc. Amer. Math. Soc. 57 (1976), 238-242.

[6]

G. Meinardus, Approximation of Functions: Theory and Numerical methods, Springer Verlag Berlin, Heidelberg, New York, 1967.

On Bernstein Type Inequalities ...

55

[7] G. Opfer, G. Schober, Richardson's iteration for nonsymmetric matrices, Linear Algebra Appl. 58 (1984), 343-361. [8] G. P61ya, G. SzegS, Aufgaben und Lehrsdtze aus der Analysis, Vol. I., 4th ed., Springer Verlag Berlin, Heidelberg, New York, 1970. [9] Q.I. Rahman, Some inequalities for polynomials, Proc. Amer. Math. Soe. 56 (1976), 225-230. [10] T.J. Rivlin, H.S. Shapiro, A unified approach to certain problems of approximation and minimization i J. Soc. Indust. Appl. Math. 9 (1961), 670-699. [11] W. Rogosinski, G. SzegS, Ober die Abschnitte yon Potenzreihen, die in einem Kreise beschrdnkt bleiben, Math. Z. 28 (1928), 73-94. [12] V.I. Smirnov, N.A. Lebedev, Function~ of a Complex Variable, Iliffe Books, London, 1968. [13] G. SzegS, Orthogonal polynomials, Amer. Math. Soc. Colloq. Publ., Vol. 23. Providence, R. I.: Amer. Math. Soc., 4th ed., 1975. Received: May 15, 1989

Computational Methods and Function Theory

Proceedings, Valparaiso 1989 St. Ruscheweyh, E.B. Saff, L. C. Salinas, R.S. Varga (eds.) Lecture Notes in Mathematics 1435, pp. 57-70 (~) Springer Berlin Heidelberg 1990

Conformal

Mapping

and

Fourier-Jacobi

Approximations D a v i d M. Hough 1 IPS, ETH-Zentrum, CH-8092 Ziirich

A b s t r a c t . Let f denote the conformal map of a domain interior (exterior) to a closed Jordan curve onto the interior (exterior) of the unit circle. In this paper, we explain how the corner singularities of the of the derivative of the boundary correspondence function can be represented by Jacobi weight functions, and study the convergence properties of an associated Fourier-Jacobi method for approximating this derivative. The practical significance of this work is that some of the best known methods for approximating f are based on integral equations for either the boundazy correspondence function or its derivative.

1. I n t r o d u c t i o n In this section the mapping problem is defined and a particular boundary integral representation for the conformal map is introduced. The contents of the rest of the paper are outlined at the end of this section. Let 0/2 := UN=I Fk denote a piecewise analytic Jordan curve in the complex plane whose component analytic arcs 11, F 2 , . . . , FN are defined by

rk := {z: z = Ck(t), -1 _< t _< 1},

k = 1,2,... ,N,

where ~k is analytic on a domain containing [-1, 1] and satisfies ¢~(t)#0,

-1
k=l,2,...,N.

It is assumed that these arcs are numbered consecutively, so that ~ ( 1 ) = ~k+1(-1) with (N+I - 41, that the positive direction on e a c h / ' k keeps int 0/2 on the left and that 0 E int 0/2. Here, int 0/2 and ext 0/2 denote respectively the bounded and unbounded components of the plane whose common boundary is the curve 0/2. Every corner point IOn leave of absence from the Department of Mathematics, Coventry Polytechnic, UK

D.M. Hough

58

on 0(2 m u s t be a m e m b e r of {¢k(1)}N=l but it is not necessary t h a t every m e m b e r of {~k(1)}N=l be a corner point since, for example, Fk and/~k+l could be subarcs of a single analytic arc. Let (2 denote one of the simply-connected domains int 0(2 or ext 0(2 and let f denote the eonformal m a p f : (2 ~ D where D is either the interior or exterior of the unit circle, i.e. {w:lwl l } i f ( 2 = e x t 0 ~ 2 . For the case J2 = int 0(2 it is assumed without loss of generality that f ( 0 ) = 0, and, similarly, for the case (2 _= ext OO that f(c~)

=

oo.

In the latter case, in order that f be one-to-one on (2, f must have a simple pole at c~. T h e orientation of the domain D is conventionally fixed by requiring that if(z) be real and positive either at the point z = 0, in the case 12 = int 0(2, or at the point z = ~ , in the case $2 ~ ext 0(2. The inner radius , b, of int 0~2 and the capacity, c, of 0(2 are then defined by (1)

b :~--~[f'(0)] -1 > 0

, (2 ~ int 052,

c := [f'(cx~)]-' > 0

, (2 -= ext 0(2.

and It is well known that the function f exists uniquely, is analytic almost everywhere in (2 and is continuous on 0(2, its only singularities at finite points in Y2 being branch point singularities at corner points on 0(2 T h e b o u n d a r y correspondence function Ok associated with the arc Fk is defined by

Ok(t) := arg(fo~k(t)),

(2)

where arg m u s t be defined so that each Ok is continuous on [--1, 1]. T h e functions N {Ok}k=1 completely define the m a p f and they, or their derivatives, are the f u n d a m e n t a l quantitites which are to be approximated in a number of integral equation methods for the numerical determination of f; see Henrici [3, §16.6-7]. If z = Ck(t) E 0(2 then it follows immediately from (2) that

folk(t) = exp(iOk(t) ),

(3)

whilst if z E Y2 then there are various boundary integral formulations t h a t m a y be used for the calculation of f(z). T h e b o u n d a r y integral formulation considered here is

(4)

f(z) =

z exp(iw - K(z)) c-' e x p ( K ( z ) )

if (2 = int 0~2, if (2 - ext 0(2,

Conformal Mapping and Fourier-gacobi Approximations

59

where w is a real constant, K : Y2 -+ C is defined by

(5)

N

1

K(z) := ~ j l uk(t) log(z - (k(t)) dt k=l

1

and uk : [ - 1 , 1] -+ R is defined by (6)

:=

o'k(t).'

see [2], [3, §16.61 and [51 for further discussion of representations similar to (4). The domain of definition of K may be extended to ~ provided the correct branch is taken for the logarithm appearing in (5). That is, if z ~ Fk then, as a function of the parameter t, log(z -- ~k(t)) must be continuous on [-1,11 whilst if z = £k(r) E -Fk then

(7)

log((k(r) - (k(t)) :=

lim log(z - ~k(t)) z-¢,(,).,~a 0

if r ¢ t,

if r = t. D

Hence, the formula (4) may be used to calculate f(z) for all z E 9 . Note that, in view of (1), w satisfies (8)

log b + iw = K(O)

and that, since If(z)l = 1 for all z C 0 9 , it follows from (4) that (9)

c = exp(Re{K(~)})

for any chosen point ~ E 0 9 . Also note that an immediate consequence of the definition (6) is that the functions {uk}k=t N satisfy (10)

~ l j F l uk(t)dt = l

In the next section it is explained how the presence of corners at the ends of the arc /'k induces end point singularities in the corresponding density function uk. In particular it is shown that these singularities are represented by a Jacobi weight function

wk(t) = (1 - t ) ~ k ( l + t ) a" , where ak , /3k are related to the interior angles of 9 at 4k(4-1). Also in §2 the FourierJacobi polynomial approximations to vk/wk are introduced and it is shown that these approximations converge almost uniformly on [-1, 1]. In §3 it is proved that the Fourier-Jacobi partial sums produce a uniformly convergent sequence of approximations to the map f . In §4 a brief outline of the collocation m e t h o d for the approximate solution of Symm-type integral equations is given and it is shown how this may be viewed as a method for estimating the Fourier-Jacobi partial sums.

D.M.

60 2. S i n g u l a r i t i e s

of

N {/]k)k=l

Hough

and Fourier-Jacobi polynomial

approximations

In order to avoid the unnecessary use of subscripts in this section, we let F, (, 0 and t, denote respectively a typical arc from the set {Fk}N=I and the associated functions (k, Ok and vk. From (3) and (6), u can be expressed directly in terms of f as

i(f°C)'(t) 2~r/o¢(t) "

v(t)=

Since f o ~ is analytic and non-zero on the open interval ( - 1 , 1) it follows that y is also analytic on ( - 1 , 1). However, since the arc end points ~(=kl) are usually corner points of 0Q, fo~ and hence v are not usually analytic at ±1. Using the results of Lehman [8], who derives the asymptotic expansion of the map f in the vicinity of a corner formed by analytic arcs, we have previously established the corresponding asymptotic expansion for the density v in the vicinity of the end points ±1; see Hough and Papamichael [6]. In order to describe this expansion, let Ar and #Tr denote the angles interior to ~2 at the points ¢(1) and ¢ ( - 1 ) respectively and define a:=-l+A

-1,

j3 := - 1 + #-1.

It is assumed always that {A, #} E (0, 2) so that

Then, from the results given in [6] it follows that there exists a number 6 with 0 < 6 < 1 such that (11)

u(t)= { (1-t)~(aCx(1-t)+x-(1-t)) (l+t)Z(bCz(l+t)+x+(l+t))

ifl-6
where a, b are constants, X+ are differentiable on [0, 6] and satisfy X:L(0) > 0 aad ¢.y is defined explicitly on [0, 6] for any 7 6 (0, 2) by

¢,(t):=

{

t 1/~ tlogt 0

if 7 > 1 , ifT=l, if7
and satisfies ¢.t(0) = 0. Clearly, ¢~ also satisfies a H61der condition of order g(7) on [0, 6] where (12)

~(7) :=

7 -1 ifT>l, 1 - e for arbitrarily small e > 0 if 7 = 1 , 1 ifT
so that ½ < 6(7 ) _< 1. From (11) it is clear that on each arc F it is always possible to express u as the product

61

Conformal Mapping and Fourier-Jacobi Approximations

(13)

u=w¢

where (14)

w ( t ) := (1

-

t) (1 +

is the classical Jacobi weight function and ¢ is a smoother function than u. The decomposition (13) is useful in that in order to construct approximations to u, which, from (11), lies in L2[-1, 1], it is only necessary to approximate ¢ which is Hhlder continuous, as follows. P r o p o s i t i o n 2.1. W i t h notation as above, for each arc F the quotient function ¢ := u / w satisfies ¢ E H~*[-1,1] with a* E (1,1] defined by { rain(A-a,# -I ) if # > 1 or A > 1 , c~* : =

irA 0 otherwise.

1

1--e

Proof. From (11), (13) and (14) it is clear that if t • [1 - 5, 1] then ¢ ( t ) = (1 +

- t) + x - ( 1 -

0).

Sums and products of Hhlder continuous funtions are also Hhtder continuous. Both (1 + t) -~ and X- are in H111 - 5,1] and hence, since min(g(A), 1) = g(),), it follows that ¢ EHI(~)[1 - 6, t]. Similarly, ¢ EHe(")[-1,-1 + 6] and ¢ EH:[-1 + 6, 1 - 5]. Hence ¢ E H~'[-1, 1] where a* := rain(g(#), 1,g(A)). From (12), this definition of a* is equivalent to that stated in the proposition. • Naturally, if A < 1 and # < 1 then it can be deduced from the expansion in [6] that a certain order of derivative of ¢ is also Hhlder continuous on [-1, 1], the order and corresponding Hhlder index being determined by A and #. The present work is concerned with the possibility of constructing polynomial approximations to ¢. Proposition 2.1 guarantees that ¢ is smooth enough for there to exist a sequence of polynomial approximations that converges uniformly to ¢ on [-1, 1]. The presence of the Jacobi weight function in the decomposition (13) naturally suggests that for numerical work it may be convenient to use Jacobi polynomials as basis functions in any polynomial approximation to ¢, and this certainly turns out to be the case for the numerical method outlined in §4. Moreover, it is also natural to consider the properties of the Fourier-Jacobi polynomial approximation of degree n defined by (15)

& := ~ (¢,Pm)Pm, m=O

where pm is the orthonormal Jacobi polynomial of degree m associated with the weight (14) and the inner product is defined by

(1 t

(¢,÷/:: J--I f W¢~" dr.

D.M. Hough

62

The equiconvergence theorem of Szeg5 [14, Theorem 9.1.2] may be used to prove oo converges almost uniformly to ¢ on [-1, 1], as follows. that the sequence { S -},=1 P r o p o s i t i o n 2.2. The Fourier-Jacobi approximation S, defined by (15) satisfies lira

[¢(t) -

&(t)l

= o

for all t in ( - 1 , 1) and the convergence is uniform on any interval [-1 + 6, 1 - 6], where 6 is any fixed number in (0, 1). Proof. Define indices ~+ > 0 and/3+ > 0 by ~+

• --

2a + 1 4

/3+

,

. -

2/3 + I 4 '

and define # : [-1, 1] -~ R by ¢~(t) : = (1 - t)~+(1 + t ) O + ¢ ( t ) .

Clearly, from Proposition 2.1, • is H51der continuous on [- 1, 1]. Let s, be the polynomial of degree n formed by the nth partial sum of the Fourier-Chebyshev approximation to on [-1, 1]; i.e., s~ocos is the nth partial sum of the classical Fourier cosine series expansion of #ocos on [-Tr, ~r]. For any t E ( - 1 , 1), ¢(t) - S~(t) =

(1 - t)-~+(1 + t)-0+(4~(t) - s~(t))

+ (i

+

-

-

&(O.

Therefore, if 6 is any fixed number in (0,1) it follows that (17)

max tE[--1+5,1--61

I¢(t) - S~(t)I < 6-°+{ ~' --

+

max

te[--1+6,1--5]

max

t~[-l+&l-6]

l ¢ ( t ) - s.(t)[

t(1

&(t)l

•

But # is Hblder continuous on [-1, 1] and hence the Dini- Lipschitz criterion ensures that s, converges uniformly to # on [-1, 1]; see., e.g., Rivlin [11, §3.4]. Also, ¢ satisfies the conditions of Szeg6's equiconvergence theorem mentioned above, which states that the second term on the right of (17) tends uniformly to O on [-1 + 6,1 - 6]. Hence, given any e > 0 there exists an integer ni = nl(e, 6) such that max

t¢[-~+a,,-~]

I¢(t) - &(t)l

<

for all n _> nl. This proves the proposition, since uniform convergence on [-1 + 5, t - 6] certainly implies pointwise convergence on (-1,1). •

The fact that Sn is the best approximation of degree n to ¢ in the norm associated with the inner product (16) is sufficient to guarantee certain uniform convergence properties for the corresponding approximation to the conformal map f , as is explained in the next section.

ConformM Mapping and Fourier-Jacol~ Approximations 3. T h e

uniform

convergence

of approximations

63

to the map

f

In this section we revert to the use of subscripts to identify quantities associated with a particular arc Fk. A number of preliminary definitions are required, as follows. Let wk denote the Jacobi weight function associated with the arc Fk and let Ck be the corresponding quotient function ¢k := v~/wk; see (13), (14). Also, let L~[-1, 1] denote the set of complex valued functions g defined on [-1, 1] such that v/-~lg] EL2[-1, 1]. L~ is a Hilbert space with inner product

wkg]~dt

(g, h)k := 1

and corresponding norm

Itgll~ := VJ-~

wktgpdt.

For functions f which are defined on Y) we also intoduce the uniform norms IlflI~ := s u p l f ( z ) l ,

]]fII0a := sup If(z)l.

zED

zEO,Q

In the case where f is analytic on ~2 with a continuous extension to 0~2, the m a x i m u m modulus principle implies that the above two norms are identical. Given any z E Y2 let l, : 0Y) --- C be defined by

(18)

zz(~) :=

{

log(z-~) 0

-

ifze~Q,~e0f2,z#~,

if z = ~ e 0 ~ ,

where the logarithm branch is the same as that described prior to (7), so that log(z - (k(t)) = Izo(k(t). It can be proved that lzo(k eL~ for all finite z E ~2. Now let us suppose that each function Ck is approximated by a real polynomial Pk,s of degree n and that a corresponding approximation fn to f is generated from (4) by (19)

f~(z)

:=

z exp(iws - Ks(z)) Cn1 exp(K~(z))

if ~2 - int 0/2, if ~2 --= ext OQ,

where, using the above inner product notation, N

(20)

Ks(z)

:= ~(~oCk, Pk,T,)~ • k=l

T h e estimate w,~ and approximate inner radius, b~, are obtained from (8) as (21)

log b,~ + iWs = Kn(O)

whilst c,~ is determined using (9) as

(22)

c~ = exp(~{Ks(~)}).

Observe that in the case ~2 --extO~2, Izo(k ~- logz as z -~ c~ so that

D.M. Hough

64 N

K~(z) ~ (~{1,Pk,~)k)log z . k=l

Thus, it follows from (19), that in order for fn(z) to have a simple pole at oo the polynomials Pk,~ must satisfy N

(23)

E
i.e., the approximations {wkPk,,~}N=i to {vk}N=I must satisfy the counterpart of condition

(10). P r o p o s i t i o n 3.1. Let ~ = int0~2 and 1et {Pk,~}~_l be sequences of polynomial approximations that satisfy (24)

l i m lick - Pk,~llk = 0 ,

k = 1,2,...,N.

Then lira IIf

n~oo

-

f~ll~

=

lira n~oo

111 - @ll~ ]

=

0.

Proof. It is clear from the definition (19) that f~ is analytic on £2 with continuous extension to 0~2. Hence, this same property is shared by both the error f - f,~ and the relative error ( f - f n ) / f , since it is readily shown that the singularity in the relative error at the origin is removable. Therefore, using the maximum modulus principle together with the fact that If(z)l = 1 for all z C c9~2 gives Ill - AII~ = II/-

AIIo~ =

Ilx -

II0n = Ill - -711~ ;

i.eo

(25)

llf - Arl~ = II1 - @11~ = II1 - ~ 1 1 0 ~ . Equations (4) and (19) imply that

(26)

1

f~(z) _ 1 - exp(i(w~ - w) - (K,~ - K ) ( z ) ) .

f(z)

Also, from (5), (20) and the use of the Cauchy-Schwarz inequality it follows that for z COD N

I(Kn - K)(z)t

= l~kl k=l

(27)

N

_<

~lll~oCklikltPk,~ -- ¢kllk k=l

< ME~ ,

ConformM Mapping and Fourier-Jacobi Approximations

65

where M := sup k m a x IlI=o¢~11~ zEOD =t,...,~,~

and N

(18) k=l

Similarly, from (8)and (21), (29)

lw~ - ~1 : lira { ( K n - K ) ( 0 ) } I

_ I(K~ - K)(0)I

_< MoE~, where Mo :=

max IlloO~kllk.

k=l,...,N

Now, it is readily established that if ]zl # 0 then

I1 - exp(z)l < Jzlexp(Izl) and [z[ exp(lz[) is a monotonically increasing function of Iz[. Combining this result with (26),(27) and (29) implies that

(30)

111 - @[10a <- (M + Mo)E,~exp((M + Mo)E~),

with equality only if E~ = 0. But from (28) it is clear that if HCk - Pk,.llk ~ 0 for k = 1 , . . . , N as n -~ oo then E,, --* 0 as n --+ c~ which, using (25) and (30), proves the proposition. • The above proposition cannot hold fully for the case f2 = ext0f2 since the error f - f , ~ has a pole at co. However, that part of Proposition 3.1 relating to the convergence of the relative error norm carries over to the exterior domain. oo be sequences of polynomial P r o p o s i t i o n 3.2. Let f2 = ext0f2 and let {p k,~}~=l

approximations that satisfy the convergence conditions (24) together with the condition (23). Then (31)

0.

Proof.. Since the polynomials {Pk,n}N=I satisfy the conditions (23) for each n it follows from (19) that fn(z) ~- c~lz as z --* oo so that the relative error ( f - f,~)/f is analytic at oo with (32)

1

f,~(ee) f(ec)

__

Cn- 1

C -1C -1

D.M. Hough

66

Since f is analytic and non-zero on S?\oe and f . is ane:lytic on ~ \ ~ it is clear that the relative error is analytic on ~ and has a continuous extension to 0 ~ . Hence the maximum modulus principle applies to give I[1 -

= II1 - - 11o .

From this point onwards the details are practically identical to those given in the proof of Proposition 3.1 and need only be outlined very briefly. Corresponding to (26), using (9) and (22), there is

A(z) f(z)

- 1 - exp(Re{(K - K~)(~)} + (K,~ - K ) ( z ) )

and from this it may be deduced that (33)

2ME,exp(2ME,).

II1 -

A number of remarks can be made about the above results, as follows. In the first ptace, it has been observed in various numerical methods for estimating f for the case ~2 - ext0f2 that the error in the estimate for c is always of significantly smaller magnitude than the maximum relative error in the estimate for f itself; see [13], [9] and [6]. The simple result (32) goes some way towards explaining this observation since it states that the relative error in c~ 1 as an estimate for c -1 is in fact the value of an analytic function at an interior point of its domain of analyticity; hence I(c-I Cnl)/C-1 ] "< [l(f - f~)/fl]g. This result is independent of the precise method of calculating f , , provided only that f , is analytic on ~2\c~ and f,~(z) ~ c~lz as z ~ c¢. Secondly, again for the case f2 ~_ ext0f2, it will be clear that if, say, f2* is any bounded domain in ext ag2 then it is possible to prove that f , converges uniformly to f on/2*, provided the conditions of Proposition 3.2 are satisfied. Thirdly, in the above Propositions the polynomial approximation to Ck m a y be allowed to take a different degree, say nk, for each k = 1 , . . . , N. Provided (24) and the counterpart of (23) hold, then the corresponding approximation to f will have the uniform convergence properties stated in Propositions 3.1, 3.2 as n := m i n { n l , . . . , nN} ---+oo. Finally, the Fourier-Jacobi polynomial Sk,, of degree n approximating Ck is --

n

m=O

where pk,.~ is the orthonormal Jaeobi polynomial of degree m associated with the weight wk, k = 1 , . . . , N . It is well known that Sk,,~ is the best polynomial approximation of degree n to ¢k in the norm II-IIk.Hence, the choice Pk,n = Sk,. for k = 1 , . . . , N minimises the quantity E . defined by (28) and therefore also minimises the bounds appearing in (30) and (33). Moreover, since ¢k eL~ and {pk,.}.~=0 is a basis for L~, this choice also satisfies the condition (24).

ConformM Mapping and Fourier-Jacobi Approximations

67

In theory then, the use of the Fourier-Jacobi polynomial approximations to the quotient functions Ck forms a very satisfactory approach to determining approximations to the map f . In practise, it may be a non-trivial task to devise a numerical scheme for estimating the Jaeobi coefficients in (34) and which has proven convergence properties as the scheme is refined.

4. T h e

Symm-Jacobi collocation

method

A possible method for estimating the Fourier-Jacobi approximation (34) is via the numerical solution of Symm's integral equations. These equations arise in a natural way from the representation (4) by imposing the condition that lf(z)l = 1 for all z • 012; i.e.,

(35)

Re{K(z)} = ~ l°glzl if 12 = int 012, if 12 = ext 012, j , z • OY2. t log c

Let R~ denote the real part of the logarithm function l~ defined by (18); i.e., R (C) : = log tz - ¢1.

Then, with notation as in §3, we may write the equation (35) together with the condition (10) as N =

, z • 012,

k----1

(36) N

= 1, k=l

where bz :=

log [z} if I2 - int 012, 0 if f2 - ext 012.

The integral equation and side condition (36) are to be solved for the quotient functions {¢k}N=a and the constant 7. For a piecewise analytic Jordan curve, the analysis of Gaier [2] shows that the unique HSlder continuous solutions of (36) are given by Ck := O'k/(27rwk ) , k = 1 , . . . , N, and the unique value of the scalar 7 is defined by

7:=

0 -logc

if 12 _= int 012, if 12 =- ext 012;

see also Reichel [10]. In numerical conformal mapping, integral equations of the first kind with a logarithmic kernel are generally known as Symm-type equations, because this kind of formulation was first used by Symm [12], [13]. In fact the equations (36) are not the same as those originally considered by Symm. The inclusion of the scalar 7 and the side condition for the interior mapping problem is a device introduced in a more

D.M. Hough

68

general context by Hsiao and MacCamy [7] in order to produce a system which has a unique solution irrespective of the scale of the domain [2. We have seen in Proposition 3.2 that it is essential in the ease/2 - ext 0/2 to impose the side condition on a numerical solution in order to produce the correct mapping behaviour at oo. The equation (36), without the side condition, for the exterior mapping problem is due to Gaier [2] and has the advantage that its solutions {¢k} are generally in smoother HSlder continuity classes than the corresponding solutions to Symm's original formulation [13]; see [6] for some discussion of this. The truncation error in the Fourier-Jacobi approximation of degree n to the kernel function R.o~k of (36) is Ez~,k := R~o~k - ~ (R.o~k,pk,m)kpk,m .

(37)

m=0

Writing Ck = Sk,~+(¢k-Sk,~) in (36) and using (37) together with the fact that Ck--Sk,n is orthogonal with respect to (., ')k to all polynomials of degree at most n it follows that N

~ ( R z o ~ k , Sk,,~)k+7+ez~ =b~

, zE0/2,

k=l

(as) N

~2<1,s~,~>~

= 1.

k=l

Here, the satisfaction of the side condition follows from (34) by using the orthonormality of the Jacobi basis polynomials and e~ is defined by N

~n := ~ (

E :,~,¢~ n -

&,~)~.

k=l

Applying the Cauchy-Schwarz inequality to the previous definition gives N

~ I I E L I I ~ I I ¢ ~ - &.ll~ •

141-

k=l

B u t IIE2,~l[k--+ 0 a n d lick - & , ~ l l ~ - - + 0 for each k as n - - + ~

so t h a t le21--+0 as n--+ ~ .

This suggests that a simple approach towards estimating the polynomials { S k , n } kN = l is therefore to reduce (38) to a linear algebraic system by collocating at an appropriate number of points on 0/2 and ignoring the contribution of e~ at these points, as follows. Let t " ~i l ilNn+N =ll be a set of distinct points on 0/?, with n + 1 points located on each are Fk, k = 1 , . . . , N. Then the polynomials of degree n, say Pk,n, which estimate the Fourier-Jaeobi partial sums Sk,,~, k = 1 , . . . , N, satisfy the equations N

~(Rz,

OCk, P k , ~ ) k + ' y ~

= bz,

k=l

(39) N

=1, k----I

, z~ e 0 / 2 ,

ConformM Mapping and Fourier-Jacobi Approximations

69

where 7n is an estimate for 7. Using the orthonormal Jacobi polynomials as a basis for the representation of Pk,n, i.e., n

(40)

Pk,n = Y~.ak,mPk,m, ram0

in (39) produces a (Nn + N + 1)×(Nn + Y + 1) linear algebraic system for the unknowns ak,m, k = l , . . . , N , m = 0 , . . . , n , and%, It is appropriate to mention here that previous methods which aim to treat the endpoint singularities of uk via the numerical solution of Symm-type equations have been based on the use of piecewise approximations combined with a variety of techniques for dealing with the singularities. For example, in [6] we construct approximations to uk using cubic B-spline basis functions augmented by special singular basis functions with support restricted to subintervals adjacent to the end points on [-1, 1]. Hoidn [4] considers a technique for the systematic non- analytic reparametrisation of l"k in such a way that the modified density is sufficiently smooth to allow the possibility of constructing global spline approximations. Costabel and Stephan [1] analyse the use of linear spline approximations with suitably graded meshes near the ends of [-1, 1]. It must be said that the latter method is currently the most mathematically respectable since a complete convergence analysis is given. Nevertheless, the orthogonal polynomial approach provides a simple and reasonably elegant means of accomodating end point singularities and it is also relatively straightforward to implement. The preliminary numerical implementation described in [5] produced excellent approximations to f on several test problems and only involved the solution of relatively small linear algebraic systems. Although there is no convergence analysis currently available for the Symm-Jacobi collocation method, it can still be used with confidence for practical mapping computations since reliable a posteriori error indicators are available to assess the accuracy of f~. In particular the the error in modulus [1 -[f~(z)l [ can be computed for z C 012 and f,(z) for z C 012 can be compared with the alternative representation obtained via (3). The most important computational difficulty for the practical implementation of the scheme (39), (40) is that of producing accurate and efficient methods for the estimation of the inner products (R~04k,pk,m)k which appear in (37) and which form the matrix elements in the linear algebraic system. Space does not allow us to go into details here, other than to note that the underlying connection between the Jacobi polynomials, the logarithmic kernel and the Jacobi functions of the second kind allows effective use to be made of three term recurrence relations. A full computer implementation of the method, using more accurate quadrature techniques than those used in [5], is currently nearing completion and it is intended that a report describing all computational aspects will be prepared shortly.

References

[1]

M. Costabel, E.P. Stephan, On the convergence of collocation methods for boundary integral equations on polygons, Math. Comp., 49 (1987), 461-478.

D.M. Hough

70

[2] D. Gaier, Integralgleichungen erster Art und konforme Abbildung, Math. Z., 147 (1976), 113-129. [3] P. Henrici, Applied and Computational Complex Analysis III, Wiley, New York, 1986.

[4] H.P. Hoidn, A reparametrisation method to determine conformal maps., In L.N. Trefethen, editor, Numerical Conformal Mapping, 155-161, North-Holland, Amsterdam, 1986.

[5] D.M. Hough, Jacobi polynomial solutions of first kind integral equations for numerical conformal mapping, J. Comput. Appl. Math., 13 (1985), 359-369.

[6] D.M. Hough, N. Papamichael, An integral equation method for the numerical conformal mapping of interior, exterior and doubly-connected domains, Numer. Math., 41 (1983), 287-307.

[7] G. Hsiao, R.C. MacCamy, Solution of boundary value problems by integral equations of the first kind, SIAM Rev., 15 (1973), 687-705.

Is] R.S. Lehman, Development of the mapping function at an analytic corner, Pacific J. Math, 7 (1957), 1437-1449.

[9] N. Papamichael, C.A. Kokkinos, Numerical conformal mapping of exterior domains, Comput. Meths. Appl. Mech. Engrg., 31 (1982), 189-203.

[10] L. Reichel, On polynomial approximation in the complex plane with application to conformal mapping, Math. Comp, 44 (1985), 425-433.

[11] T.J. Rivlin, The Chebyshev Polynomials, Wiley, New York, 1974. [12] G.T. Symm, An integral equation method in conformal mapping, Numer. Math., 9 (1966), 250-258.

[13] G.T. Syrnm, Numerical conformal mapping of exterior domains, Numer. Math., 10 (1967), 437-445. [14] G. Szeg6, Orthogonal Polynomials, American Mathematical Society, New York, 1975. Received: April 6, 1989.

Computational Methods and Function Theory Proceedings, Valpara/so 1989 St. Ruscheweyh, E.B. Saff, L. C. Salinas, R.S. Varga (eds.) Lecture Notes in Mathematics 1435, pp. 71-79 (~) Springer Berlin Heidelberg 1990

Numerical

Solutions of the Schiller Equation 1 J.A. Hummel

Department of Mathematics, University of Maryland College Park, Maryland, USA

1. Introduction Variational methods are powerful tools for studying the properties of classes of univalent fimctions. These methods can be apphed to many specific classes and lead to a Sehiffer Differential Equation which the extremal function must satisfy. When the functions considered are univalent in the unit disk A, the differential equation is usually of the form (1)

_. ~dz 2 R( z ) - - ~

=

Q(w)dw 2

where R and Q are rational functions. This differential equation is an equality between quadratic differentials and its solution often requires the use of properties of these quadratic differentials. Equation (1) is actually a functional differential equation. That is, the equation itself involves parameters which depend on the unknown function which solves the extremal problem. In the past, the equation was often solved by exhibiting all possible solutions of (1) and choosing from these one which actually produced the extreme value. This required that (1) have solutions which could be written in terms of elementary functions which was true if (1) was simple enough, or was a perfect square [2]. More recently, the problems being studied became more complex and integration of either side (or more properly, the square root of either side) of (1) would lead to elliptic or hyperelliptic integrals. With the availability of computers, we may take the attitude that a function defined by such an integral is just as good as one of the elementary functions. We merely have to compute it as needed rather than look it up in a table. We of course lose some of the relations between the elementary functions, so it becomes more difficult to select the proper values of the parameters to produce the extremal function. 1Work on this paper was supported in part by a grant from the National Science Foundation.

J.A. H u m m e l

72

2. An example The problems can be illustrated by considering a specific example [4] which is just complex enough to show some of the difficulties. Let two non-zero points, a and b, be given in the unit disk A. We wish to find a domain D C A with a, b E D and 0 ~ D such t h a t the hyperbolic distance between a and b is m i n i m u m (or equivalently find such a D for which the Green's function g(a, b) is a minimum). We can also invert the problem and search for a function f ( z ) analytic, univalent, and non-zero in the unit disk which m a p s A into A, and such that f(O) = a, f ( r ) = b where 0 < r < 1 and r is a m i n i m u m . Using the variational method derived in [3], this leads to the Schiffer differential equation of the form (1) for the extremal function w = f ( z ) with rz(1 - r 2) R(z)

= (1 - r z ) ( T

- z)'

(2) 1 [ b2bl_ Q (w ) = -~ [ b - w

bb--1 (1-r2)a 1 - bw + ~-~(1 - a w )

( 1 - r 2 ) a 2] -~l ~ : -~) J

where bl = 1/f'(r), al = f'(0), and we assume without loss of generality t h a t a > 0. The unit circle in the z plane is a trajectory of the quadratic differential on the left hand side of (1) since R(z) < 0 on this circle. In general, when we consider a class of functions which are univalent in the unit disk, the unit circle in the z plane will be a trajectory. Thus it is convenient to write the left hand side of (1) with the expression dz2 / z 2 separated out. It further follows from the variational method for this particular example, that w2Q(w) is real and negative when [w[ = 1 and that it must have a zero of order 2 at some point w0 on the unit circle. It therefore follows that Q(w) must be of the form (3)

Q(w) =

c2 (w w(b - w)(a - w)(bw - 1)(aw - 1)'

where [w01 -- t and c is real. Thus equation (1) involves the known p a r a m e t e r s a (real) and b (complex), and the three unknown parameters r (real), c (real), and w0 (lw0[ -- 1), reduced f r o m the five unknown real parameters in r, al, and bl, in (2). T h e reduction results from the properties of Q(w) following from the variational method. This t y p e of reduction is typical of the situation arrived at when the variational m e t h o d is used.

3. Defining the conformal map Equation (1) is separable. Each side can be integrated to obtain a multiple valued function. T h e desired m a p p i n g is implicitely determined by the equality between the two integrals. But, how can the unknown parameters be determined? Let us write dE22 =

dz2 / R(z)-Tg- and then define Y2(z) -- da'2. Apart from the zeros and poles of dY22, ~ ( z ) is a locally univalent function which maps trajectory arcs of d/22 to horizontal lines.

NumericM Solutions of the Schiffer Equation

73

For this particular example, we observe from (2) that the line segment from 0 to r is a trajectory of dJ~ 2. Thus, the interior of the unit circle less this line segment is a Rir~g Domain and if we cut this domain along the conjugate trajectory from r to 1, then f~(z) will be univalent in A1 = A\[0, 1) and will m a p this domain onto the interior of a rectangle R , in the s9 plane. In the same way, we can set df2~ = Q(w)dw ~ and obtain a multivalued m a p p i n g from the w plane into the f2~ plane. However, the function w = f ( z ) which solves the desired problem must satisfy the differential equation and hence must m a p A1 onto a domain in the w plane bounded by trajectory arcs and conjugate trajectory arcs of dO~ which must m a p onto a rectangle R~ in t h e / ? ~ plane, congruent to the rectangle Rz. To solve the problem, we first a t t e m p t to find parameters so that the trajectory structures of dO 2 and d/?~ are of the same type. Usually, the left hand side of (1) will have only a small n u m b e r of possible trajectory structures as in this case where the only possible structure of d J22 is a ring domain. We look for w0 so that df2~ will have this structure also. (Note that since c is real and positive, changes in c will not change the trajectory structure of dJ2~.) When the structure of dr2 2 is more complicated, the problem of determining the parameters so that dO~ has the same structure can be quite difficult, but it usually reduces to finding p a r a m e t e r values so that certain integrals axe real. Further, depending on the class being considered, other conditions will have to be satisfied. Since f ( z ) # 0 in A in our example, one of the trajectories of df2~ must join w0 and 0. T h e other two simple poles of dO~, at a and b, must be joined by a trajectory 7 which is the image under f of the line segment (0, r) in the z plane. Thus, all we need to do is find a value for w0 so that .f.yd J2 is real. Then the trajectory from 0 m u s t go to w0 and the structure of the resulting dO~ must be that of a ring domain. Since tw0t = 1, the value of this integral depends on only a single real parameter, and thus this is an easy problem to solve numerically provided we know 3' (or a curve homotopic to 7 in A less the three singularities of d/?~ at a, b, and 0). This is particularly easy in this case since d f ~ contains the factor ~0(w - w0) 2 and hence

where df2] = c2dw2/[w(b - w)(a - w)(aw - 1)(bw - 1)]. Thus, w0 is determined by the values of the two integrals which do not depend on wo. The unknown p a r a m e t e r c is real and hence cancels out in the determination of w0. In other, more general, cases it would be necessary to solve a system of equations in order to make the structures the same. If dO 2 is complicated, this might have to be done for a n u m b e r of possible structures. The second p a r t of the problem is to find values of the p a r a m e t e r s so t h a t the images of A in the z plane and f ( A ) in the w plane are identical Riemann surfaces in the (2 and ~?~ planes respectively. In our example, this only requires that the moduli of the ring domains be the same because the parameter c can then be adjusted so that the sizes of the rectangles are identical. In the z plane, the width of this rectangle is the integral of dO around tzl = 1, or twice the integral of dO along the real axis from 0 to r. T h e height of the rectangle is the imaginary part of the integral of dY? from r to 1. The modulus #~ is the ratio of these and is a monotone increasing function of r. In

J.A. Hummel

74

the w plane, the width of the rectangle is twice the integral of d12~ along 7. T h e height is the imaginary part of the integral of dJ2~ along a curve 7' from 0 to a which does not intersect 7- Once Wo is determined, these integrals can be cMculated and we can solve for r so that #~ equals the modulus of the ring domain in the w plane. Thus, we can easily solve the superficially bigger problem of finding the minimum for each given homotopy class of 7-

4. Solving

the extremal problem

The final problem is to select the particular solution which solves the extremal problem. In this case, that means to select the proper homotopy class for 7. It can be shown from the theory of quadratic differentials, [5], [6], that this is the class containing the line segment joining a to b (with appropriate changes if a, b, and 0 are collinear). However, even if such information is not available it is often possible to select the proper solution using the available numerical information. In [4] it was shown that only a finite number of homotopy classes are feasible for a given a and b. Thus it is easy to pick out the actual extremal value from these. This type of reasoning would usually be applicable in more complex cases where no theoretical results are available. In order to solve for the w0 which makes the integral (4) purely real for a given homotopy class of 7, we must reduce the integrals to manageable form. We observe that any 7 joining a and b is homologically equivalent in A\{0, a, b) to n171 + n272 where 71 and 72 are the line segments joining 0 to a and 0 to b respectively (with appropriate changes if b > 0) and nl and n2 are odd integers with signs chosen properly to give the same integral. T h a t is, if we compute

Ij-- f

(5)

wdt21,

Jj = f

d~21, j = 1 , 2 ,

fixing a sign for each one, then, given 7, there exist a pair of odd integers nl and n2 such that, setting I = nlI1 + n212 and J = nl J1 + n2J2, f~ dYl = Wol/2I _ Wol/2~J. We will call this pair of integers nl, n2, the signature of the arc V-2.3561945 1, 0, 0 1, 1

0.7853982 1, 0, 2 1,-1

Figure 1

Figure 2

Numerical Solutlons of the Schiffer Equation

75

The equation Imf~ dr2 = 0 is homogeneous in nl and n2. T h a t is, the pairs nl, n2 and knl, kn2 will result in the same w0 satisfying the above equation. Thus, every relatively prime nl and n2 will define a w0. Three interesting questions present themselves: First, does every pair of relatively prime odd integers correspond to an actual homotopy class? Second, can there be more than one homotopy class with the same signature nl, n2? And third, can a pair of odd integers which are not relatively prime be the signature of some 7? These questions were not considered in [4] since we were interested only in the extremal case, which was known to exist. First, let us look at some examples. Figures 1 and 2 show the actual trajectory of (3) which joins Iwl = 1 to 0 for the value of 00 = argw0 shown next to it for the case when a = 0.5 and b -- 0.5i. The corresponding trajectories 3' joining a to b are also shown. The w0 were actually computed by solving (4) for the signatures 1,1 and 1,-1 respectively (these signatures are given on the bottom line next to the figure) even though the fact that lal = Ibl means arg w0 = -37r/4 and 7r/4 respectively and the trajectory joining 0 to w0 is a straight line segment. Since the trajectory of main interest is the one which joins w0 to 0 (being part of the boundary of f ( A ) ) it might seem better to introduce the line segment 70 from -1 to 0 along the negative real axis, add j = 0 to the definitions in (5) and set I = Io+nlI1 +n2I~ and J = Jo + nlJ1 + n~J2. Any trajectory 7 joining Iwl = 1 to 0 must then define a pair of even integers nl and n2 such that Imfz dr2 =Im[wol/2I - wJ2J]. One can then set this equal to zero and solve for w0. This process works, but gives much less precise correspondence between the "signature", 1, nl, n~ of this curve and the trajectory structure. (These three integer signatures are also shown in the figures.) Note that if we slide the end point of the arc joining ]w I = 1 to 0 in Fig. 1 clockwise from 00 to ~r, the signature would become 1,2,2. In fact, upon calculation, the signatures 1,2,2 and 1,0,0 both result in the same ~0. (Observe that this process of sliding the initial point of 70 along the unit circle does not change the signature of the curve joining a to b.) Fig. 2 is for signatures 1,-1 or 1,0,2. The signature 1,2,0 obtained by decreasing 00 to -~r also gives the same 00. 2.2920053 I,-2~ 0 3, 1

-0.7212090

1, 0~-2 1,3

Figure 3

Figure 4

J.A. Hummel

76

Wrap the 7 in Figure 2 once more around the circle, decreasing 00 to - 3 % resulting in a three integer signature 1,4,2. Calculation of 00 for this signature gives 0o = 2.2920053 with the results shown in Figure 3. The trajectories defined by the differential equation are in a different homotopy class, with signatures 1,-2,0 and 3,1. Starting with signature 1,2,0 and increasing 00 to 3~r would give a signature 1,2,4. This defines 00 = -0.7212909 with the results shown in the fourth figure. This too is actually in a new h o m o t o p y class, with signature 1,3, and 1,0,-2 (or 1,2,4). Thus, use of the signature of the curves joining lwl = 1 to 0 seems to cause difficulty. However, the curves joining a to b are much better behaved. T h e o r e m 1. The set of possible homotopy classes o£simple arcs 7 joining a and b in A\{0, a, b} is in one to one correspondence with the set of reIativeIy prime odd integers nl,n2 with nl > O. Proo£ Any 3' can be viewed as starting at the point a and proceeding first to either 0 or b. If to b, we can deform it from a to 0 and then to b. In either case, we assign a positive sign to the first segment of the path. If it continues to b, we assign a plus 1 to the path from 0 to b when the path does not go around O. Thus the line segment from a to b has signature 1,1 as in Figure 1. f"

"II

f"

"1

!

'

'

I°0

!

! ! I.

*b

ob [

\,

I.,,o

I

*a

L

! J

J

Figure 5

L

J

Figure 6

Every simple arc joining a and b in A\{0, a, b} has a signature nl, n2 with nl and n2 odd and n1 > 0. We need only show that each relatively prime odd pair nl, n2 with nl > 0 determines an arc and that there are no arcs for which the signature is not a relatively prime pair. The Dehn-Thurston theorem [1] asserts that the arc 7 can be represented up to homotopy by a special system of arcs in a decomposition of the region into two pairs of pants and a belt as shown symbolically in Fig. 5 where we decompose A\{0, a, b} into three regions by two disjoint Jordan curves containing a and b in their interior and 0 in their exterior. These curves are represented by the dashed lines in Fig. 5. (The second leg of the outer pair of pants is the boundary of the unit disk, and is not shown.) Except for the special case of the homotopy class of the line segment joining a to b (which will be just that), the Dehn-Thurston theorem tells us that the structure in the inner region consists of two arcs joining a and b to the belt and some number, say k - 1, of disjoint arcs with both end points on the belt, separating a from b. The outer

N u m e r i c a l S o l u t i o n s o f t h e Schiffer E q u a t i o n

77

region will contain k disjoint arcs with both end points on the belt, separating 0 from [w[ = 1. The two sides will be connected by 2k disjoint arcs crossing the belt. Figure 5 shows the case of k = 2 with direct connections across the belt, resulting in a signature of 1,-3. For a general k, direct connection across the belt defines a curve with signature 1 , - ( 2 k - 1). T h e only remaining variable is the number of D e h n Twists in the belt. This can be any integer, positive or negative. Fig. 6 shows the result of a Dehn Twist of +1. Each arc crossing the belt has been moved up one position on the left. The arc connecting to the top position on the left goes around the belt to connect to the b o t t o m left position. The signature is nov,, 3,-1. T h a t is, each integer in the signature has been increased by two. Another Dehn Twist would similarly result in a signature of 5,1. It is easy to see that the result of each Dehn Twist is equivalent to adding an arc going completely around the points a and b. Half of this is the new arc around the belt, while the other half is made up of the totality of the rest of the shifts. A twist in the reverse direction would subtract two from each n u m b e r in the signature. If nl becomes negative, we would no longer have the standard form for nl, n2, but the results remain valid. (For some initial configurations, the results of the positive or negative twists will be reversed, but in every case, a twist in one direction will add two while the reverse twist will substract two.) Let nt, n2 be any pair of odd integers with nl > 0. By interchanging the roles of a and b if necessary, we m a y assume without loss of generality that n2 _< nl. Set j = (nl - 1)/2 and substract 2j from b o t h nl and n2 to give the pair 1 , m where m = n2 - n l + 1. T h e n m < 1. Suppose first that rn < 0. Letting k = (1 - m ) / 2 and making direct connections across the belt as described above there exists a curve joining a to b with signature 1, m. We now m a k e a Dehn twist of + j to obtain a configuration with signature n l , n 2 . The question is, does this configuration represent a single arc joining a to b? We represent the junctures of the arcs on the right and left sides of the belt by/?4 and Li respectively, where i is an integer 0 < i < 2k - 1 and the arcs are n u m b e r e d from the b o t t o m to the top. Thus R0 and Rk are the end points of the arcs from the points a and b respectively. In the left hand region an arc connects Li to L 2 k - l - i for each i, and on the right Ri is connected to R2k-i except when i = 0 or k. Now a Dehn shift of + j results in an arc from Ri to R2j+l+i (mod 2k) by Ri -~ Li+j --+ L 2 k - l - i - j --+ R 2 k - l - i - 2 j -~ -R2j+I+i.

But 2j + 1 = nl, so after the twist of j , we have the arc defined by a --~ Ro --+ Rnl ---+R2nl -~ Ran1 --* . . . -'+ Rpnl --+ b

which terminates when p n l - k (mod 2k). This certainly holds when p = k, since nl is odd, but m a y hold for a smaller p. Indeed if (nl, k) = q, then let p = k / q . We see that p a l - k ( m o d 2k) and this will be true for no smaller p. If nl and n2 are relatively prime, then (nl, k) = 1 and the resulting arc joining a to b includes all of the pieces crossing the belt and we have constructed an arc with signature n l , n2. If (nl, n2) = q > 1, then since 2k = nx - n2, (nl, k) = q also, and the curve system will contain an arc joining a to b with signature n l / p , n2/q and one or more closed curves. So, if ( n l , n 2 ) = 1, there exists an arc with signature nl, n2 while if we start with a standard system of arcs

J.A. Hummel

78

connected directly across as described above with signature 1, m (m < 0), then no set of Dehn Twists will result in an arc with a signature nl, n2 for which (nl, n2) > 1. Next, suppose that there exists an arc joining a to b which has signature nl, n2 with nl > 0, nl _> n2 and (hi, n2) > 1. Do a Dehn Twist of - j where 2j = nl - 1. This results in an arc system having signature 1, m with ra _< 1 and containing an arc joining a to b. If m = 1, then a and b are connected directly and there can be no arcs separating a a n d b. T h u s there are no crossings of the belt and Dehn twists will have no effect. T h a t is, there are no h o m o t o p y classes with signature nl, n2, nl > 1. It. remains only to show that the only possible system with signature 1 , m , m > 0, is one of the t y p e described above with direct connections across the belt joining a system of k arcs on the left to the corresponding standard system on the left. T h e first integer in the signature being 1 requires that the connections across the belt be direct or direct plus a twist of 4-1 since otherwise there will be more than one arc around b o t h a and b. A twist of - 1 clearly requires that m > 0. A twist of +1 would give a first integer > 1 also, since there would be 1 from the connection to Li and two more from the arc going around the belt to connect L0 to R2k-1. Thus the only feasible case is the set of direct connections and the theorem is proved. •

5. R e m a r k s E v e r y simple curve joining a and b in A\{0, a, b} defines a signature nl, nz and every signature defines a 00. However, there are only a countable n u m b e r of such signatures. Hence, almost all 00 must give rise to a trajectory structure in which the unit disk is a density domain. How is it that density domains do not seem to give rise to problems in the calculations, particularly since numerical methods are of necessity not exact? The answer is that we are looking for information about the m a p p p i n g and most of this information is a continuous function of the p a r a m e t e r 00. It is i m p o r t a n t to verify t h a t a 00 exists defining the solution to the problem. Then nearby 80 will define an approximation to the solution. The exception to this principle occurs when we consider properties t h a t depend on the homotopy class of 7, such as the length of the arc 7. For example, the curves with signature n, n + 2 with n odd and n, n + 2 relatively prime define angles 80 which converge to the 80 corresponding to the signature 1,1, but the lengths of the 3' tend to infinity. T h e above discussion is necessarily somewhat vague since each problem will be different a n d one is faced with m a n y difficult topological problems, each which m u s t be solved on its own merits. However, it is clear that the Schiffer Differential Equation can usually be solved numerically with the help of standard numerical integration and multidimensional zero finding techniques. As a closing remark, we observe that one of the biggest problems in the numerical solution of the Schiffer Differential Equation is in determining the argument of the functions being integrated. For example if we integrate R(w)l/2dw along some path, it is essential t h a t the argument of the integrand be continuous. The complex square root in F O R T R A N or other computer languages has a j u m p if the variable crosses the negative real axis and this can easily lead to erroneous results. The best way of overcoming

Numerical Solutions of the Schiffer Equation

79

this problem is to do the computation of individual factors separately, choosing forms which do not cross the negative real axis along the path of integration. For example, the argument of the factor (b - w) in (3) will change by at most ~r along a line segment, so, if it is multiplied by a constant as necessary so that at least one point along the path makes (b - w) real and positive, there will be no jump in the argument. The alternative is to adjust the argument at each point to make sure that it is continuous.

References

[1]

A. Fathi, F. Laudenbach, V. Poenaru, et al., Travaux de Thurston sur les surfaces, Asterisque, (1979), 66-67.

[2]

P.R. Garabedian, M. Schiffer, The local maximum theorem for the coefficients of univalent functions, Arch. Rational Mech. Anal., 26 (1967), 1-32.

[3]

J. Hummel, B. Pinchuck, Variations for bounded nonvanishin9 univalent functions, J. Analyse Math., 44 (1984/85), 183-199.

[4]

J. Hummel, B. Pinchuk, A minimal distance problem in conformal mapping, Complex Variables, 9 (1987), 211-220.

[5]

J. Krzy~, An extremal length problem and its applications, Proc. of the NRL Conference on Classical Function Theory (1970), Math. Research Center, Naval Research Laboratory, Washington D.C., 143-155.

[8]

L. Liao, Certain extremal problems concerning module and harmonic measure, J. AnMyse Math., 40 (1981), 1-42.

Received: April 5, 1989.

Computational Methods and Function Theory Proceedings, Valparaiso 1989 St. Ruscheweyh, E.B. Sail', L. C. Salinas, R.S. Varga (eds.) Lecture Notes in Mathematics 1435, pp. 81-87 (~) Springer Berlin Heidelberg 1990

Behavior of the Lagrange Interpolants in the Roots of Unity K.G. Ivanov 1 I n s t i t u t e of Mathematics, Bulgarian A c a d e m y of Science Sofia, 1090, Bulgaria and

E.B. S a ~ I n s t i t u t e for Constructive Mathematics, D e p a r t m e n t of M a t h e m a t i c s University of South Florida, T a m p a Florida 33620, USA

Dedicated to R.S. Varga on the occasion of his sixtieth birthday.

A b s t r a c t . Let A0 be the class of functions f analytic in the open unit disk Izl < 1, continuous on Izl < 1, but not analytic on Izl < 1. We investigate the behavior of the Lagrange polynomial interpolants Ln-1 (f, z) to f in the n - t h roots of unity. In contrast with the properties of the partial sums of the Maclaurin expansion, we show that for any w, with lwl > 1, there exists a g E A0 such that L~-l(g,w) = 0 for all n. We also analyze the size of the coefficients of L~-l(f, z) and the asymptotic behavior of the zeros of the L~-l(f, z).

1. Convergence Let f ( z ) = ~,k=Oakz ~ k be continuous on D1 : = {z E C : lz] < 1}. T h e n the L a g r a n g e interpolant to f at the n - t h roots of unity e ( ~ ) , k = O, 1 , . . . , n 1,e(x) : = e 2~i*, can be w r i t t e n as

1The research of this author was conducted while visiting the University of South Florida. 2The research of this author was supported, in part, by the National Science Foundation under grant DMS-881-4026.

K.G. Ivanov and E.B. Saff

82

L,~_~(f, z) - z~ - 1

(1.1)

k

k

n

e(~)f(e(g)) k=o z - - - e ~k-) -

n

j)k

n-1

E c ( j , n ) zj, j=o

where

(12)

c(j,n):=-

1~

e

f

e

k

,j=0,1,...,n-1.

rt k=0

When f is analytic on D1 (that is, f is analytic on Izl _< l + e for some e > 0), several results concerning Walsh's theory of equiconvergence describe the very close behavior of the sequence of Lagrange interpolants {L,~_~(f, z)} and the sequence of partial sums n--1 { s n - l ( f , z)} of its Taylor series, s n - l ( f , z) := ~k=0 akzk" For example, for such f ' s and for any z C C, the sequences {L~-l(f,z)}~ and {s~_~(f, z)}~° either both converge or both diverge (hence the term equiconvergence). But when f belongs to A0 ....... the set of all functions continuous on D1, analytic on o

the interior D], but not analytic on D~ - - , the behavior of the two sequences may be different. Of course, both {L~_~(f, z)}~and {s~_~(f,z)}~°converge (to f(z)) when Iz[ < 1. When Izl = 1 there are several examples of f E A0 such that {Ln-~(f, z)}~converges but {sn-l(f, z)}~diverges (the first goes back to du Bois-Reymond who constructed a function f C A0 with a divergent Maclaurin series at z = 1, but L~_~(f, 1) = f(1)). Conversely, {L~_~(f, z)}~°may diverge at a point on Izl -- 1 where {S~-l(f, z)}~converges (if f is continuous and of bounded variation on Izl = 1, then S~-l(f) converges uniformly to f, but Ln_~(f, z) can diverge for appropriate f and z, e.g. f(z) = (1 - l o g ( 1 + z)) -1/2 at z -- - 1 ) When Izl > 1, then {.%-1(f, z)}~ necessarily diverges (the terms a,~z'~ do not tend to zero). Surprisingly, it is still possible for {L~_~(f, z ) } ~ t o converge for some z with ]z I > 1, as the corollary of the following theorem shows. T h e o r e m 1. Let A be any subset of N and Iet m C N. The foI1owing are equivMent: (a) There exists an f C Ao such that the first m coet~cients c(j, n),j = n - 1,..., n rn of L,~-l(f, z) are zero for every n E A. (b) There exist distinct points wj, lwjt > 1,j -- 1 , 2 , . . . , m , and g E Ao such that L~-I(g, wj) -- 0 for every j = 1, 2 , . . . , m and every n C A.

Prod. (1.3)

(b). From (12) we have ~_e k=0

f(e 7t

(k)) = 0 ,

s=l,2,...,m,

neA.

,e

Let wj,j -- 1 , . . . ,rn, be any rn different points in where p(z) := 1-Ij~__l(Z- wj). Setting

Izl

> 1 and let g(z) := f(z)p(z),

( x v = 0 Or" 1

and using (1.1) and (1.3) we get that for any n C A and j = 1 , 2 , . . . ,rn

Behavior of the Lagrange Interpolants in the Roots o£ Unity

Ln-l(g, wj) -

83

w')-l'~-ln k:o~-'f( e(n))e k k'---P(e--(~))(n)wj-e(k k)

wj

1 Ef(e(k))e(k) ~_~(e~(k ) - w , )

-

n-~

n

k=0

n

n

(_~,

(k)

n

(1.4) w~ - 1 n - 1 12

j-1 92

f(e '-))e

k=0

~

7"~

w ....

s=l

n

,~-lf ( e k e ks k=0

(b) ::~ ( a ) . Keeping the notation from the first part of the proof we again set f(z) :=

g(z)/p(z). T h e n f C A0 because the wj's are outside D1. From (1.4) we have ~W .... jc(n-s,n)=O,

j=l,2,...,m,

for any n e A. But D e t ( W . . . . j)j~=l,~_-i = H,~ 1, there is a g E Ao such that L n - l ( g , w ) = 0 for every n E N.

Proof, According to Theorem 1 it is enough to find f E A0 for which all leading coefficients of the Lagrange interpolants are zero. For the function

k----1

where # is the Mhbius function (of number theory), we know (see [1],[5]) that

~F(e k=0

=0,

heN.

"~

Hence for f(z) := F(z)/z we have £:~-~f(e(~))e(,~) = 0 for any n, which proves the corollary. In this case, g ( z ) = Z~= 1 ( @ - - •U J -~(k+l)~ k+l ] z k -- W. • R e m a r k 1. We do not know whether there exists a g E A0 such that L,~-l(g, wj) = 0, j = 1,2, for every n E N, where [wj] > 1,Wl ~ w~. R e m a r k 2. Any g satisfying Theorem 1 or Corollary 2 will not be smooth. For example, no function with absolutely convergent Maelaurin series on tz] = 1 can satisfy Corollary 2.

K.G. Ivanov and E.B. Saff

84 2. Coefficients

and

the

Distribution

of Zeros

According to a theorem of Jentzsch [6], for any function f E A0, every point on the boundary of O1 is a limit point of the zeros of {S~-l(f, z)}~. One can say even more - the zeros of a special subsequence {s~_~(f, z)} tend weakly to the uniform distribution on the unit circle {z : [z I = 1} (see Szeg5 [7]). The same behavior can be observed for the zeros of the best polynomial approximants to f E A0 (see [2,3]). It is natural to ask whether the sequence of Lagrange interpolants {L~_~(f, z)}~° also possesses this property. A crucial step in establishing the above mentioned facts is the proof that the leading coefficients of the full sequence of polynomials are not geometrically small. For example, for the partial sums of Taylor series, this means limsup [a~l 1/~ = 1, n ---*oo

which is equivalent to f E A0, provided f E C(D1). One cannot expect the same behavior for the leading coefficients of L~-I (f). Indeed, as the example function f E A0 from the proof of Corollary 2 shows, we may have c(n - 1,n) = 0 for every n. But results similar to Jentzsch's and Szegb's theorems still can be established by utilizing the following statement, which is a special case of Theorem 1 in G r o t h m a n n [4]. Let pm be an algebraic polynomial of exact degree n(m). Define the zero-measure v~ associated with pm as vm(A) := # of zeros of pm in A

for any Borel set A C C, where the zeros are counted with their multiplicity. T h e o r e m A, ([4]) Let A be a sequence of positive integers and assume that the following three conditions hold for the sequence {Pm}mEA Of algebraic polynomiMs: (i) l i m s u p ( s u p - ~ l

m~

~zED1I¢(rFt)

loglpm(z)[) < 0 ; o

(ii) for every compact set M CD1, limoo vm( M )

rnEA

= 0;

(iii) there is a compact set K C_ C \ D 1 with liminf sup ~-~ L,~EK

log Ipm(z)l - log Iz

> O.

Then, in the weak-star topology, um tends to the uniform distribution AdO on the unit circle as m --* oc, m E A.

This leads us to investigating

Behavior of the Lagrange Interpolants in the Roots of Unity (r(f,O) := t i m s u p n--*oo

max

(i-O)n<j
85

Ic(j,n)l ~/~

for 0 E (0, 1], where c ( j , n ) is defined in (1.2). Obviously a(f, 1) = 1 and a is an increasing function of 0 for any f E A0. We shall prove Theorem

3. For any f E Ao, we have a(f, 1/3) = 1.

Proof. For any r E N using (1.2) we get (0 < j < n) r-I Ez=o c(j + In, rn)

1 =

-

rn-1

I*

E

rTt k = 0

-

E

rTt k = 0

- k ( j + In ) ) --

l=O

I rn-1

(2.i)

r-1

( ~~) ) f(~

1 "-1

rrt

rTt

( - t ~~ 'j ) ~

r~

: l=0

( ~ -

~gl) r

(nm))(-~)

rTt m=O

Let us assume that a(f, 1/3) < 1. Fix q, such that a(f, 1/3) < q < 1. Then for any n > no and a n y j, gn_2 < j < n, we have

Ic(j, n)I < q%

(2.2)

Fix l :> no such that 1-~q ~ < ~1 q1, s -- [~], and ]at I > qZ ( f E Ao implies that limsupla~l 1/~ = 1). From the continuity of f and (1.2) we get

n--,oo

i=1

Therefore Ic(l, n)l > ½q~for any n _> nl. Let us fix m E N such t h a t 2 . 3 m- 5 _> ni. Then 1 t

Ic(t, 2- 3 "~. 5)I > ~q.

(2.3)

Now our aim is, using (2.1) and (2.2), to obtain an estimate contradicting From (2.1) we get (2.4)

c(l,2s) = c(/,s) - c(l + s,2s),

(2.5)

c(l, 6n) + c(l + 2n, 6n) + c(l + 4n, 6n) = c(l, 2n),

and (2.6)

c(l + 2~, 6~) + 41 + 5~, 6~) -- c(z + 2., 3~).

From (2.5) and (2.6) we get

(2.3).

K.G. Ivanov and E.B. Saf[

86

c(l, 6n) = c(l, 2n) + {c(I + 5n, 6n) - c(1 + 4n, 6n) - c(l + 2n, 3n)}.

(2.7)

F r o m (2.7) with n -- 3 k - ' s , k -- 1, 2 , . . . , m, and (2.4) we o b t a i n

c(I, 2 . 3 m s ) = c(l, s) - c(l + s, 28) + a

(2.8) where m

a = ~{c(l+

- c(l+ 4 . 3 k - ' s , 2 . 3 k s ) - c(1 + 2 . 3 k - l s , 3 k s ) } .

5.3k-'s,2.3ks)

k=l

It is easy to see t h a t all t e r m s on the right-hand side of (2.8) are of the t y p e c(j, n) with ~n < j < n, n > s > I > no. By applying (2.2) in (2.8) we get

Ic(t,2.3m

)t

< qs +

+ ELi{2q (3ks) + q3k,)

<

+ 3q3V(1 - q) < 5 q V ( 1 - q) <

+

11

This e s t i m a t e contradicts (2.3) and proves the theorem.

•

If one a s s u m e s t h a t f E A0 has an absolutely convergent M a c l a u r i n series on Izl = 1, t h e n lira s u p n _ ~ Ic(n - 1, n)I 1/~ = 1 (cf. [5]). This implies t h a t a ( f , 0) --- 1,0 C (0, 1], for such f ' s . T h e o r e m 3 and the above observation give some evidence to the following. Conjecture.

For any f E Ao and any 0 < 0 < 1, we have a ( f , O) = 1.

Now we can establish T h e o r e m 4. If the above conjecture is true, then for a n y f E A0 there is a subsequence {nj} such that the zero m e a s u r e s ~'w (corresponding to L,~j-I(I)) tend (in the weak-star topoIogy) to the uniform distribution on the unit circle as j ~ oc.

Proof We are going to apply T h e o r e m A with {pro} an a p p r o p r i a t e subsequence L ~ , - l ( f ) of the L a g r a n g e interpolants. T h e subsequence Lnj-l(f) is chosen so t h a t condition (iii) is satisfied. Let r > 1 be fixed. Assume t h a t there exists an e, 0 < e < 1/2, such t h a t limsup

sup

log[L~_~(f,z)l-logr

<-3elogr,

where ~(n) is the precise degree of Ln-~(f, z). T h e n for n > No a n d for every z, [z[ = r, we have 1 ~(n) log I L ~ - , ( f , z)[ - log r < - 2 e log r, t h a t is,

[L,~-l(f,z)[ < r (~-2~)~(~) <_ r (~-2~)~. Hence for a n y j, (1 - e)n _< j < n, we have

Behavior of the Lagrange Interpolants in the Roots of Unity 1 ~

87

L~_l(f,z)z_J_ld z <_r(l_2~)n_j ~_ r_~"

This inequality implies that ¢(f, e) _< r -~, which contradicts the conjecture. Therefore

Hence there exists a subsequence {nj} such that liminf sup

loglL ,-,(f,z)l-logr

)]

_> O.

Consequently, condition (iii) of Theorem A is fulfilled for the sequence {Lw_~(f)}. The other two conditions of Theorem A are easily seen to be satisfied (even for the whole sequence of Lagrange interpolants). Indeed, condition (i) follows from the trivial estimate I] L~-l(f)liD,-< n I[ f lID, (see (1.1)) and g(n) ~ ~n (see Theorem 3 - - we do not need the Conjecture here). Condition (ii) follows from the facts that {L,~_I(f)} approximates f uniformly on any O compact set M CD1 and that f can have only finitely many zeros on M. Hence Theorem 4 follows from Theorem A. •

References [1] G.R. Blakley, I. Borosh, C.K. Chui, A two dimensional mean problem, J. Approx. Theory 22 (1978) 11-26. [2] H.-P. Blatt and E.B. Saff, Behavior of zeros of polynomials of near best approximation, J. Approx. Theory 46 (1986), 323-344. [3] H.-P. Blatt, E.B. Saff, M. Simkani, Jentzsch-Szeg5 type theorems for best approximants, Journ. London Math. Soc. (2) 38 (1988) 307-316. [4] R. Grothmann, On the zeros of sequences of polynomials, J. Approx. Theory (to appear). [5] K.G. Ivanov, T.J. Rivlin, E.B. Saff, The representation of]unctions in terms of their divided differences at Chebyshev node.~ and roots of unity, (to appear). [6] R. Jentzsch, "Untersuchungen zur Theorie Analytischer Funktionen", Inauguraldissertation, Berlin, 1914. [7] G. SzegS, Uber die Nullstellen yon Polynomen, die in einem Kreis gleichm5flig konvergieren, Sitzungsber. Berl. Math. Ges. 21 (1922) 53-64. Received: October 10, 1989

Computational Methods and Function Theory Proceedings, ValparMso 1989 St. Ruscheweyh, E.B. Saff, L. C. Salinas, R.S. Varga (eds.) Lecture Notes in Mathematics 1435, pp. 89-101 (~) Springer Berlin Heidelberg 1990

O r t h o g o n a l P o l y n o m i a l s , Chain S e q u e n c e s , Three-term Recurrence Relations and C o n t i n u e d Fractions Lisa Jacobsen Division of Mathematical Sciences, The University of Trondheim N-7034 Trondheim, Norway

1. Introduction Let H denote the set of all (complex) polynomials, and l e t / : : H --* C be a linear functional. We say that {P,(z)}~=o; P, E H is a sequence of orthogonal polynomials (OPS) with respect to £; iff (i) (ii) (iii)

deg P , = n £:[P,. P,,] = 0 /:[P~] # 0

for n = 0, 1 , 2 , . . . , if n # m, and for n = O, 1 , 2 , . . . .

Such a sequence does not always exist. Indeed, it is simple to see that £: permits an OPS iff

(1.1)

An:=

#o

#1

..

#.

#1

#2

• •

#n+l

""

~0

forn=0,

t,2,...

,

.. .°

where {#k} are the moments #k = [[zk]. On the other hand, if t~ permits an 0 P S {Pn(z)}, then each P,~(z) is unique up to a constant factor. So, in the following we m a y assume that all P . are normalized to be monic; i.e. to have leading coefficients 1. Letting

Lisa Jacobsen

90

(1.2)

.

n = 1,2,3,...

-

where/41_ 1 : = 1, one can also show t h a t if {P~(z)} exists, then it is a unique solution of the three-term recurrence relation (1.3)

Pn(z)~-(Z-Cn)Pn-l(Z)-)%~Pn-2(z);

Xn¢O

forn=l,2,3,...

where P - l ( Z ) := 0, Po(z) := 1 and ~1 is arbitrary. It is often practical to choose -'~1 : = - # 0 . By the so-called F a v a r d ' s theorem, [2, T h e o r e m 4.4, p. 21] one also has t h a t a sequence of p o l y n o m i a l s satisfying some t h r e e - t e r m recurrence relation (1.3) with the said initial values, is an O P S for some linear functional £. For fuller i n f o r m a t i o n on o r t h o g o n a l p o l y n o m i a l s we refer to C h a p t e r 1 in C h i h a r a ' s b o o k [2]. A s s o c i a t e d with a t h r e e - t e r m recurrence relation (1.4)

Xn = bnXn-1 + anXn-2;

an ¢ 0,

an, bn E C

for n = 1 , 2 , 3 , . . .

there is always a continued fraction (1.5)

Kan al a2 a3 bn " - bl + b 2 + b 3 + . . . +

=

al bl+

a2 ; a3 b2 + - b3 + ...

an ~ 0 .

For one thing one has (by induction) that its modified approximants (1.6)

Sn(w) . - al a2 an An ÷ A~_lW. bl ÷ b2 ÷ . . . ÷ bn ÷ w = Bn ÷ B n - l W '

n E N

has coefficients {A,~} and {Bn} which are solutions of (1.4) with initial values A-1 = 1, A0 = 0 a n d B-1 = 0, B0 = 1. {An} and {Bn} are called the canonical n u m e r a t o r s a n d d e n o m i n a t o r s of K(a,Jb, O. Moreover, if {Xn} is a non-trivial solution of (1.4); Xn e C, then {Xn/X,~_I} is well defined in ~; = c u {oc}, and one has (1.7)

Xo . . X-1

.

al . b: - X~

al b: + b2

No

a2

al

X2 . . . .

a2

an

bl + b2 + . . . + bn -

X1

X---L-~

Xn-1

T h e sequence { - X , ~ / X ~ _ l } is called a tail sequence for K(an/b~). It satisfies (1.8)

to = Sn(t~) - An + An_lt~ B~ + Bn-lt~

for all n E N;

t~

=

-Xn/Xn-1,

and (1.9)

tn-1 = Sn(t~)

where s~(w) . - b~an +w

for all n E N.

We say t h a t K ( a J b , ) converges if its sequence of classical a p p r o x i m a n t s S , ( 0 ) = A,/B,~ converges in C. Its value is then f = l i m A , / B n . Clearly K ( a , / b , ) converges to f if a n d only if its n - t h tail

Orthogonal PolynomiMs, Chain Sequences, ...

91

0o

(1.10)

K (a=+k/b,+k)= an+l

an+2

"b~+,+ b~+~+..

k=l

converges to f(") for all n, where {f(")} is a tail-sequence for K(aJb,~) with f(0) = f . For more information on continued fractions, see Chapter 2 in the bQok by Jones and T h r o n [6]. We see immediately that the OPS {P~(z)} in (1.3) gives the canonica~ denominators of the continued fraction (1.11)

A, Z--C

A2 1 --

Z--C

ka 2-

Z--C

A~# O, 3-

...

a so-called J-fraction or Jacobi-fraction. Further, the (quasi-orthogonal) polynomials (1.12)

P*(z,w,~) := P~(z) + w , P ~ _ i ( z ) ;

w, e C

are the canonical denominators of S,(wn). We shall mainly use this connection to derive information on the location of the zeros of P,~(z) and P~(z, w~), a very important issue. In the special case where all An > 0 and all c~ E R, one has proved that all the zeros of each P~(z) are real and simple. To locate an interval containing all zeros of all P~(z), the concept of chain sequence~ has been a useful tool. { ,~}~=1 is said to be a chain sequence if (1.13)

an := ( 1 - g~_~)g~;

0<_g0
0
forn=l,2,3,...

for some such sequence {g~}. {gn} is then called a parameter sequence for {an}. At this conference the question was raised by Paul Nevai of whether chain sequences could be generalized to yield information in the complex case (A~ E C, c,~ E C). In Section 3 we shall see that this is indeed so. The basis for the arguments to be used is the following simple observation: (1.14)

S,~(w) = o0 Ca { B,~ + Bn-lw = 0 or w = c~ and Bn-1 = 0.

(Proof. A= and B , cannot both be zero since by induction on n one has (1.15)

A,B,_I - B,A~-I = ( - 1 ) ~-1 f i aj # 0. j=l

Further, As + A~_aw and B~ + B=_lw cannot both be zero since (1.16)

(An + A~-aw)B,_I - (B~ + B~_~w)A~_a = A=B=-I - B,A,_~ # O.i)

This observation (1.14) means that locating zero-free sets for Pn(z) (or P~(z, w~); w, # c~) is equivalent to finding sets where S~(0) (or S~(w,)) is finite. This can again be done by means of element sets and various types of value sets for continued fractions. These are explained in Section 2. Finally, the close connection between tail sequences for continued fractions and parameter sequences for chain sequences is discussed in Section 4.

Lisa Jacobsen

92

2. E l e m e n t a n d v a l u e s e t s for c o n t i n u e d f r a c t i o n s Following [6] we say that { f 2 n } ~ is a sequence of element sets for a continued fraction K(an/b=) iff (an, bn) • f2, C C 2 for all n. Even though each f2n does not have to contain more than the one point (an, b~), it is often of advantage to consider larger sets ~2,~. Further, we say that {V,n}n=0 ~ is a sequence of pre value sets corresponding to { n,~} iff O # Vn c C and

(2.1)

an

G Vn-1; (an, bn) • ~n =* - -+ bn v.

Vn • N.

The importance of these sets lies in the fact that if (ak, bk) 6 ~2k for all k and w~ E Vn, then (2.2)

a2 an Am + An-xwn bl + b2 + . . . + bn + w-----~- Bn + Bn-lW,~ E V0.

Sn(Wn) .-- a l

So, if Wn 7~ OO and oc ~ Vo, then B, +B~-lwn 7~ O. If in particular 0 E Vn, or more generally (2.3)

(a~, bn) E Y2n ~ an/bn E Vn-l,

then A,JB,~ E Vo, so that ~ ti~ V0 =~ Bn # 0. If (2.3) holds for all n, we say that {P~} is a sequence of value sets corresponding to {~n}. Still another type of value sets is the following: { n}n=0 iS called a sequence of value sets for tails corresponding to {Y2n) iff (2.4)

(a~,b~) E T2~ => -bn +

a~ C Wn; V n E N . Wn-1

This means that if {tn} is a tail sequence for K(aJbn) with tn--1 E 14z~-1, then tn E W,~ and thus tk E Wk for all k >_ n. By (1.7) it follows that { - B , / B n - 1 } is a tail sequence for K(aJbn). Since by (1.15) B~ and Bn-1 cannot both be zero, it follows that Bn-1 # 0 if -B~/B~_~ E Wn and ~ ¢f Wn. Likewise, Bn # 0 if - B ~ / B ~ - I E Wn and 0 ¢~ W,, Finding pre value sets {Vn} and finding value sets for tails {Wn} corresponding to a sequence {~2n} of element sets, is really the same thing. Indeed, {Vn}; Vn # 0, C for all n, is a sequence of pre value sets for {f2~} if and only if {C\Vn} is a sequence of value sets for tails for {f2n}. (Proof. s~(w) := a~/(b~ + w) is a non-singular linear fractional transformation, and thus a bijective mapping of C onto C. Hence sn(l~) C_C_Vn-1 ~ Vn C

$[l(Vn-l) ~=}C \ V n ~ 8nl(C\Vn-l).') Another matter is that starting with element sets {~?n} and trying to find corresponding pre value sets {Vn} is tough going. The "right" thing to do is to cheat, and start with {Vn} or {W~) and then find a sequence {Qn} of element sets to which they correspond. Since we are actually mapping Vn or Wn by means of linear fractional transformations Sn or s~-1, it is natural to let Vn of Wn be halfplanes, interiors of circles or exterior of circles. Let us look at some well known examples.

93

O r t h o g o n M PolynomiMs, Chain Sequences, . . .

Example 2.1. Let Vn:= {z E C; [zi <_ ( 1 - g,)}

for n = 0 , 1 , 2 , . . .

where 0 _< go < 1, 0 < g. < 1 for n E N. Then {V~} is a sequence of value regions for {Y2,} where I2, := E , x { 1} and

En := {Z e C; [Z] <~ (1 --gn-1)gn}. This example originally dates back to Worpitzky 1865, [13], but the present form is due to Wall [12, Theorem 11.1, p. 45-46]. Clearly, the radii of E~ form a chain sequence. Since all/2,~ have the form E , x {1}, {Y2,} is a sequence of element sets for continued fractions K ( a , / 1 ) ; as E E,. Every continued fraction K(a,,/bn) with all b, # 0 can be brought to the form K ( d , / 1 ) by the equivalence transformation (2.5)

K(a,~/b,~) "~ K(dn/1) ¢* dn = a,~/b,,bn-1, bo := 1.

The connection between K(a,~/b,~) and K ( d J 1 ) is then that (2.6)

Sn(W) .-- al a2 a,~ _ dl d2 d,~ bl + b2 + . . . + bn + w 1 + 1 + . . . + 1 + w/b,~ - " T,~(w/b,~),

so that S n ( w ) -- ¢c ¢:¢ T~(w/b,~) = ~ . The next example is part of the most general version for the celebrated parabola theorem. It is due to Jones and Thron, [5],[6, Theorem 4.4, p. 68]. (The first parabola theorem dates back to Scott and Wall [10], 1940).

Example 2.2. Let (2.7)

Vn:={zEC;

Re(ze-'¢")_>-pn}; p~:>0, ¢,~ER

for n -- 0,1, 2, . . .

Then {V~} is a sequence of value regions corresponding to (2.8)

~2, := {(an, b,) E C2; la.I - Re(a,e -'(¢"+~"-~)) _< 2pn_l(Re(b,~e -'¢") - p,)}

for n = 1, 2, 3, . . . . It is called a parabola theorem since for fixed b,, (as, b,) E l?, if and only if a , is contained in some region bounded by a parabola. A more familar form of the parabola theorem is obtained if we choose M1 ¢ , = ¢, and require all b, to be 1: Let ¢ be a fixed real number such that -zr/2 < ¢ < ~r/2, and let (2.9)

V,~:={zEC;

Re(ze-'¢)>-(1-g~)cos¢}

for n -- 0,1, 2, . . .

where 0 _< go < 1 and 0 < gn < 1 for n E N. Then {V~} is a sequence of value regions corresponding to {12~} given by Y2~ :-- En x {1} where (2.10)

E. := {z e ¢; Izl- Re(ze

< 2(1 -g,~-,)gn cos 2 ¢}.

Here, p, in (2.8) is replaced by (1 - g,)cos ¢ to make it easier to recognize the chain sequence (1 - g,~-l)gn in the definition of E,.

Lisa Jacobsen

94 Note that the half planes V~ do not contain e¢.

Example 2.3. (2.11)

Let

V~:={zEC;

Iz-F~[
/'~cC,

p~>0

for n = 0,1, 2, . . . .

T h e n { Vn } is a sequence of pre value regions corresponding to (2.12)

~,~ := {(an, bn) C C2; ta,~(b~-~-~) - F,~_~([b,~ +/',~[~ - P~)I

+ [a,~lpn <_ p,~-l([bn + Fnl 2 - p~)} for n = 1, 2, 3 , . . . . This was essentially proved by Lane, [7], [6, Theorem 4.3, p. 67], although he was interested only in cases where 0 E V,. This result has in later years been referred to as (part of) the oval theorem, since for fixed b,, (a,, bn) E $2, if and only if an is contained in a domain bounded by a cartesian oval. If we require that all b, = 1, the element sets Y2, := E , x {1} are given by

(2.13)

En

:=

if Fn_~

0,

{z ~ c;

Izl _< p._~(ll + F~t - p . ) }

{an¢ c c;

Pn Pn-1 l if F~-i # 0, i¢ - 11 + rtln -+- - - ~ I¢I < Irn-~i"

=

where (2.14)

p~

an=Fn-,(l+F~)(1-il+F~]2)

forn=l,2,3,....

However, in order that E~ # $ we need the conditions (2.15)

[l+_Fn]>p~

and

[fn_l[p,~<]l+f,~[p,~_~

forn=l,2,3,....

For information on cartesian ovals, see [4]. Let it only be mentioned here that one easily sees that cgEn is a cartesian oval with axis along the ray a r g z = argan, loci at z = 0 and z = an, and that ~,~ E E~. For m a n y more examples, see Chapter 4 in Jones' and Thron's book [6].

3. Zerofree regions for orthogonal polynomials Let us start with a simple case. Let all An > 0 and all c~ E R in (1.3) so that all zeros of P,~(z) are real. Via the equivalence transformation (2.5), the J-fraction (1.11) assumes the form (3.1)

K(dn(z)/1)

where

d~(z) :=-

z - c~'

dn(z) :=

(z - Cn_l)(Z - ca)

for n ~ 2, as long as z ~ Cn for all n. From Example 2.1 and (2.6) it follows then that all Pn(Z) ~ 0 if Z ~ Cn for all n and ]d~(z)[ < (1 - g,-x)g~ for all n. It is simple to prove that if 0 < Ida(z)[ __< (1 - gn-1)g~, where {(1 - gn-1)g,} is a chain sequence, then also

Orthogonal Polynomials, Chain Sequences, ...

95

{Ida(z)[} is a chain sequence for this particular z. (See for instance [2, Theorem 5.7,p. 97].) So:

O b s e r v a t i o n 3.1. All P,~(z) # 0/or all z E C such that {Ida(z)[} is a chain sequence. This is of course already well known, [2, Theorem 2.1, p. 108]. Note however that Observation 3.1 also holds if An and cn are complex numbers. As a very simple corollary we find for instance:

Icnl

C o r o l l a r y 3.2. If M and N are two positive constants such that [An[ <_ M 2 and < N for all n, then all zeros of all Pn(z) lie in the disk Iz[ < 2 M + N .

Proof. If [z[ >_ 2M + N, then [d~(z)[ < M 2 / ( ] z [ - N ) 2 < 1/4. The result then follows co 1 since {OLn}~=~ with all an := 1/4 is a chain sequence; 1/4 = (1 - 1 / 2 ) . 3" • This is also well known, compare Wall [12, Theorem 26.6, p. 112]. Note also that since Pn(z)/I'I~=l(z - ck) are the canonical denominators of K(d~(z)/1), we also have that Pn(z) d- (z - Cn)Wn(z)P~-I (Z) 7~ 0 for n ----1, 2, 3 , . . . for [z[ _> 2 M + N as long as lwn(z)[ < 1/2. Let us now turn to the parabola theorem (2.9)-(2.10) from Example 2.2. For convenience we exend the definition of chain sequences as follows: {an} chain sequence (3.2)

=ez {¢)n} chain sequence. 0<6~_
Then we have immediately: O b s e r v a t i o n 3.3. A11 Pn(z) # 0 for all z e C for which {6n(z, ¢)}~=1 given by (3.3)

6n(z, ¢) := Ida(z)[ - Re(dn(z)e-2{¢) 2 cos 2 ¢

is a chain sequence for some ¢ 6 < -7r/2, 7r/2 >. Naturally ¢ may vary with z. If we fix ¢ to be 0, then this observation still represents a generalization of Observation 3.1. In some cases the condition is easier to check if we write 6n(z, ¢) in the form

(3.4)

[!m(~e-'¢).] 6n(z,¢) -- [ cos ¢

~ ] "

T h a t these two expressions for 6n(z, ~) really are identical is easily checked. Also from Observation 3.3 one can derive corollaries, such as for instance: C o r o l l a r y 3.4. Let all cn = 0 and all An be real, and let M be a positive constant such that An < M 2 for all n. Then all zeros of all Pn(z) are contained in the strip IRez[ < 2M.

96

Lisa Jacobsen

Proof. From Observation 3.3 and (3.4) it follows that all P , ( z ) # 0 for each z which satisfies

(3.5)

I m ~

_< cos2¢

E

C

for all n

for some ¢ E< -~r/2, r / 2 >. By choosing ¢ = - arg z if] arg z[ < ;r/2, and ¢ = 7r-arg z or ¢ = -Tr + argz if ;r/2 < I argzl < to get ¢ e < - ~ / 2 , ~/2 >, (3.5) reduces to Im - ~ / ] z [

_< [Rez[/2lz[

for alln,

which holds if IRez[ > 2M.

•

Again we can also get results for quasi-orthogonal polynomials since P~(z) + zw~(z)Pn_l(z) # 0

for n = 1 , 2 , 3 , . . .

for I R e ; I _> 2M as l o n g as Re(w~(z)e -'¢) > - ½ c o s ¢ . Because of the form of the J-fraction (1.11) it is usually simpler to apply the parabola theorem (2.7)-(2.8): O b s e r v a t i o n 3.5. A11 P~(z) ~ 0 t'or all z E C for which {~/n(z, {¢k})}~=1 given by

I:,.I + (3.6)

Re(~ne-'(¢°+'~°-l))

7n(z, {¢k}) := 2Re[(z - c,~)e-i¢-] • Re[(z - c,~_1)e-'¢--1]

is a chain sequence for some real sequence {~,}.

C o r o l l a r y 3.6. Let A,~ and cn be reai for all n, and let M1, M2 and N be three positive constants such that - M ~ <_ A~ <_ M~

and

- N <_ c~ <_ N

for aJl n.

Then a11 zeros of MI P,~(z) are contained in the domain D consisting of all z = x + iy E C such that 2 m a x { M 2 ( l x [ - N),Mxly[} > ( [ x [ - N): + y 2 i f l x [ - N ~_O,

(3.7)

/

i l Ixl - N < O.

IYl < 2M,

2M,

o

........

Figure 1

N + 2M2-

97

OrthogonM Polynomieds, Chain Sequences, . . .

This domain D is symmetric about the real and the imaginary axis, it is bounded and (depending on the constants M1, M2 and N) may took something like Fig. 1. P r o o f o f Corollary 3.6. Let z = x + iy E C \ D be fixed, and choose

¢,~ = ~b =

arg(z-N) arg(z + N ) ~sgn(y)

ifx>N, if x _< - N , if - N < x < N.

Assume first that x _> N. Then cos ¢ > 0, so that ~.(z, {¢})

=

JAn[ + A. cos 2¢

2[Iz-NI +(N-c.)eosV][Iz-

NI + ( N -

c~_l)cos¢]

< 1~.[ + 1. cos 2¢ 2[z_NI2 -

Hence all 7,(z, {~b}) < 1/4, and thus {7,(z, {¢})} is a chain sequence if M~cos 2 ¢ < 1 fz -

Np

•

This clearly holds since e '¢ =

-

4

and

M~sin 2¢ < 1 Iz -

z-N x -N+iy Iz - N i Iz - ~ l

N[~

-

4

- cos ¢ + i sin ¢.

The proof goes similarly if x <: - N . If - N < x < N, then cos ¢ = 0 so that ~'~(z, {¢}) - IA~[ -A____________<~ 1 2y 2 - 4

if

M~ < 1 y2 - 4

Let us finally look at the cartesian ovals of Example 2.3. If we choose all F,, = 0 in (2.13), then we are back to the Worpitzky case of Example 2.1. If we choose all F,, = F and all p,~ = IF + 1/2[ in (2.13), and let F ~ ee in such a way that arg(F(1 + F)) = 2¢ is kept constant, then we are back to a version of the parabola case in Example 2.2. (See [4, Theorems 8.1C, 8.2C, p. 120-122].) So, the oval theorem is closely connected to these two previous cases. O b s e r v a t i o n 3.7. L e t at1 c~ = 0 and tet all A~ be contained in the cartesian ovM

(3.S)

E := { t ( e C; I¢ - 1t + ~t¢] < t}

for given n u m b e r s A E C, 0 < s < 1 and t >_ 1. T h e n all zeros o f aH P~(z) are contained in the d o m a i n

(3.9)

D:={zE

C; 1 < ~ + ~ / 1 - 4 A / ( z ~ ( 1 - s 2 ) ) t --~/1 4A/(z2(1 s2)) < ~}"

Proof. Let z E C \ D . Then

Lisa Jacobsen

98

F:=2(~/1-4A/(z~(1-s2))-l),

whereRe .~..>0

satisfies

z+rI_>t

and

->1

F ( z + F)(1 - s 2) = - A .

s

For ~ = ~¢~ E E it therefore follows that

/9 2

-A~=r(z+r)(1

iz+rl2)¢~

where

p=~lz+rl

and

I¢~ - 11 + ,leaf = t¢~ - 11 +

p ~ + r l _
so t h a t

la,,(, + r ) - a(z + r)l + lanlp < p(Iz + r l 2 - p~)

and the result follows from (2.11)-(2.12). R e m a r k s 3.8. 1. The method still works if 0 < s < t < 1, but then 0 ~ Vn, where V= is given by (2.11). This means that we get zero-free regions for P,~(z, w ( z ) ) where

tw(z) -

r(z)T _< p(z) = ~lz + r(z)l.

2. If we allow c, # 0 we can still apply the same idea. Clearly, (-)~n, z - cn) satisfies (2.12) with Y = F ( z - c,), p = p(z - c,~) as above, if z - c, e C \ D and AN E E. Hence, if .~, E E and all c= E G for some complex set G, then no P,~(z) has any zeros in

H:

~

cEG

{z; z - c c C \ D } .

Orthogonal Polynomieds, Chain Sequences, ...

99

4. Tail s e q u e n c e s a n d c h a i n s e q u e n c e s Let {an } be the chain sequence given by (1.13) with parameter sequence {g~}. That is, a s = (1 - gn-1)gn for all n. Further, let ts := - ( 1 - g~) for n = 0, 1, 2 , . . . . Then - a s = tn-l(1 + t~) for all n, so that {in} is a tail sequence for the continued fraction K ( - a j 1 ) . Hence, {as} is a chain sequence if and only if K ( - a J 1 ) has a tail sequence {ts} with (4.1)

-1
and

-l
forn=l,2,3,....

Several of the known properties for parameter sequences of chain sequences are really properties of tail sequences for continued fractions. Let me mention some examples: T h e o r e m 4.1. (Auric 1907 [1], [8, Satz 2.45, p. 96], Waadeland 1984 [11], J. 1986 [3]). Let {tn} be a tail sequence for the continued fraction K(an/bs) with all tn # oo. Then K ( a J b s ) converges if and only if (4.2)

~

h

bn + t,

k=O n = l

--ts

converges to a value P C 4. I f K ( a , / b ~ ) converges, then it converges to t0(1 - 1/P). For

every Vo E 4, (4.3)

VN := tN(1 + 1/TN);

TN . -

Vo

to

~t n

~[ to j;l= -b n-+ t+n

N-1

N

__t s

E II bn+ts

k=O n = k + l

for N=0,1,2,... is also a tail sequence for K(an/bn). P r o o f From formula (2.3) in [3] it follows that 1 1 [ tN N b,~+ts SN(W) -- to -- to -k~- tN r I -t~

N-~ k b~+t~] ~ I] -t~ j k=O n = l

Setting w := 0 proves the first part, and w := vg, so that SN(VN) = CO, proves the last part. • T h e o r e m 4.2. (Pincherle 1894 [9], [6, Theorem 5.7, p. 164], J. 1986 [3].) Let {t,} and {Us} be two tail sequences for the continued fraction K(a,~/b,0 with to ~ Uo and all t~, u~ # oc. Then K(a~/b~) converges if and only if f i tn/un

(4.4)

n=0

converges to a value Q c C.

If K(an/bn) converges, then it converges to (to - ~oq)/(1 - q). For every Vo e 4, (4.5)

vg := UN(vo -- to) -- tNOY(vo -- Uo) Vo - to - QN(vo - uo) '

N - 1 tn

ON := 1-I - s=0 us

Lisa Jacobsen

100

is also a tail sequence for K(aJbn). Proof. From formula (2.4) in [3] it follows that SN(W)

- - to --

u o

W -- t N - - Q N . -

Hence, the result follows by choosing w := 0 and w := v N.

[]

R e m a r k 4.3. In the proof of Observation 3.7 we used that t~ = F for all n gives a tail sequence for the continued fraction K((-A/(1 - s2))/z). (Confer with the fact that g,~ = 1/2 for all n is a parameter sequence for the chain sequence an = 1/4 for all n.)

References

[1]

A. Auric, Recherches sur les fractions continues alggbraiques, J. Math. Pures Appl. (6) 3 (1907).

[2]

T. S. Chihara, Introduction to Orthogonal Polynomials, Mathematics and Its Applications Ser., Gordon, 1978.

[3]

L. Jacobsen, Composition of linear fractional transformations in terms of tail sequences, Proc. Amer. Math. Soc. (1) 97 (1986), 97-104.

[4]

L. Jacobsen and W. J. Thron, Oval convergence regions and circular limit regions for continued fractions K(an/1), Lecture Notes in Math., Springer-Verlag 1199 (1986), 90-126.

[5]

W.B. Jones, W. J. Thron, Convergence of continued fractions, Canad. J. Math. 20 (1968), 1037-1055.

[6]

W.B. Jones, W.J. Thron, Continued Fractions. Analytic Theory and Applications. Encyclopedia of mathematics and its applications, 11, Addison-Wesley, 1980.

[7]

R.E. Lane, The convergence and values of periodic continued fractions, Bull. Amer. Math. Soc. 51 (1945), 246-250.

[8]

O. Perron, Die Lehre yon den Kettenbriichen, Band II, Teubner, Stuttgart, 1957.

[9]

S. Pincherle, Delle funzioni ipergeometriche e di vari questioni ad esse attinenti, Giorn. Mat. Battaglini 32 (1894), 209-291.

[lo]

W.T. Scott, H.S. Wall, A convergence theorem for continued fractions, Trans. Amer. Math. Soc 47 (1940), 155-172.

[11]

H. Waadeland, Tales about tails, Proc. Amer. Soc. (1) 90 (1984), 57-64.

[12]

H.S. Wall, Analytic Theory of Continued Fractions, Van Nostrand, New York, 1948.

Orthogonal Polynomials, Chain Sequences, ...

[13] J.D. Worpitzky,

101

Untersuchungen ~ber die Entwicklung der monodromen und monogenen Funktionen dutch Kettenbrgche, Friedrichs-Gymrmsium und Realschule,

Berlin, Jabxesbericht 1865, 3-39. Received: August~ 7, 1989

Computational Methods and Function Theory Proceedings, Valparaiso 1989 St. Ruscheweyh, E.B. Saff, L. C. Salinas, R.S. Varga (eds.) Lecture Notes in Mathematics 1435, pp. 103-115 ~) Springer Berlin Heidelberg 1990

O n T h u r s t o n ' s F o r m u l a t i o n a n d P r o o f of A n d r e e v ' s Theorem A1 M a r d e n 1 University of Minnesota, Minneapolis Minnesota 55455, USA and

Burt Rodin 2 University of California, San Diego La Jolla, California 92093, USA

In C h a p t e r 13 of his notes [4], W. Thurston states a general result, T h e o r e m 13.7.1 (see also Corollary 13.6.2), regarding the existence and uniqueness of circle packings of prescribed combinatorial type on closed surfaces. This theorem treats the cases of genus g = 1 and g > 2; it is pointed out that the case g = 0, which is not proved in these notes, is a result of E.M. Andreev [1,2]. It is implicit in T h u r s t o n ' s notes that the continuity m e t h o d used there to prove T h e o r e m 13.7.1 in the cases of genus g = 1 and g > 2 could be modified to give a proof of the g = 0 case. Such a proof would be very different form Andreev's. T h e purpose of the present p a p e r is to present such a proof. We separate the statement of the g = 0 case into two parts: T h e o r e m A below, which deals with standard circle packings, and T h e o r e m B, which allows the circles to intersect at prescribed angles. These theorems have apphcations to conformal mapping [3]. 1. A circle packing on the Riemann sphere or in the plane is a collection of closed disks or the R i e m a n n sphere on in the Euclidean plane with the p r o p e r t y t h a t the interiors of the disks are disjoint. The nerve of such a circle packing is the graph which has a vertex for each disk and an edge connects two vertices if and only if the corresponding closed disks intersect. T h u r s t o n ' s theorem, in the case of circle packings, is the following:

1Research partially supported by the NSF. 2Research partially supported by the NSF and DARPA/ACMP.

A. Marden and B. Rodin

104

T h e o r e m A. Let T be a triangulation of the Riemann sphere P. There exists a c/rc/e packing of P whose nerve is isomorphic to the one dimensional skeleton of T ; any two such circle packings are images of each other under some linear fractional transformation or its complex conjugate. 2. We begin the proof with some formulas for the Euler characteristic. Let V, E , and F denote the number of vertices, edges, and faces in the triangulation T . Then V-E+F=2.

(1)

Since 3 F = 2E, we can eliminate E in (1) and obtain (2)

2V = Y + 4.

It will be convenient to label the vertices of the triangulation T by vl, v 2 , . . . , vv. Let r = ( r l , r 2 , . . . , r v ) be a vector of V positive numbers. Then r determines a polygonal structure on the topological 2-sphere ]T] as follows. Associate to each face of T, with vertices (vl, vj, vk) say, the Euclidean triangle determined by the centers of three mutually (externally) tangent circles of radii rl, rj and rk. Transfer the Euclidean metric on this Euclidean triangle to the associated face of T. Note that the metric is well defined on an edge which is common to two different faces. In this way ITI becomes a locally Euclidean space with cone type singularities at the vertices; we denote this space by T~. The curvature of T-~ at the vertex vi, denoted by ~r(vi), is defined as follows. Consider all faces of ~ which have vi as one of their vertices. Let a(vi) be the sum of each angle at vi in each of these triangles. Then

(3) Let us note that V

~., n~(v,) = 4re.

(4)

i=l

Indeed, the left hand side reduces to 27rV - ~ 8, where ~ 0 denotes the sum of all angles of all triangles of T~. This sum is equal to TrF. An application of Equation (2) now establishes (4). 3. If A > 0 then ~ and ~r~r are similar in the sense that corresponding angles are equal. Therefore nT(vi) = t~aT(vi). It turns out to be advantageous to normalize the map r ~ f ( r ) = (~T(vl), sz~(v2),..., ~ ( v v ) ) by restricting its domain to the simplex (5)

A_-- { ( r l , r 2 , . . . , r v ) ) c R Y : r 1

> 0 , r2 > 0 , . . . r y

>0&rl+r2T...+rv--

It follows form (4) that the range of f can be taken as the hyperplane (6)

Y = { ( y l , y 2 , . . . , y y ) ¢ n V : Yl + Y2 + . . . + yv = 4r}.

1}.

On Thurston's Formulation and Proof of Andreev's Theorem

105

4. For convenience of notation, assume that vl, v2, va are the vertices of a single face r0 of T . We now prove that the existence assertion of Theorem A will follow once it is shown that the point Po = (47r/3, 4~r/3, 47r/3, 0 , . . . , 0), for example, lies in the image of the map f:A-*Y. To see this, suppose f(ro) = po. If we remove that face r0 from T~0 then the remaining triangles can be placed isometrically in the plane, one by one, in an orientation preserving manner, keeping identified edges coincident. Since the curvature is zero at each interior vertex of this complex it can be shown that we obtain in this way an isometric embedding of T~0 less To onto a triangle in the plane. {To prove that this is so, one can first show that the process of placing adjacent faces in the plane yields a well defined isometry once the image of an initial face is fixed. For suppose a sequence of adjacent faces are placed in the plane in this way and suppose the first face in the sequence is the same as the last. Then the placement of the first and of the last face will agree-this is clearly true if the sequence of faces surrounds only one interior vertex of T~0 less r0, and can be shown to be true in general by induction on the number of such vertices. T h e second step in the proof is to use the fact that this placement process provides a locMly isometric embedding of ~0 less r0 into the plane and is an actual embedding of the boundary of T~0 less r0. It is easy to see that a local embedding of a topological disk into the plane which is an actual embedding on the boundary must by a global embedding. One concludes that this placement process is a global isometric embedding of T~0 less r0 onto a triangle in the plane}. We have constructed an isometric embedding, call it ¢, of T~0 less r0 onto a triangle A B C in the plane. It follows from the definition of T~0 that if we center a circle of radius ri at the point ¢(vi) we obtain a circle packing in the plane whose nerve is isomorphic to the one dimensional skeleton of T. Stereographic projection transforms this packing to a packing of the Riemann sphere with the same property. It will be useful when we discuss uniqueness to observe that triangle A B C is necessarily equilateral. To verify this, weld another copy of triangle A B C to this one along corresponding edges. One then obtains an isometric image of all of T~0. We can calculate the curvature at the vertex ¢(vl) which, we may assume, corresponds to the point A, directly from the definition (3) using this isometric image. We see that the curvture at A is 27r less the sum a ( A ) of all angles in this isometric image with this vertex A. This sum a ( A ) is clearly twice the angular measure re(A) of angle A in triangle A B C . On the other hand, we know by the definition of p0 that the curvature must turn out to be 47r/3. Thus 47r/3 = 27r - 2m(A). Hence re(A) = ~r/3. Similarly, m ( B ) = m ( C ) = 7r/3 and so A B C is equilateral. 5. We now show that f : A --, Y is one to one. Let r' = (rtl,r~ . . . . ,r~v) and II r" = (7"1, r 2l ! , . . . , r ~ ) be distinct points in A. Let Y0 be the set of vertices v; of T for which r 'i < r i" . Note that the definition (5) of A implies that V0 is a nonempty proper subset of the set of all vertices of T.

A. Marden and B. Rodin

106

Consider a vertex v E ])0 together with all the faces of T} which have v as vertex. In each such face there is an angle at v, and we classify this angle as type a if it is the only angle in this face which has its vertex in Y0, of type/3 if two vertices in this face are in Y0, and of type 7 if all three vertices of this face are in Y0. Now E x~,,(vi) = E (2~r - a(v)) vEVo vEVo

(7) =

2

lV01 -

of type a) - ~_,(Zs of type fl) - ~_,(Zs of type -y).

Consider three mutually tangent circles in the plane and their triangle of centers. If one of the circles shrinks and the other two either expand or stay the same size, and if the three circles always remain mutually tangent, then the angle in the triangle of centers with vertex at the center of the shrinking circle will (strictly) increase. If two of the circles shrink and the other either expands or stays the same size, then in the triangle of centers the sum of the two angles which have their vertices at the centers of the shrinking circles will increase. These observations show that if r" is replaced by r' in equations (7) then

(8)

E

> E

vEVo

vEVo

Indeed, in passing from r" to # the radii at v E 1)0 shrink and so the first two quantities in of type /~), of type 7) E ( / s of type a),

E(Ls

E(Zs

will each increase, and the third will remain constant. Since ~0 is a n o n e m p t y proper subset of vertices, not all angles are of type 3'. It follows that the inequality in (8) is strict and that f : A ---+y must be one-to-one. 6. We now examine the behaviour of f ( r ) as r tends to a boundary point s = (Sl, s 2 , . . . , sv) of A. It will turn o u t - a n d this seems very remarkable-that f cannot be extended continously to the boundary of zl, yet the set of accumulation points of f ( r ) as r tends to the boundary of A form the boundary of a polyhedron. Let F0 be the set of vertices vl in T for which 8i = 0; F0 is a nonempty proper subset of V. We classify the angles of T~ into types a,/~, 7 as above. Then as r ~ s we have E ( L s of type o~) --+ (9)

Z ( Z s of type/3)

--+ ~I/~I/2,

E ( / s of type v)

--~ ~rl-yl/3,

where Ixl denotes the number of angles of type x. Therefore equation (7) yields (10)

lira ~ ~ ( v ) = 2~r[~2o]- 7rIa ] ~-~ ~eVo

~r]fl] ~r]T] 2 3

On Thurston's Formulation and Proof of Andreev's Theorem

107

= 2rc]ld0I - re- (no. of faces with a vertex in ]do). From (8) and (10) we see that the image f(Zl) of f : A ~ T lies in the bounded convex polyhedron ]I0 formed by intersecting Y with the half spaces (11)

~ ] yi > 2~rlII- rr-(no, of faces with a vertex in ldi - {vi: i E I}) iEI

as I varies over all nonempty proper subsets of { 1 , 2 , . . . , V}. We have also seen that the accumulation points of f ( r ) as r ---* 0A lie on the hyperplanes (12)

~ Yi = 2zr[I[ -- zr. (no. of faces with a vertex in 121 =-- {vi: i E I}). iEI

which form the boundary of Y0. 7. We know that f : A --~ Y0 is a continuous 1-1 mapping. Hence, by Invariance of the Domain, f is a homeomorphism. We also know that f ( r ) -~ OYo as r --* 0A. It follows by elementary topology that f : A ~ Y0 is surjective. Indeed, merely pass to the one point compactifications and apply the simple fact that if X, Y are Hausdorff spaces with X compact and connected and Y connected, and if ¢ : X -* Y is continuous and open, then ¢ is surjective. 8. We complete the proof of the existence part of Theorem A by showing that p0 = ( 4 r r / 3 , 4 r r / 3 , 4 7 r / 3 , 0 , . . . , 0 ) is in the image Y0 of f : A ~ y (see Section 4). According to (11), this can be done by showing that for every nonempty proper subset

Xof {1,2,...,v},

(13)

~Pi

> 2~r[I[- 7r. (no. of faces with a vertex in 12x =- {vi : i E

I}),

iEI

where Po = (Pl, P 2 , . . . , pv) = (47r/3, 4zr/3,4~r/3, 0 , . . . , 0). If [I[ = V - 1 then every face has a vertex in VI - {vi : i E I}. Therefore the right hand side of (13) is, by (2), (14)

27r(V - 1) - 7r. F = 27r.

For subsets I of this cardinality the left hand side of (13) becomes

(15)

Y~Pi = iEI

8¢r/3 or 12~/3.

Thus Po satisfies (13) when IZl = v - 1. If I/t = v - 2 similar reasoning shows that the right hand side of (13) is zero while the left hand side is at least 47r/3. Thus po satisfies (13) in this case also. We shall show that p0 satisfies (13) v/hen 1 < ] I I _< V - 3 by proving that the right hand side of (13) will be negative in these cases. First we rewrite the right hand side in a more invariant form. Let F1, F2, Fa denote, respectively, the number of faces of T which have exactly 1,2,3 vertices in Idl. Then the right side of (13) is ~r(2]I I - F1 - F~ - F3). Let

A. Marden and B. R o d i n

108

: O(e,-j) /

........~.~

F i g u r e 1. T h e angle of intersection of two disks

E2 d e n o t e the n u m b e r of edges of T which have b o t h of their b o u n d a r y vertices in 1;z. T h e n the simplicial complex TI spanned by the vertices of Yl has Euler characteristic X0 = II] - E2 + F3. Since 3F3 + F2 = 2E2, we can eliminate E2 from the expression for X0 a n d o b t a i n 2X0 = 2]II - F3 - F~. Therefore the condition (13) t h a t ( P l , P 2 , . . . , p y ) lies in the image of f can be rewritten as (17)

~-~p, > ~r(2X0 - F1) iEl

for every n o n e m p t y p r o p e r subset I of { 1 , 2 , . . . , V}. We wish to show t h a t the right h a n d side of (17) is negative for 1 <: ]I I ~ V - 3. For t h a t p u r p o s e we m a y assume that 7"1 is connected. T h e Euler characteristic of a c o n n e c t e d simplicial 2-complex can be interpreted as 2 - 2g - n where g is the genus a n d n = 1, 2 , . . . is the connectivity. In our case g = 0 and X0 = 2 - n. If n >__3 there is n o t h i n g to prove. If n = 1 or 2, one of the c o m p o n e n t s of the c o m p l e m e n t of T / c o n t a i n s at least two vertices in 1) - 1;I. Therefore we can find an edge (x', y') where the vertices x' a n d y' are in 1; - 1;I. We can even choose x' and yr so t h a t there is an edge (y~, a') with a' in 1;I- If we examine the star of y' we can find a triangle face (x, y, a) of T with a E ]7I a n d x , y E 1; -- 1;i. N o w look at the union of the star of x and the star of y. If all the vertices adjacent to x a n d y belong to 1;I the { x , y } is a c o m p o n e n t of the c o m p l e m e n t of TI. Since there are at least three vertices in 1; - 1;I we m u s t be in the case n = 2. Therefore t h e right h a n d side of (17) is negative in this case since ( x , y , a ) is an F~ t y p e triangle. In the r e m a i n i n g case the set of vertices adjacent to x a n d to y contains a E ))I a n d some vertex z ( # x , y ) E Y - )21. It follows t h a t there are three F1 t y p e triangles in the u n i o n of the stars of x and y. 9. T h e p r o o f of the existence part of T h e o r e m A is now complete. W e have seen t h a t a given triangulation T of the R i e m a n n sphere P can be realized as the nerve of a circle p a c k i n g of P . By means of a linear fractional t r a n s f o r m a t i o n of P we can always a r r a n g e the realization so t h a t any three preassigned m u t u a l l y t a n g e n t circles will have

On Thurston's Formulation and Proof of Andreev's Theorem

..,'""

109

02

la . . . . . . . . . . . . . . . . . . . . . . . . .

12

F i g u r e 2. Prescribed intersection angles

equal radii; c~ will be an interior point of the face whose vertices are their centers. The nerve of this packing on the finite complex plane forms a triangulation of an equilateral triangle by straight line segments. Let the circles which correspond to the vertices vl, v 2 , . . . , vv have radii rl, r 2 , . . . , rv respectively. We may assume that rl + r2 + . . . + rv = 1. For r0 = (rl, r 2 , . . . , rv) we can calculate the curvatures f(ro) of T~0 by means of this triangulated equilateral triangle as was done in the last paragraph of Section 4. If vl, v2, v3 correspond to the vertices of the equilateral triangle we find that f(r0) = (47r/3, 47r/3, 4rr/3, 0 , . . . , 0). By the injectivity of f , r0 is uniquely determined. Therefore the radii of the circles in this normalized circle packing are uniquely determined. It is clear that two circle packings with the same abstract nerve and with corresponding radii equal are (proper of improper) rigid motions of each other. This proves that all circle packings which realize the same triangulation of a 2-sphere axe linear fractional transformations of each other followed possibly by a reflection. This completes the proof of Theorem A. 10. We now consider a generalization of Theorem A in the spirit of T h e o r e m 13.7.1 in Thurston's notes (loc. cit) for the case g = 0. Let T be a triangulation of the 2-sphere, let g be the set of edges in T , and let 6) : £ --~ [0, ~r/2] be any function. A family C of closed disks on the Riemann sphere P or in the plane will be said to realize the d a t a T , 6) provided the following conditions are satisfied: (a) the nerve of C is isomorphic to T , and (b) two disks C~ and Cj in C intersect if and only if their angle of intersection has radian measure 6)(eij), where e U is the edge of T which spans the vertices corresponding to Ci and Cj. (The angle of intersection of two disks is the one in the exterior of the two disks; see Figure 1.) We shall prove the following result. T h e o r e m B . Let T be a triangulation of the 2-sphere. Let 6) : g --+ R be a function defined on the edges of T with the property that 0 _< 6)(e) < ~r/2 for aI1 e 6 $. Assume that 0 has the following two properties: (i) If el + e2 + e3 is a cycle of edges in T then 6)(el) -t- O(e2) -{- O(e3) < 7r, and Oi) if el -t- e2 4- e3 + e4 is a cycle of distinct edges in T

A. Marden and B. Rodin

110

pl

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

A1

h

Figure 3

then {9( el ) + {9( ) + {9( e3 ) + {9( e, ) < Then there exists a ramily C or rou d disks on the Riemann sphere which realizes the data T , {9. This family C is uniquely determined up to a linear fractional transformation or its conjugate. 11. The proof proceeds as before except for several additional complications. The first complication arises in constructing T~. The metric on a face with vertices vi, vj, vk should be the Euclidean metric of the triangle of centers of three disks which intersect pairwise at the nonobtuse angles {9(e~j), {9(ejk), and {9(eke) and which have the radii prescribed by r. The following lemma from [4] guarantees the existence of such a configuration. L e m m a 2. For any three nonobtuse angles O1, 02, O3 aJld any three positive numbers Pl, P2, P3, there is a unique configuration in the plane consisting of three disks having these radii and intersecting in these angles. The proof of Lemma 2 refers to Figure 2. Determine the sides ll, 12, 13 of the desired triangle of centers as follows. Side 11 is the length of the third side of a triangle which has sides p2 and p3 and included angle ~r - 01. Sides 11 and 12 are determined similarly. To see that ll, 12, la satisfy the triangle inequality note that property (i) of Theorem B implies I1 ~ p2 + p3 _< 12d-13, and similarly for 12 and la, because the angles of intersection are nonobtuse. Thus the configuration shown in Figure 2 can always he constructed and is uniquely determined. 12. Having constructed T~ we proceed as before to define the curvatures at the vertices vi, v 2 , . . . , vv and thereby obtain the map (18)

r -~ f ( r ) = (~r(Vl), ~ ( v 2 ) , . - . , ~ ( v v ) ) : ~ -~ Z.

The previous proof (Section 5) that f is one to one can be imitated with the help of the following 1emma from [4].

On Th urston's Formulation and Proof of Andreev's Theorem

111

A1

,(OA,/Opl)

Figure 4

L e m m a 3. Consider three circ/es in the plane which intersect pairwise in nonobtuse angles. If one radius decreases and the other two remain the same, and if the circles continue to intersect each other at the originM angles then, in the triangle of centers, the angle at the vertex of the shrinking circle will increase and the other two angles will decrease, Let the radii be pa, p2, p3 and let the triangle of centers have sides of lengths 11,12, I3 and vertices at A1, A2, A3. If we fix A2 as the origin, so HA~H = 13, then we can differentiate A1 = 13u, where u is a unit vector, to obtain

OA___Z_ Ot~ A___L+ 13B,

(19)

op, lla, ll

Op,

where B is orthogonal to Aa. Note that (Figure 3) 13 = Pa cos 71 + P2 cos 72 and so (20)

013 _

71

Opl

Pl ~

"/2

sin 71 - P2 ~Pl sin 72 + cos "h-

Since p2 sin 72 = pl sin 71 =- h, (20) becomes (21)

013 _

op,

h (07,

07~)

\opl + -~p~) + cos71.

T h e t e r m in parentheses is zero since 71 + 72 is a constant equal to the fixed angle 013 of intersection of circles 1 and 2. Therefore ~px = cos 7a- Thus (22)

013

pl ~

= pl cos 7~,

A. Marden and B. Rodin

112

A2

A1

.ff

!

fix Opl

Figure 5

which is the distance form A1 to the point of intersection of the radical axis of circles 1 and 2 (that is, the line through their points of intersection) with the line joining their 0A1 .

centers. It follows t h a t pX-~p1 1S the vector with its tail at A1 and its tip on the radical axis of circles 1 and 2. By symmetry, its tip is also on the radical axis of circles 1 and 3. 0A1

Hence p l - ~ p1 is the vector from the common point of intersection of the three radical axes to the point A1 (Figures 4 and 6; in the context of Theorem B the configuration of Figure 6 will not occur because condition (ii) in that theorem insures that the three disks have empty intersection). Therefore, if Pl decreases and pl and p2 remain unchanged, the vertex A1 will move toward the intersection point Q of the radical axes. If we show that Q lies in the triangle of centers it will follow that the angle at A1 decreases and the proof of L e m m a 3 will be complete. If Q did not lie in the triangle of centers then one of the sides of that triangle would separate Q from the vertex not on that side. Suppose side A2A3 separates Q from A1 as in Figure 6. First note that the circle centered at A1 does not intersect side A2A3 because if it did then the distance A2A3 would be greater than the sum of lengths of the tangents from A2 and A3 to circle 1 and the lengths of these tangents are upper bounds for r2 and r3 since circles 2 and 3 must intersect circle 1 in nonobtuse angles (see Figure 6). Thus in Figure 7 the circle centered at A~ in the upper half plane determined by A2A3 cannot enter the lower plane. It follows that the radical axis of circles 1 and 2 cannot enter the shaded region of Figure 7. This contradicts the fact that Q lies on that radical axis. Therefore Q must lie in the triangle of centers.

On Thurston's Formulation and Proof of Andreev's Theorem

113

13. We now consider the modifications to Section 6 that are needed for the present case. Consider three circles which intersect pairwise in nonobtuse angles. Let the radius 1.b. for P2 + P3

u.b. for p, " ~

.........~

................"~x,u.b. for pa

Figure 6 of circle 1 shrink to zero while the intersection angles and the other two radii remain constant. Then in the triangle of centers the angle at the center of circle 1 will increase to the limiting value 7r - (0(e)), where e is the opposite edge. If the radii of two circles shrink to zero then in the triangle of centers the sum of the two angles at the centers of these two circles will tend to the limiting value ~r. When all three radii shrink to zero we use the fact that the sum of the three angles is constantly equal to 7r. Thus the three equations (9) are to be replaced by

(23)

E(Z~ of type a)

--+ E(~-- O(e(a)),

E ( Z s of type/~)

~

~1¢~1/2,

E ( Z s of type 7) Equation (10) is replaced by

(24)

!ira Z ;

= 2

IVol-

-

vEVo

2

3

= 2rr11201 - 7r. (no. of faces with a vertex in 120) + y~ O(e(a)). ct

We conclude that the image Y0 of f : A -~ y is the hyperplane formed by intersecting Y with the half spaces (25)

~ yi > 2zrlI] - 7r- (no. of faces with a vertex in PI) + Y~, O(e(a)) iE1

a

for each nonempty proper subset I of {1, 2 , . . . , V}. 14. As in Section 8, the existence assertion of Theorem B will follow from showing that Po = (4~r/3, 4rr/3, 4rr/3, 0 , . . . , 0) is in the image Yo of f : A --, y . According to (25) this is equivalent to showing that for each nonempty proper subset I of {1, 2, 3 , . . . , V}, (26)

~pl iEI

> 2~1I I - ~. (no. of faces with a vertex in 12I) + E O(e(o~)) a

114

A. Marden and B. Rodin

::::::::::

iiiiii :::::1:::: :::::::::: :::::::::: ::::::::::

Figure 7

where p0 = ( p l , p 2 , . . . , p v ) ---- (47r/3, 4~r/3,4~r/3, 0 , . . . , 0 ) and where the s u m m a t i o n denotes the sum over angles of type a of the value of O at the edge opposite the angle of type a. If [I[ = V - 1, (26) holds. Indeed, there are no angles of type a in this case, so the earlier calculation (Equations (14) and (15)) applies. If IZl -- v - 2 the left hand side of (26) is at least 4~r/3. The first two terms on the right hand side cancel by (2) and the summation term is at most 7r since there can be at most two terms in the sum. Thus (26) holds in this case as well. For the cases 1 < IZl < y - 3, rewrite (26) in the form (27)

~pi iEl

> ~r(2Xo - F~) + ~

O(e(a)) o:

= 2~rXo - ~ ( ~ r - O ( e ) ) e

in the same way that (13) was rewritten as (17); here the sum of F1 terms (each t e r m satisfies ~r/2 _< ~r - O(e) _< 7r) is taken over the edges in the F1 type triangles opposite the angles of t y p e (~. These edges form a 1-cycle in the m o d 2 homology of T . As before, we have to show that the right hand side of (27) is negative and we m a y assume t h a t the complex spanned by 12x is connected (X0 <_ 1). The negativity is clear if Xo -- 0. T h e reasoning in Section 8 following (17) showed t h a t if Xo = 0 then/;'1 >__1, and if X0 = 1 then F1 >_ 3. Therefore the right hand side is negative if Xo = 0; it will also be negative if X0 = 1 and F1 > 5. The remaining cases X0 = 1 and F1 = 3 or 4 are covered b y properties (i) and (ii) of Theorem B. In case F1 = 4 one can eliminate the case t h a t the four edges are not distinct because in that case 12 - 121 will consist of the vertices on these edges. Since they do not span a triangle face, one of the vertices vl, v2, va is in 12I. Therefore the left hand side of (27) is at least 47r/3 and the desired inequality (27) holds, even though the right hand side m a y be zero.

On Thurston's Formulation and Proof of Andreev's Theorem

115

References

[1] E.M. Andreev,

On convex polyhedra in Lobacevskii spaces, Math. USSR Sb. 10

(1970), 413-440.

[2j

E.M. Andreev, On convex polyhedra of finite volume in Lobacevskii space, Mat. Sb., Nor. Ser. 83 (1970), 256-261) (Russian); Math. USSR, Sb. 12 (1970), 255-259 (English).

[3]

B. Rodin, D. Sullivan, The convergence of circle packings to the Riemann mapping, J. Diff. Geom. 26 (1987), 349-360.

[4]

W.P. Thurston, The Geometry and Topology of 3-manifolds, Princeton University Notes, Princeton, New Jersey, 1980.

[5]

W.P. Thurston, The finite Riemann mapping theorem, invited address, International Symposium in Celebration of the Proof of the Bieberbach Conjecture. Purdue University, March 1985.

Received: December 27, 1989

Computational Methods and Function Theory Proceedings, Valparaiso 1989 St. Ruscheweyh,E.B. Saff, L. C. Salinas, R.S. Varga (eds.) Lecture Notes in Mathematics 1435, pp. 117-129 (~) Springer Berlin Heidelberg 1990

Hyperbolic Geometry in Spherically k-convex Regions Diego Mejla Departamento de MatemAticas, Universidad Nacional Apartado A6reo 568, MedelIfn, Colombia and

David Minda 1, 2 Department of Mathematics, University of Cincinnati Cincinnati, Ohio 45221-0025, USA

1. I n t r o d u c t i o n This paper is the second of a proposed trilogy dealing with the concept of k-convexity in various geometries. The first paper [MM] dealt with k-convexity in euclidean geometry. This paper presents a generalization of most of the results of [MM] to k-convexity in spherical geometry. In a proposed third paper we plan to treat the concept of k-convexity in hyperbolic geometry. We assume that the reader is familiar with [MM] and frequently omit details of proofs when they are similar to proofs of analogous results in [MM]. In fact, since many of the proofs in [MM] are truly geometric in nature, it is often clear how to extend the proof from euclidean geometry to the case of spherical geometry. A Jordan region ~2 on the Riemann sphere P is called spherically k-convex if the spherical curvature of the boundary is at least k at each point of 0f2. This assumes that the boundary of f2 is smooth. A definition of spherical k-convexity that applies to arbitrary regions is given in section 3. Our program is to give sharp spherical estimates for various hyperbolic quantities in spherically k-convex regions. These estimates lead to sharp distortion and covering theorems (including the Bloch-Landau constant and the Koebe set) for the 1Research partially supported by NSF Grant No. DMS-8801439. ~I want to thank the Universityof Cincinnati and the Taft Foundation for partially supporting my sabbatical leave during the 1988-89 academic year and the Universityof California, San Diego for its hospitality during this period.

D. Mefia and D. Minda

118

family Ks(k, ce) of normalized (f(O) = O, f'(O) = oL) conformal mappings of the unit disk D onto spherically k-convex regions. For the special case of spherically convex regions (the ease k = O) some of these results were established in [MI] and [M3].

2. Spherical

geometry

We begin by recalling some basic facts about spherical geometry on the Riemann sphere P; see [M,] for more details. The spherical metric is Ae(z)ldz I = [dz[/(1 + [z[2); it has Gaussian curvature +4. The group of rotations of the Riemann sphere is Rot(P)=

{T(z)= .izo T~-az

:8•R'

a•e

}.

The spherical metric is invariant under the group Rot(P); that is, T*(Ap(z)ldz]) = Ae(z)ldzl, or, equivalently, IT'I/(1 + ITI 2) = 1/(1 + Iz12). The spherical distance between a, b • P is defined by

de(a, b) = infJ~ Ae(z)ldzl, where the infimum is taken over all paths 7 on P which join a and b. Moreover, this infimum is actually a minimum and is attained for the shorter arc 5 of any great circle through a and b. The arc 5 is unique unless a and b are antipodal points in which case both of the subarcs of any great circle through a and b is a possible choice for 5. Any path 5 between a and b such that

de(a, b) -- f~ Ae(z)]dz I is called a spherical geodesic. The explicit formula for the spherical distance is

dr,(a, b) = { arctan(ta - bl/ll + hal) arctan(1/lal)

if a, b • C if a • C, b = cx~.

Observe that de(a, b) < 7r/2. Also, the spherical distance is invariant under the group of rotations of P. Sometimes it is more convenient to employ a related quantity in place of the spherical distance. Set Ee(a, b) = tan dp(a, b) and note that this quantity is also invariant under the group Rot(P). Let D,(a, 9) denote the spherical disk with center a and radius ~. We shall also make use of the notion of spherical curvature. We briefly recall this concept; for more details the reader should consult [M1]. Suppose 7 is a C 2 curve on P with nonvanishing tangent at the point a • P. The spherical curvature of 3' at a is

k,(a, 7) = k(a, 7) - (O/On) log Ap(a) Ae(a) where k(a, 7) is the euclidean curvature of 7 at a and n = n(a) is the unit normal at a which makes an angle + ~ / 2 with the tangent vector at a. For example, if 7 is the spherical circle {z • P : dp(a, z) = ~}, ~ • (0, ~/2), oriented so that the center a lies on

Hyperbolic Geometry in Spherically k-convex Regions

119

the left-hand side of 7, then ks(z, 3') = 2 cot(2~0). Observe that the spherical curvature of 7 is positive for ~ E (0, ~r/4) and negative for ~ C ( r / 4 , 7r/2); the value ~ = 7r/4 yields a great circle which is a spherical geodesic and has zero spherical curvature. If f is a meromorphic function and 7 is a path, then for z E 7

k s ( / ( z ) , / o "y)f#(z) = ks(z,~/)Ap(z) + Im

1 + I z + "f'(z)

1 + lf(z)pj

'

where t(z) is the unit tangent to 3' at z. Recall that f # ( z ) = [f'(z)l/(l+ If(z)l 2) denotes the spherical derivative of a meromorphic function. Now we turn to an issue involving hyperbolic geometry for regions on the Riemann sphere. For a hyperbolic region ~2 on the Riemann sphere P it is natural to consider the spherical density of the hyperbolic metric in place of the density of the hyperbolic metric. For a hyperbolic region ~ on the Riemann sphere the spherical density is

#•(z) - A'(z)ldzt =

Ap(z)tdzt

(I

+ [zl2)A~(z)

which is a continuous function on f2 and is invariant under rotations of P. We shall frequently make use of the spherical distance to the boundary of the region. Let ¢9(z) = rain{alp(z, c) : c E 0f2}. This is the spherical distance from z to 0f2 and is clearly invariant under rotations of P. In some applications our formulas become much simpler if we employ a related quantity in place of ee(z). Define Ea(z) = t a n s y ( z ) . This quantity is invariant under rotations of P. We reformulate several basic results for the hyperbolic metric in terms of the spherical density. These results are well known for the euclidean density of the hyperbolic metric.

Principle of Hyperbolic Metric. Suppose ~2 is a hyperbolic region on P. If f is meromorphic on the unit disk D and f(D) C ~2, then # n ( f ( z ) ) f # ( z ) <_ AD(z) = 1/(1 - [z[ 2) for z E D with equality if and only if f is a conformal mapping o l d onto ~. Monotonicity Property. Suppose f2 and z3 are hyperbolic regmns on P and ~ C A. Then #n(z) ~_ #n(z) for z E ~2 with equality if and only i l l ) = A.

3. Geometric properties of spherically k-convex regions In this section we introduce the concept of a spherically k-convex region and study some of its basic properties. The proofs of some of the results are so similar to the analogous results for euclidean k-convex regions that we sometimes omit details; see [MM] for the details in the case of euclidean k-convexity. Suppose that k > 0, a, b E P and de(a, b) < arctan(2/k). Then there are two distinct closed spherical disks O--1and 0---2,each of radius ½arctan(2/k), such that a, b E OD---j(j = 1,2); note that 0Dj has constant spherical curvature k. Let Sk[a, b] = D1 M 92. The boundary of Sk[a, b] consists of two closed circular arcs/'1 and F2, each with constant spherical curvature k. We define S0[a, b] to be the spherical geodesic between a and

120

D. Mejfa and D. Minda

b when de(a, b) < rr/2. Also, for de(a, b) = arctan(2/k) we let Sk[a, b] be the closed spherical disk with center at the midpoint of the spherical geodesic joining a and b and radius ½arctan(2/k). Then for 0 _< k' < k < 2 / t a n ( d e ( a , b)),. Sk,[a, b] C Sk[a, b]. D e f i n i t i o n . Let k E [0, oo). A region 12 C P is called spherically k-convex if for any pair of points a, b C 12 with de(a, b) < arctan(2/k) we have Sk[a, b] C 12. Clearly, spherically 0-convex is equivalent to spherical convexity, so we shall employ the phrase spherical k-convexity only when k > 0 and use spherical convexity instead of spherical O-convexity. Note that if 12 is spherically k-convex, then it is also spherically k'convex for 0 < k' <_ k. In particular, a spherically k-convex region is always spherically convex and simply connected. For each k > 0 any spherical disk of radius ½arctan(2/k) is spherically k-convex but not spherically k'-convex for any k' > k. The intersection of a finite number of spherically k-convex regions is spherically k-convex and the union of an increasing sequence of spherically k-convex regions is again spherically k-convex. L e m m a 0. Suppose that 12 is a simply connected region on P with de(a, b) < ~r/2 for all a, b C 12. If at each point c E 012 there is a supporting great circle t'or all points of f) in a sumciently small neighborhood of c, then f) is spherically convex. Proof. This proof is adaptation of the proof of the analogous result for euclidean convexity [S]. Let a, b E f2. We want to show that the spherical geodesic connecting a and b is contained in D. By rotating 3"?if necessary, we may asume that a = 0. Because 12 contains no antipodal points, 12 must lie in C. Then 0 and b can be joined by a polygonal curve H with straight sides which is contained in ~2. Let 0 --- z0, zl, z 2 , . . . , z,,, z,,+l = b be the vertices of H in the order in which they are met in traversing H from 0 to b. We show that the vertices can be removed one at time, so that eventually the polygon, while remaining inside/-2, becomes the straight line [0, b], the spherical geodesic connecting 0 to b. Suppose that [0, zk] C 12. We want to show that [0, zk+l] C f2. If 0, zk and zk+l are collinear, we are done. Suppose [0, Zk+l] ~ 12. Consider the set of all segments [0, p], where p ranges over [zk, zk+l]; let O(p) denote the angle between [0,p] and [0, zk]. There is a smallest angle O(po) such that p0 E (zk, zk+~] and [0,p0] contains a point of 012. Let c E 012 A [0, p0] be the point in this set that is closest to the origin. Then all points of the closed euclidean triangle A with vertices 0, zk and P0 are in 12, except for c and possibly other points of [c, p0]. But then the point c fails to have a supporting great circle locally since the great circle through 0 and c contains points of 12 arbitrarily close to c and any other great circle through c meets the triangle A inside. • P r o p o s i t i o n 1. Suppose that 12 is a simply connected region on P bounded by a closed C 2 Jordan curve 012 such that de(a, b) < ~r/2 for all a, b E 12. If ks(c, 012) > k > 0 for ali c E 012, then 12 is spherically k-convex. Proof We begin by showing that 12 is spherically convex. By the preceding lemma, it is sufficient to show that there is a locally supporting great circle at each point c E 012. We m a y assume that c = 0. Then k(0,012) = ks(O, 0Y2) > 0. Hence, k((, 012) > 0 for all ~ C 012 in a neighbourhood of 0, so 12 has a locally supporting euclidean straight line at 0 [S, p. 46]. This straight line through 0 is also a great circle, so f2 has a locally supporting great circle at c.

Hyperbolic Geometry in Spherically k-convex Regions

121

Next, we show that [2 is spherically k-convex:. Fix a, b E /2. Let r be the s u p r e m u m of all t _> 0 such that St[a,b] C/2. Note that r <_."2/tandF,(a,b). S i n c e / 2 is spherically convex, we know that r > 0. We want to show that r > k. If r = 2 / t a n d e ( a , b), t h e n / 2 contains the closed spherical disk with center at the midpoint of the spherical geodesic joining a and b and radius ½de(a, b). Let D be the largest spherical disk with the same center t h a t is contained in /2. Then OD meets 0/2 at some point c. By rotating W if necessary, we m a y assume c = 0. The comparison principle for euclidean curvature [G, p. 28] implies t h a t k(0, 0/2) _< k(0, OD). Consequently, k < ks(O, 0/2) < k~(O, OD) < r. T h e remaining case is r < 2 / t a n de(a, b). Consider the two circular arcs /"1 and /~2 of spherical curvature r which bound S,[a, b]. At least one of these two arcs, say 1"1, meets 0/2 in some point c. As before we m a y assume c = 0. The comparison principle for euclidean curvature now gives k(0, 0~2) < k(0, F1), so that k _< r . We need to show strict inequality. B e c a u s e / 2 is open, we can select points a' and b~ i n / 2 so t h a t a and b lie strictly between a ~ and b~ on the spherical geodesic in /2 joining these latter two points. Let v ~ be defined relative to a' and b' in the same manner that r was defined for a and b. T h e n for a ~ and b~ near a and b, respectively, ~-~ < v. Since k < r ~just as k _< T, we obtain k < r. I

Proposition 2. Suppose/2 is a spherically k-convex region. Then for a n y a E /2 and c e a/2, S~[a,c]\{c} C / 2 . Corollary.

I f / 2 is a spherically k-convex region, then intSk[c, a~ C f2 for c, d E 0/2.

L e m m a 1. Suppose D is an open spherical disk of radius ½a r c t a n ( 2 / k ) and B is an open spherical disk such that c E OB M OD and B and D are externally tangent at c. If de(a, c) < a r c t a n ( 2 / k ) and a ¢ -D, then Sk[a, b]\{c} V1B # ¢.

Proposition 3. Suppose 12 is a spherically k-convex region. Assume a E / 2 , c E 0/2 and de(a, c) = ¢~(a). If D is the spherical disk of radius ½a r c t a n ( 2 / k ) that is tangent to the circle {z E P : dp(z, a) = en(a)} at c and contains a in its interior, t h e n / 2 C D. Proposition 4. Suppose/2 is a spherically k-convex region. Let a E P\/2, c E 0/2 and dp(a, c) = e~(a). IY D is the spherical disk of'radius ½arctan(2/k) that is tangent to the circle {z E P : d~,(z, a) = ~ ( a ) } at c and that does not meet the disk V,(a,¢~(a)), then f2 C D.

4. Lower bound for the spherical density of the hyperbolic metric in a spherically k-convex region We obtain a sharp lower bound for the spherical density in terms of the spherical distance to the b o u n d a r y of the region.

Theorem

1. Suppose/2 is a spherically k-convex region. Then t'or z E / 2

>_

1 + E~(z) Ea(z)[2 - kEn(z)]

with equality at a point if and only if/2 is a spherical disk with radius ½a r c t a n ( 2 / k ) .

D. Mejla and D. Minda

122

Proof. If D is a spherical disk of radius ½arctan(2/k) = arctan(v/-~ + 4 - k)/2, we show t h a t equality holds at each point of D. Since # a ( z ) is invariant under rotations of P, we m a y assume without loss of generality that D is centered at the origin. We set r = ( v ~ + 4 - k)/2 or k = (1 - r2)/r. Then ED(Z) = (r -- Izl)/(1 + r]z[), or Izl = (r -- ED(Z))/(1 + rED(z)), and so r r(1 + (1 + #D(Z) = (1 + Izl2) r 2 _ iz P -- ED(z)[2r -- (1 -- r2)ED(z)] = ED(Z)[2 -- kED(z)]" This shows t h a t equality holds at each point of D. Now consider any spherically k-convex region $2. Fix a E $2. Select c E 052 with E~(a) = tan de(a, c). Let F be the spherical circle with radius ½a r c t a n ( 2 / k ) through c t h a t is tangent to the spherical disk D~(a, arctan Ea(a)) at c and whose interior D contains a. T h e n Proposition 3 implies that $2 C D; also ED(a) = E~(a). T h e monotonicity p r o p e r t y of the hyperbolic metric yields #~(a) > #D(a) with equality if and only if $2 = D. Because ED(a) = Ea(a), this inequality in conjunction with the above formula for #D(Z) completes the proof. • C o r o l l a r y 1. Suppose that $2 is a spherically k-convex region. If f is meromorphic in D and f ( D ) C $2, then for z 6 D (1 - [ z l ~ ) f # ( z ) <_ Ea(f(z))[2 - k E e ( f ( z ) ) ]

1 + E~(f(z)) EquMity holds at a point if and only if $2 is a spherical disk with radius 1 a r c t a n ( 2 / k ) and f is a conformd mapping of D onto $2. Proof. The principle of hyperbolic metric gives # a ( f ( z ) ) f # ( z ) <_ AD(z) = 1/(1 - [ z [ 2) for z E D with equality if and only if f is a eonformal m a p p i n g of D onto $2. The theorem implies that

1 + E~(f(z)) <_ # a ( f ( z ) ) E~(f(z))[2 - kE~(f(z))] with equality if and only if $2 is a spherical disk of radius i arctan(2/k). By combining the two preceding inequalities and the necessary and sufficent conditions for equality, we obtain the corollary. • D e f i n i t i o n . Let Ks(k, a) denote the family of all holomorphie functions f defined on D such that f is univalent, f(0) = 0, f ' ( 0 ) -- a and f ( D ) is a spherically k-convex region. If f E K , ( k , a) and $2 = / ( D ) , then the preceding corollary with z = 0 produces a = [if(0)[ < h(Es2(O)), where h(t) = t(2 - kt)/(1 + t2). Note that h(t) is increasing on the interval 0 < t < ( v ~ + 4 - k)/2 = r and h(r) = r. Because $2 is spherically k-convex, we know that E~2(0) _< r. Therefore, a _ r whenever f E Ks(k, a). Moreover, a = r if and only if f ( z ) = rz.

Hyperbolic Geometry in SphericM1y k-convex Regions

123

E x a m p l e . Set fk(z) = az/(1 - ~/1 - a(a + k) z). Then fk e Ks(k, a) since fk(D) is a spherical disk of radius ½arctan(2/k). Note that f k ( - 1 ) = - a / ( 1 + ~/1 - a ( a + k)) and fk(1) = a/(1 - ~/1 - a ( a + k)). The largest spherical disk contained in fk(D) and centered at the origin has radius a / ( 1 + ~/1 - a(a + k)). C o r o l l a r y 2. Supposef e Ks(k, ) Then either { w : Iwl < is contained in f ( D ) or f(z) = e-i° fk(ei°z) for some 0 e R.

-

+ k))}

Proof. Set £2 = f ( D ) and apply the preceding corollary with z = 0 to obtain

= If'(O)l ~ E~(0)(2 - kEn(0)) 1 + EL(0 ) This yields E a ( 0 ) _> a/(1 + ~/1 - a(a + k)) with equality if and only if ~ is a spherical disk of radius i arctan(2/k). In the case of equality, f is a conformal mapping of D onto a spherical disk of radius 1 arctan(2/k) that contains the origin and whose boundary is externally tangent to the circle {w : Iwl -- ~/(1 + x/1 - a ( a + k))}. In this case it is straightforward to check that f must have the prescribed form. •

5. T h e spherical B l o c h - L a n d a u c o n s t a n t for t h e family

Ks(k, a)

We derive a sharp lower bound for the spherical density of the hyperbolic metric for a spherically k-convex region in terms of a uniform upper bound on ca. We use an extremal region which has been employed in [M1] and [MM]. Suppose k > 0, 0 E (0, 7r/2) and N = tan0. Let R = ~/N(2 - kN)/(k + 2N); R is selected so that the circle through - R , iN and R has spherical radius ½arctan(2/k) = arctan(v/-~-+ 4 - k)/2, or equivalently, spherical curvature k. Let S = S(N) = i n t S k [ - R , R ] . Note that for N = ( k v / ~ T 4 - k)/2 the set S is actually a spherical disk. In all cases, S contains the disk {z : [z I < N}, but no larger disk centered at the origin, and S is contained in the disk D = {z : Izl < R}. Each of the two circular arcs bounding S makes an angle 2T with the segment [ - R , R], where qp = arctan( N / R). We also introduce a certain collection of triangular spherically k-convex regions. Let T = T ( N ) denote the family of all spherically k-convex regions that contain {z : Izl < N} and are bounded by three distinct circular arcs each of spherical radius ½aretan(2/k) and having the property that the full circles are tangent to Izl = N and contain {z : ]zl < N} in their interior. Each of these circular arcs will meet OD in diametrically opposite points and has euclidean radius k' = (1 + N2)/(k + 2N). Therefore, each region A in T is both spherically k-convex and euclidean k'-convex. From [MM, Lemma 2] we obtain the following result. L e m m a 2. If A E T, then for z E A, #a(z) > (zr/4qO)#D(Z) > (zr/4Tn). T h e o r e m 2. Suppose f2 is a spherically k-convex region. Let 0 = max{¢~(z) : z E g2} and N = tan 0. Then

D. Mej[a and D. Minda

124

k+2N

1

#9(z) >_ ~ V N ~ Z - f f - N )

/ N ( k + 2N)" arctan V 2 : k N

Equality holds at a point a E t71 if and only if there is a rotation T of P such that Y2 = T(S) and a = T(O). Proof. Select a E $2 with Ea(a) = N. From Proposition 3 we see that 1 2 x/~ +42 ¢~(a) <__~ arctan ~ = arctan

k

with equality if and only if 12 is a spherical disk with center a and radius ½ arctan(2/k). Hence, N _ ( v / f i + 4 - k)/2 with equality if and only if Y2 is a spherical disk with center a and radius ½arctan(2/k). First, suppose N = (x/rfi + 4 - k)/2. Then Y? is a spherical disk centered at a and so (see the first part of the proof of Theorem 1)

1 + E~(z) = Eg(z)[2 - kEn(z)]

The right-hand side of this identity is a strictly decreasing function of E~(z), so we obtain I+N 2 ,.(z)

___ N[2 -

kN]

with strict inequality unless z = a. This is the desired result in this case. Now, assume that 0 < N < (A/r~ + 4 - k)/2. We may suppose that a = 0 since all quantities involved are invariant under rotations of P. Let I = {z : Izl = N and z E 0~}. The set I is nonempty and closed. A result of Blaschke [B] for euclidean convexity readily extends to spherical convexity and implies that I cannot be contained in a closed subarc of the circle [z[ = N with angular length strictly less than ~r. Now the proof completely parallels that of [MM, Thin. 4], so all further details are omitted." The function

gk(t) = v

,/t(k + 2o

k + 2t a r c t a n v 2 - k t

is strictly increasing on [0, ( v / f i + 4 - k)/2] with maximum value iT

gk(( k2x~V-~ - k ) / 2 ) = ~ ((v/-fi + 4 - k ) / 2 ) . Hence, for a e [0, ( v / ~ + 4 - k)/2] the equation gk(t) = a~r/4 has a unique solution g ( a ) • [0, ( v / ~ + 4 - k)/2]. C o r o l l a r y ( B l o c h - L a n d a u c o n s t a n t for K~(k, a)). Let f • K~(k, a). Then either f(D) contains an open spherical disk with radius strictly larger than a r c t a n N ( a ) or else f(z) = e-~¢F(eiCz) for some ¢ E R, where

F ( z ) : q N ( a ) ( 2 -+k N 2N(a) ( a ) ) ' t a n h ( 2 4 N ( a ) ( 2 - kk N + (2N(a) a))'k

'°g ---~Jl+z' 1

Hyperbolic Geometry in Spherically k-convex Regions

belongs to K,( k,

125

and maps D conform 1.v onto

Proof. Set ~2 = f ( D ) and N = m a x { E ~ ( z ) : z E ~}. If N > N ( a ) , then we are done. Assume N _< N ( a ) . Then gk(N) <_ g k ( N ( a ) ) = a r r / i . Since AM(0) = 1 / f ' ( 0 ) = l / a , the theorem with z = I ( 0 ) = 0 gives 1 / a > r:/4gk(N), or gk(N) >_ oer~/4. Now gk(N) = art/4, so N = N ( a ) . Thus, equality holds in the theorem at the origin, so a9 is just a rotation of S(N(oO). Since F e K s ( k , a) and maps D onto S ( N ( a ) ) , we conclude f ( z ) = e-lC'F(eiCz) for some ~b e R. •

6.

Hyperbolic convexity in spherically k-convex regions

We show that the intersection of a spherical disk and a spherically k-convex region is hyperbolically convex provided the center of the disk is sufficiently close to the region. E x a m p l e . Let D = {z : Iz-(i/k)] <_ I / k } and F ' = OD. Note that D is spherically kconvex since F ' has constant spherical curvature k. For a > 0 we have E = ED(--ia) = a. Suppose G is the spherical circle with center - i a and radius arctan O, where

(1)

E =

kO 2

1

+

+o2)2 + k2o2"

T h e euclidean center of F is b = - i E ( 1 + O2)/(1 - 0 2 E 2) and the euclidean radius is r = O(1 + E2)/(1 - 02E2). Necessary and sufficient for the circles F and F ' to be orthogonal is r 2 = [bl 2 + (2lb]/k). Straightforward calculations show that this condition is equivalent to (1). Therefore, F and F ' are orthogonal, so F A D is a hyperbolic geodesic in D and the hyperbolic half-plane D N {z : Ep(a, z) < O} is a hyperbolically convex subset of D. On the other hand, if E strictly exceeds the right side of (1), then D n {z : Ee(a, z) < O} is not a hyperbolically convex subset of D. This example shows that the following theorem is sharp. T h e o r e m 3. Suppose ~ is a spherically k-convex region on P. Let a E P\-~, E = E~(a), 0 > O, R = {z : E e ( a , z ) < 0}, F = OR a n d j denote reflection in F. If

(2)

E <

kO 2 1 + 02 + ~/(1 + 02) 2 + k202'

then j ( D \ R ) C ~. In particular, ~ ;~ {z : Ee(a, z) < O} is a hyperbolically convex subset of ~.

Proof. Select c E 0 ~ with Ep(a, c) = Era(a) = E. By making use of rotational invarianee, we m a y assume that c = 0 and that {z : Ep(a, z) < E} is tangent to the real axis at the origin and contained in the lower half-plane. Let D denote the disk of the preceding example and F ' = OD. Proposition 4 implies D C D. With the normalization c = 0, the proof of T h e o r e m 3 when equality holds in (2) parallels the proof of T h e o r e m 5 of [MM]. Observe that the euclidean and spherical curvature coincide at the origin. As in [MM] the case of strict inequality can be reduced to the case of equality. •

D. Mefia and D. Minda

126 7. Applications

to spherical

curvature

T h e o r e m 4. Suppose that 1? is a sphericalIy k-convex region, 7 is a hyperbolic geodesic in f2 and zo E 7. Let a denote one of the spherical centers of the spherical circle of curvature/'or 7 at Zo and 0 the tangent for the spherical radius of this spherical circle of curvature. Then

Eel(a) >_

kO ~ 1 + o5 +

o )2 + k202"

Equedity holds if and only if 12 is a spherical disk of radius ½a r c t a n ( 2 / k ) a n d 7 is a circular arc orthogonM to 017. Proof. For k = 0, the ease of spherical convexity, this result was established in [M1]. Since the proof for k > 0 parallels the case k = 0 and is similar to the proof of the analogous result for euclidean k-convexity [MM, Thm. 6], the details are omitted. • C o r o l l a r y 1. Suppose 12 is a spherically k-convex region, 3` is a hyperbolic geodesic in 12 and Zo E 3`. Then

Ik.(Zo, 3`)1 -<

2(1 - E~(zo) - kEo(zo)) E, (zo)(2 - kE, (Zo))

Equality implies that 17 is a spherical disk of radius ½arctan 2k and 7 is a circular arc orthogonal to 012. Proof. T h e r e is no loss of generality in assuming that ks(zo, 3`) > 0. Let F be the circle of curvature for 3` at z0. Suppose a is a spherical center of T' such that the spherical radius of F with respect to a is ~0 E (0, ~r/4]. Let 5 = co(a). T h e spherical geodesic through z0 and a meets 012 at some point c, so we have e~(Zo) <_ dt,(Zo, c) = de(zo, a) - dp(c,a) < ¢p - 5. If E = E~(zo), D = t a n ~ and F = t a n ~ , then this gives D < ( F - E ) / ( 1 + E F ) . The theorem gives kO 2

F - E - -


1+O 2+~/(1+O2) 2+k202-

-

I+EF"

This yields F _> V/(1 + E2 - kE)2 + k2E4 - (1 - E 2 - k E )

E(2 - k s ) so that

k s ( z o , 7 ) - I _ _ - F2 < 2 ( 1 - E 2 - k E ) F E(2 - k E ) If equality holds here, then equality holds in the preceding inequalities, so 12 is a spherical disk of radius ½a r c t a n ( 2 / k ) and 3' is a circular arc orthogonal to 012. • C o r o l l a r y 2. Let 1? be a spherically k-convex region. Then for z E 1"2

(i)

(t + Izl2)lVlog

(z)l <

2(1 - E ~ ( z ) - k E n ( z ) ) Ea(z)(2 - kEn(z))

127

Hyperbolic Geometry in Spherically k-convex Regions and

(1 + IzP)lVlog#,~(z)l _< 2x/[~(z)]~ : [1+ k#a(z)]

(ii)

with equMity if and only if $2 is a spherical disk of radius ½arctan(2/k). Proof. It is straightforward to show that equality holds in both (i) and (ii) for a spherical disk of radius i arctan(2/k). Because of the rotational invariance of the quantities involved, it suffices to establish (i) in the special case z = 0. In this case the proof is similar to that of Corollary 2(i) to Theorem 6 in [MM]. We establish (ii) as follows. Theorem 1 gives E o ( z ) >_

#~(z) + ~[~(z)]~

[1 + k#,(z)]

with equality if and only if $2 is a spherical disk of radius ½arctan(2/k). Since the function h(t) = 2(1 - t 2 - kt)/[t(2 - kt)] is strictly decreasing on its domain, we obtain

1

h(E~(z)) <_ h #a(z) + \/[#~(z)] 2 - [1 + kp~(z)]

) = 2~[#,(z)p - [1 + k#.(z)].

Thus, (i) implies (ii).

8. A p p l i c a t i o n s

to spherically

k-convex

mappings

We now apply our results to the study of conformal mappings of D onto spherically k-convex regions. K.W. Bauer in his thesis [Ba] investigated, among other topics, spherically k-convex mappings and found some of the results given in this section, especially those given in Theorem 6. T h e o r e m 5. Suppose $2 is a spherically k-convex region and f : D --+ $2 is a conformal mapping. Then If"(O)l < 211 - E~(f(O)) - kEa(f(O))]

lf,(0)P

< 2~/[Ea(f(0))] 2 - [1 + kEa(f(O))]

and equMity holds if and only if $2 is a spherical disk of radius ½arctan(2/k). Proof. Since l V l o g , , ( f ( 0 ) ) l = If"(O)I/lf'(O)P, the result follows from Corollary 2 to Theorem 4. •

C o r o l l a r y . I f f ~ g s ( k , a ) , then J/"(0)l < 2%/1 - ~ ( a only if f ( z ) = e-'e fk(dez) for some 0 C R.

+ k). Equality holds if and

Proof. Set $2 = f(D). Since f'(0) = a and # o ( f ( 0 ) ) = I/If'(0)[, the result follows from the theorem. Note that the only functions in Ks(k, a) which map onto a spherical disk of radius ½arctan(2/k) are the functions e-iefk(eiez) for some 8 e R. •

D. Mefia and D. Minda

128 T h e o r e m 6. f f f E Ks(k,a), then for z 6 D

if(z)if(z)

1 -2Ztzl~

2f(z)f'(z)l
(1

Izp)f*(z)[(1-1zl~)f*(z)+k].

EquMity holds at a point if and only if f(z) = e-~°fk(e~°z) for some 0 E R. Proof. For z = 0 this reduces to the preceding corollary. In order to establish the result at the point a E D, just replace f by ( z+a~ f \ 1 + ~z/ f(~) 9(z) = ( z+a~" -

1 + f(a)f \ \ /

This function is univalent in D, g(0) = 0, 9'(0) = (1 - N2)f'(a)/(1 + [f(a)l 2) and g(D) is spherically k-convex. If we apply the preceding corollary to this function g with a replaced by g'(0), then we obtain the desired result. If f(z) = e-~°fk(e~°z), then equality holds at z =re -i°, 0 < r < 1. • C o r o l l a r y 1. Let f be holomorphic and univalent in D and normalized by f(O) = 0 and if(O) = a > O. Then f e Ks(k, a) if and only if 1 + Re ( z f " ( z )

\ if(z)

2zf_(z)f'(z)~ > kf*(z)lzl 1 + If(z)l 2 ] -

for z E D.

Proof. First, suppose f E K~(k, a). Straightforward, but tedious, calculations show that the inequality of Theorem 6 implies the inequality of the corollary. On the other hand, suppose that the above inequality holds. Consider the path 7 : z = z(t) = re ~t, t 6 [0,2~]. Since k~(z,7) = (1 -[z[2)/[z[, we obtain

(zS,,(z) 1 + Re ~ if(z)

k / f ( z ) , f o 7) =

1 + If(z)l ~ ]

izf#(z)[

Therefore, k,(f(z), f o 7) >- k for all z E 7, so f ( { z : lzl < r}) is a spherically k-convex region by Proposition 1. Then f(D) is also spherically k-convex since it is an increasing union of spherically k-convex regions. • C o r o l l a r y 2. Suppose f is holomorphic and univalent in D. Then f(D) is sphericedly k-convex if and on/y if f maps each subdisk o l d onto a sphericalIy k-convex region.

References

[Ba]

K.W. Bauer, Uber die Ab~chdtzung yon Lbsungen gewisser partieller Differentialgleichungen yore elliptischen Typus, Bonner Mathematische Schriften, 10, Bonn, 1960.

Hyperbolic Geometry in Sphericedly k-convex Regions

129

[B] W. Blaschke, Uber den gr~flten Kreis in ether konvcxen Punktmenge, Jahresber. Deutsch. Math. Verein. 23 (1914), 369-374.

[¢1 H.W. Guggenheimer, Differential Geometry, McGraw-Hill, New York, 1963. [MM] D. Mejla, D. Minda, Hyperbolic geometry in k-convex regions, Pacific J. Math. (to appear). [M1] D. Minda, The hyperbolic metric and Bloch constants for spherically convex regions, Complex Variables Theory Appl. 5 (1986), 127-140. [M2] D. Minda, A reflection principle for the hyperbolic metric and applications to geometric function theory, Complex Variables Theory Appl. 8 (1987), 129-144. [M3] D. Minda, Applications of hyperbolic convexity to euclidean and spherical convexity, J. Analyse Math. 49 (1987), 90-105. [S] J.J. Stoker, Differential Geometry, W'iley-Interscience, 1969. Received: May 14, 1989

Computational Methods and Function Theory Proceedings, Valparaiso 1989 St. Ruscheweyh, E.B. Sail', L. C. Salinas, R.S. Varga (eds.) Lecture Notes in Mathematics 1435, pp. 131-142 (~) Springer Berlin Heidelberg 1990

The

Bloch

and

Marden

Constants

D a v i d M i n d a 1' 2 D e p a r t m e n t of Mathematical Sciences, University of Cincinnati Cincinnati, Ohio 45221-0025

1. I n t r o d u c t i o n Suppose F is holomorphic in the open unit disk D. For a E D let r(a, F) denote the largest nonnegative n u m b e r r such that there is a simply connected region 1"2, with a E t2 C D, which is m a p p e d conformally onto D(F(a), r) b y F. (Here we use the notation D(b, r) for the open euclidean disk with center b and radius r. For b = 0 we simplify this to D(r).) Set r(F) = sup{r(a, r ) : a e D}. Clearly, r(a, F ) = 0 if and only if F'(a) = O. T h e Bloch constant B is defined by B = i n f { r ( F ) : F is holomorphic in D and F ' ( 0 ) = 1}. In 1924 Bloch proved that B is positive in spite of the vast collection of functions over which the infimum is taken. Landau ([L1],[L2]) gave various lower bounds for the Bloch constant the best of which is B > 0.396. Landau also showed that B = i n f { r ( F ) : F e B,

IIFII

= 1 and F ' ( 0 ) = 1},

where B is the class of all holomorphic functions F defined on D such t h a t F ( 0 ) = 0 and

IIFll = sup{(1 -Iwl2)lF'(w)l : w • D} <

~.

T h e normalization F(0) = 0 is not really essential but it is convenient. L a n d a u also gave the u p p e r b o u n d B < 0.555. Ahlfors and Grunsky lAG] presented a geometric t r e a t m e n t of this example of Landau and noted that Landau did not get the best possible u p p e r b o u n d from his example. T h e y improved the upper bound to 1Research partially supported by NSF Grant No. DMS-8801439. 2I want to thank the University of Cincinnati and the Taft Foundation for partially supporting my sabbatical leave during the 1988-89 academic year and the University of California, San Diego for its hospitality during this period.

D. Minda

132 B _< .4719... = F(1/3)F(11/12)

~/1 + v~r(U4)" In addition, they conjectured that this upper bound is the correct value of the Bloch constant. Ahlfors [All developed a powerful differential-geometric method, now called Ahlfors' Lemma and the method of ultrahyperbolic metrics, and, as one application of this method, he showed B _> v/3/4. Later, Heins [H1] introduced the notion of an SK-metric as a generalization of an ultrahyperbolic metric and developed a theory of these metrics, noting the parallel between the theories of subharmonic functions and SK-rnetrics. In particular, he obtained a sharp form of Ahlfors' Lemma; an application is the improvement of Ahlfors' lower bound to B > v~/4. Pommerenke [P], essentially using a function-theoretic version of Ahlfors' Lemma, gave another proof of Heins' lower bound for the Bloch constant and also presented a lower bound for the locally schlicht Bloch constant. Minda [M1] employed Ahlfors' differential-geometric method to study various Bloch constants. Peschl ([Pc1], [Pe2]) obtained a number of results about Bloch constants for families of locally schlicht functions by a different method. Thus, in the fifty years since Ahlfors' original paper, there has been little progress on improving the lower bound for the Bloch constant. Recently, Bonk [B] introduced a new technique in the study of the Bloch constant. His method yields a sharp lower distortion estimate on ReF' when F C B, []F][ = 1 and F'(0) = 1. This inequality immediately gives B >_ v ~ / 4 ; intriguingly, the same lower bound that Ahlfors obtained by a different method. By making a more careful use of this distortion theorem, Bonk obtains B > V~/4 + 10 -14, the first quantitative improvement on the lower bound for the Bloch constant in a half century. There is another, related constant defined for Bloch functions. This constant is defined relative to the domain of the function while the Bloch constant is defined relative to the range. For a C D let s = s ( a , F ) E [0,+oo] be the largest nonnegative number such that F is univalent in the hyperbolic disk Dh(a,s). Recall that dh(a,w) = 2 a r t a n h ( l w - al/[1 - 8w[) is the hyperbolic distance between the points a, w E D and the hyperbolic disk with center a and hyperbolic radius s is Oh(a,s) = { w e D : d h ( a , w ) < s}. Note that s ( a , F ) = 0 if and only if F'(a) = 0 and s ( a , F ) = +co precisely when F is univalent on D. Thus, s ( a , F ) is the hyperbolic radius of the largest hyperbolic disk centered at a in which F is univalent. Set s ( F ) = sup{s(a, F ) : a E D}. The Marden constant is defined by M = inf{s(F) : 0 < []F[[ < oe} = i n f { s ( F ) : [[F[[ = 1}. The latter equality holds since s ( F / m ) = s(F) for any positive constant m. The Marden constant was introduced in [M3] and named for its analogy with various Marden constants for Fuchsian groups. (The reader is warned that the pseudohyperbolic, rather than the hyperbolic, distance was employed in [M3].) Minda [M3] gave the bounds ~ + V~ - 1 . 4 0 1 . . . . 2 artanh ~1 = 1.098... _< M < 2 artanh ~/1

The upper bound is conjectured to be sharp and is obtained from the same function that Ahlfors and Grunsky conjecture is extremal for the Bloch constant. The distortion

The Bloch and Marden Constants

133

theorem of Bonk immediately gives the improved lower bound M k 2 artanh 1/x/~ = 1 . 3 1 6 . - . . Thus, from a single distortion theorem Bonk obtains good lower bounds for both the Bloch and the Marden constants. The purpose of this paper is to present a different, geometric proof of Bonk's distortion theorem. This new proof yields additional information and seems likely to extend to other situations. In particular, all of the extremal functions for Bonk's distortion theorem are obtained from this new proof; the extremal functions are two-sheeted branched coverings of D onto another disk. Also, I will give a geometric significance for the constants v ~ / 4 and 2 artanh l/v/3; they give the sharp solution to two geometric problems related to the Bloch and Marden constants. Finally, the extremal functions for Bonk's distortion theorem are related to the ultrahyperbolic metric that is employed in the proof that B > v ~ / 4 via Ahlfors' method.

2. M a i n r e s u l t s In this section we present a new, geometric proof of Bonk's distortion theorem. The following facts will be used several times. Suppose F(w) is holomorphic in D, w = S(z) is a conformal automorphism of D and G = F o S, then IIFI] = HG I]. In fact, the identity

IS'(z)l

1

-

-IS(z)l~ 1 -Izl ~

yields the stronger pointwise result

(1 -Izl=)lG'(z)l

=

(1 -Iwl~)lF'(w)l .

Also, r(S(a), F ) = r(a, G) and s(S(a), F) : .s(a, G) for all a e D. We begin by discussing the functions that are extremal for Bonk's distortion theorem. E x a m p l e 1. Set

fl(w)

1 - v~w

f i (n2

-

l)w"

-

and

Fl(w) =

// fl(w)dw.

Note that f~(0) -- 1 and FI(0) = 0, F~(0) = 1. We wish to determine the explicit values of the quantities IIF1If, r(0, F1) and s(0, F1). In order to do this, it is advantageous to express both fl and F1 in another fashion. Set 1 1 w--z+~_ z=T(w)= x/~ and w = S ( z ) - - T - l ( z ) :

D. Minda

134 Note that T and S are conformal automorphisms of D and T'(w)

2

-

Consequently, fl(w) = - ( 3 v ~ / 2 ) T ( w ) T ' ( w ) and F~(w) -- -(3v/3/4)[T(w) 2 - (1/3)]. Define G, = F~ o S. Then tlFlll = Ilaltt, a~(z) = - ( 3 v ~ / 4 ) [ z 2 - (1/3)1 and

(1 -lzl=)lai(z)i

= 3--~--~(1 -Izl=)lzl.

The function h(t) = (1 - t2)t vanishes at t = 0, 1 and attains its maximum value of 2 / 3 v ~ on the interval [0,1] uniquely at the point t = 1/v~. Thus, [IGlll = 1 and (1 - I z l = ) l a i ( z ) l = 1 if and only if Izl = I/x/g. This implies that IIFlll = 1 and (1 - Iwl~)lF£(w)l = 1 if and only if w lies on the circle through 0 and x/~/2 that is symmetric about the real axis. Next, we determine r(0, F1) and s(0, F~) geometrically. We observe that r(0, F1) = r ( - 1 / v ~ , G1) and s(0, F1 ) = s ( - 1 / v ~ , G1). Clearly, the function G1 is a two-sheeted branched covering of D onto the disk D ( v ' 3 / 4 , 3 v ~ / 4 ) with G~(0) = x/-3/4 and Gi(0 ) = 0. Thus, the branch point closest to G ~ ( - 1 / v ~ ) = 0 is v ~ / 4 , so that r ( - 1 / v ~ , G 1 ) = v ~ / 4 . Also, G1 is univalent in the half-plane {z : R e z < 0} and a i ( 0 ) = 0, so , ( - 1 / , / g , 31) = 2artanh l/v/3, the hyperbolic distance between - 1 / v ~ and the origin. We also give analytic proofs of the facts r(0, F~ ) = v ~ / 4 and s(0, F1) = 2 artanh l v @ Since (~ 1)Re(w ~)

> 1- ~

(~= -

1)lw~l

= f~(Iwl) = F~(lwJ) > 0

if [w[ < 1/v/3

and F [ ( 1 / v ~ ) = f~(1/x/~) = 0, the Wolff-Warschawski-Noshiro Theorem shows that F1 is univalent in D(1/v/-3) but in no larger disk. Thus, s(0, F1) -- 1/v/-3. Next, we show that F1 maps D ( 1 / v ~ ) conformally onto a region that contains the disk D ( v ~ / 4 ) . For w = e i ° / v ~ we have ,w

,i/,a

El(w) = ]o F;(w)&o = I

.

F;(re'°) ie'°dr,

ao

so that IF~(~)l >_ Re F1(w) ieio -- fo 1/'/5 R e F ~ ( r e i ° ) d r >_ f ~/v'5 F ; ( r ) d r = Jo

El(I/v/3) -- v/3/4.

Thus, F I ( D ( 1 / v ~ ) ) D D ( v ~ / 4 ) . Since F l ( 1 / v ~ ) = v ~ / 4 and F~(1/v~) = 0, it follows that r(0, F1) = v/3/4.

The Bloch and Marden Constants

135

It is not difficult to show that r(F) = 3 v ~ / 8 and s(F) = +co. T h e o r e m 1. Suppose that s > O, la[ = s and g is holomorphic and nonconstant on D(s) U {a}. If [g(z)] _< lg(a)l for z • D(s), then c = ag'(a)/g(a) is positive and Re~g(ta)~ > (c+l)t-(c-1) [ g(a) J - (c + l ) - ( c - 1 ) t '

t•(-1,1),

with equedity for some t • (--1, 1) if and only if g(z) = g ( a ) ~c + 1)z -- (c -- 1)a + 1)a -~---1-~" Proof. We start by reducing the general case to the special case in which a = 1 and 9(a) = 1. Set h(z) = g(az)/g(a). Then h is holomorphic in D U {1}, h(D) C D, h(1) = 1 and c = h'(1). Since h(D) C D and h(1) = 1, elementary geometric considerations show that c > 0. Set c-1 c+1 _ (c+l)z-(c-1) V(z) c- 1 (c + 1) - ( c - 1)z" 1z c+l Note that U is a conformal automorphism of D, U(1) = 1 and U'(1) = c. We need to prove that Reh(t) > U(t), t E ( - 1 , 1), with equality if and only if h = U. Set k = U - ' oh. Then k is holomorphic in DU{1}, k(D) C D, k(1) = 1 and 1 = k'(1). Recall that for r e (0, oz) Z

--

-

-

I1 - zl ~ < r} 1 A ( 1 , r ) = {z e D : {zl------1_ 7 =D(l+r,l+r

)

is a horodisk in D based at 1; that is, an open disk in D that is internally tangent to the unit circle at 1. Observe that U(Zl(1,r)) = Zl(1,cr). Julia's Lemma ([A2, pp. 7-9], [C, pp. 23-28]) asserts that for z E D [ 1 - k ( z ) [ ' < ik,(1)11117 ; ; _

II-z] =

1 -IKz)l = -

i -Iz[

I

2'

with equality for some z E D if and only if k = Rb for some b E R, where

Rb(z)

=

(I + z) - (1 + ib)(1 - z) (i + z)+~--ib)(1 z)"

Note that Rb is a conformal automorphism of D that fixes 1 and maps each horodisk A(1, r) onto itself. Julia's Lemma has an elegant geometric interpretation: either k = Rb for some b • R or else k(A(1, r)\{1}) C A(1, r) for all r • (0, oo). Thus, either h = UoRb for some b • R or else h ( A ( i , r ) \ { 1 } ) c U(A(1,r)) for all r • (0,oo). Given t • ( - 1 , 1 ) , select r = (1 - t ) / ( 1 + t ) • (0, oo). Then A(1,r) is the horodisk in D whose boundary meets the real axis in the points t and 1. From the inclusion h(A(1, r)) C U(A(t, r)) we obtain Reh(t) >_ U(t) with equality for some t if and only if

D. Minda

136

I m h ( t ) = 0 a n d h(t) = U(t), t h a t is, k(t) = t. But then equality holds in Julia's L e m m a , so k = Rb for some b • R. But if t = Rb(t) for some t • ( 1 - , 1 ) , then it follows t h a t b = 0. Since/go is the identity function, we deduce h = U. C o r o l l a r y . Suppose s > 0, [a] = s and g is holomorphic and nonconstant on D(s) U {a}. u b(z)t _ b(a)L for z • D(s) and aV(a) = g(a), then

R e { g(ta)~g(a)j >-t'

t•(-1,1),

with equMity for some t • ( - 1 , 1) if and only if g(z) = g(a)z/a. R e m a r k . For the p r o o f of T h e o r e m 1 we only require the weak f o r m of Julia's L e m m a in which g is assumed to be analytic at z = 1, not the general result dealing with the a n g u l a r derivative at z = 1. There is a simple p r o o f of Julia's L e m m a in this special case in [PS, prob. 292, p. 141]. Now, we give a geometric proof of Bonk's distortion t h e o r e m with a careful analysis of the sharpness. T h e o r e m 2. Suppose F • 13, [IFII = 1 and F ' ( 0 ) = 1. Then a e F ' ( w ) _> F;(Iw[) for [w[ < v~/2 with equality at re 'e # 0 if and only if F(w) = e'eFl(e-'ew).

Proo£ T h e r e is no h a r m in assuming t h a t w = u E (0, 1); if not, t h e n consider e-leF(eiew) to treat the case in which the point is re ie. Note t h a t 1

11 + F"(O)w + . . . I = IF'(w)l -< 1 - Iwl ~ - 1 + Iwl = + . . . implies F"(O) = O. Consider any a , 0 < a < 1. Set w = S~(z) = (z + a)/(1 + az) and G~ = F o S~. T h e n IIFII = ]IG=]I and (1 - I z [ 2 ) l G ' ( z ) l = 1 for z = - a since S~(-a) = 0 and F'(O) = 1. Thus, for ]z[ _< a we have 1

1

[G:(z)J <_ -1- - qI z l

<<- 1 - a 2 -[G',~(-a)[.

In fact, direct calculation reveals G ' ( - a ) = 1/(1 - a 2) and G"a(-a) = - 2 a / ( 1 - a2) 2, so that

aG"(-a) a'(-a)

c-

2a 2

- l-a2"

T h e preceding t h e o r e m applied to g = G" yields

R G'(-at) e

~

(c + 1)t - ( c - 1) (1 - 3a 2) + (1 + a2)t > (c+l)-(c-1)t = (l+a')+(l-aa2)t '

or

Re{F'(S=(-at))S'=(-at)} >_

(1 -

(1

-

3a 2) + (1 + a2)t

aa)[(1 + a 2) + (1 - 3a=)t] '

with equality for some t • ( - 1 , 1) if and only if

t•(--1,1),

• (-1,1),

The B1och and Marden Constants

137 1

(c + 1 ) z - ( c -

1)(-a)

G'a(z) = 1 - a ~ (c 4- 1 ) ( - a ) - (c - 1)z" Set u = Sa(--at), or t = (a - u)/a(1 --au). Observe that Sa maps the interval ( - a , a ) onto the interval (0, 2a/(1 + aS)) and that -

S'=(-at)

1

au) 2 1 - as

_ (1 -

(S:')'(u) Therefore, (*)

ReF'(u) >

(1

-

2a- (l+3a2)u au)2[2a (1 a2)u] '

u e (0, 2a/(1 + ab).

-

-

This implies that ReF'(w) >

2a - (1 + 3a~)lwl (1 -

alwl)~[2a

-

(1 -

a2)lwl]'

Iwl < 2a/(1 + a2).

Thus, ReF'(w) > 0 for lwl < 2a/(1 + 3ab. The choice a = 1/x/~ produces the largest disk on which a e F ' ( w ) > 0 and also yields ReF'(w) >_ F~(lwl) for lwl < v ~ / 2 . Note that for a = 1/v/3 we have c = 1 and so strict inequality holds in (*) for u E (0, V"3/2) unless a',(z) = - 3 v / 3 z / 2 , or equivalently, G~(z) = a , ( z ) , and so F ( w ) = El(W). Here G1 refers to the function in Example 1. Conversely, if F = F1, then for a = 1/yr3 equality holds in (*) at each point of the interval (0, v~/2). C o r o l l a r y 1. Suppose Y e I3, IlVll = 1 and F'(O) = 1. Then r(O, F) > v/3/4 and s(0, F ) ~ 2 a r t a n h 1/x/~ with strict inequality unless F ( w ) = e~°Fl(e-l°w) for some ~ER.

Proof. Suppose F ( w ) • eieF,(e-'ew) for all 0 e R. Then ReF'(w) > F{(lwl) > 0 when Iwl < v ~ / 2 . In particular, min{aeF'(w) : Iw[ = 1/v~} > 0. Thus, there exists s > 1 / v ~ such that ReF'(w) > 0 on D(s). The Wolff-Warschawski-Noshiro Theorem implies that F is univalent on D(s), so s(0, F) k 2 artanh s > 2 artanh 1/V~. Also, for = d U 4 5 we have iF(w)l > r~e ~ ] -

=

ReF'(re'e)dr > J o

F~(r)dr

4

Hence, min{lF(w)t : Iwl = 1 / 4 5 } > v ~ t 4 , so there exists r > V~14 such that F ( D ( l l v / 3 ) ) contains D(r). Because F is univalent in D(lfv/3), this implies that r(0, F ) > x/~/4. • R e m a r k . For bounded analytic functions there is a sharp analogy of Corollary 1 due to Landau, see ([H2, pp, 36-39]). The extremal functions for Landau's result are two-sheeted branched coverings of O onto itself. This result of Landau is the basis for one elementary proof of a lower bound for the Bloch constant, see ([H2, pp. 46]). C o r o l l a r y 2. B > v ~ / 4 and M > 2 artanh 1/v~.

D. Minda

138

P r o o f Let B1 = { F : F C B, t]F]I = 1 and F'(0) = 1}. The family B1 is a compact normal family. Robinson [R] showed that there exist extremal functions for the Bloch constant, that is, functions F E B1 with r ( F ) = B. Consider any such extremal function F. If F(w) ¢ ei°Fl(e-i°w) for all 0 E R, then r(F) > r(0, F ) > v ~ / 4 by the preceding corollary. On the other hand, if F(w) = e'°Fl(e-i°w) for some 0 e R, then r(F) = 3x/3/8 by Example 1. Thus, in either case, B > x/~/4. Next, we show M > 2 artanh 1/x/~ by a similar argument. Consider any sequence {F~} in B1 with s(F~) --~ M. Because B~ is a compact normal family, we may assume that Fn ~ F E B1, where the convergence is uniform on compact subsets of D. If F is univalent on Dh(a, s), then for each e > 0 there exists N such that for all n _> N, F~ is univalent on Dh(a, s - e). Hence, for n > N, s(F~) >>_s(a, F,~) >_ s(a, F ) - e. By letting n tend to infinity we obtain M > s(a, F) - e. But e > 0 is arbitrary, so M > s(a, F). This yields M >_ s(F), so M = s(F). This demostrates that there are extremal functions for the Marden constant. The remainder of the proof that M > 2 a r t a n h 1/x/3 is now analogous to the proof in the preceding paragraph and is omitted. •

3. G e o m e t r i c

interpretation

Suppose F is holomorphic and nonconstant on D. Let X = X ( F ) be the Riemann image surface of F viewed as spread over the complex plane C. In connection with the Bloch constant problem, it is natural to inquire about the possible location of the largest schlicht disk on X. One plausible guess is that the largest schlicht disk would be centered at a point where the hyperbolic metric attains its minimum value. For a plane region it makes sense to speak of the minimum of the density of the hyperbolic metric, but for a Riemann surface it makes no sense to speak of the value of a metric at a point of the surface. However, for a Riemann surface spread over C there is a. natural way to define a density for the hyperbolic metric. This density will be infinite at all branch points and at all finite boundary points. We will show that a minimum point for this density is always the center of a relatively large schlieht disk, but generally not the largest such disk on the surface. In fact, the work of Bonk yields a sharp lower bound for the radius of a schlicht disk centered at such a minimum point; this is the geometric significance of x/3/4. The result of Bonk also gives the best possible lower bound on the hyperbolic radius of a hyperbolic disk centered at a minimum point which is one-sheeted when viewed as spread over C. This is a geometric interpretation of 2 artanh 1/v/3. This is the same as the lower bound on the hyperbolic distance from a minimum point to the nearest branch point. We begin by making precise the notion of the Riemann image surface and establish some notation. Precisely, X = {(w, F(w)) : w C D} is the graph of F in C x C. Define the two natural projections of X onto the coordinates planes: ~h:X~D

1r2:X~C

~rl(w, F(w)) = w,

7r2(w, F(w)) = F(w).

The surface X is endowed with the unique conformal structure that makes both 7rl and r2 analytic functions. Note that 7h is a conformal mapping of the simply connected

The Bloch and Marden Constants

139

Riemann surface X onto D. The set of branch points of X is b(X) = { ( w , F ( w ) ) : F'(w) = 0}. The function ~r2 can be used as a local coordinate at each point of X\b(X). For convenience, let 7r~-1 = / ~ : 13 ~ X,

~'(w) = (w, F(w)), and observe that 7r2 o ~" = F. There are two natural metrics and associated distance functions on X. First, there is the hyperbolic metric )~x(w)ldwI and the associated distance function dx. Recall that the hyperbolic metric on X is the unique conformal metric on X whose pull-back under any conformal mapping of D onto X is the hyperbolic metric on D; in symbols,

=

D(w)ld

2jdwl , l = 1-7:

where /kD(w)IdwI is the ordinary hyperbolic metric on D with curvature -1. Second, there is the pull-back of the euclidean metric via rr2. Explicitely, ~c(¢)]d~l = 1IdOl is the euclidean metric on C and rr~(~c(~)ld¢[) is a metric on Zkb(X), but just a semi-metric on X itself. In other words, ~r;(~c(~)ld~l) is a continuous, nonnegative linear density on X which vanishes precisely at the branch points. Because the zeros of this semi-metric are isolated points on X, it induces a distance function d on X that is compatible with the topology of X. Note that r(a, F ) is the minimum of the distance from (a, F(a)) to b(X) and the distance from (a,F(a)) to the ideal boundary of X relative to the distance function d. Crudely speaking, it is the radius of the largest disk on X, relative to the distance function d, centered at (a, F(a)) which does not contain a branch point or meet the ideal boundary of X. Similarly, let t(a, F) be the minimum of the distance from (a, F(a)) to b(X) and the distance from (a,F(a)) to the ideal boundary of X relative to the distance function dx. In fact, the latter distance is always infinite, so t(a, F) is just the dx-distance from (a, F(a)) to the set of branch points. The geometric quantity t(a, F) was introduced and studied in [M2]. Also, observe that s(a, F) is the radius (relative to dx) of the largest disk centered at (a,F(a)) in which ~r2 is univalent. In other words, it is the radius of the largest hyperbolic disk centered at (a, F(a)) which is one-sheeted when viewed as spread over the complex plane. Clearly, s(a, F) < t(a, F). The quotient ~x(w)ld~l =

= 7r

(Ac(¢)ld¢l)

of the hyperbolic metric by the pull-back of the euclidean metric defines a positive, continuous function on X which is infinite at each branch point. The function # is the density of the hyperbolic metric when the hyperbolic metric is expressed in terms of the local parameter 7r2. Observe that ~(w, F(w)) = , ( P ( w ) ) =

F*(:~x(~)ld~l) F*(~r~(Ac(C)tdCI))

D. Minda

140

P*(;~x(~)ld~l) F'(.Xc(C)IdCI)

~D(w)ldwI IF'(~)lldwl

Here we have used F* = (~r~ o F)" = / ~ " o 7r] and

2 (X - IwP)lF'(w)l"

P*(Ax(w)ldwl)-- Ao(w)ldwl.

Set

m = re(X) = inf{u(w, F ( w ) ) : (w, F(w)) E X } . From the preceding work it is clear that re(X) = 2/ItFIf. Also, # attains its minimum value rn at the point (a, F(a)) if and only if (1 - la]:)lF'(a)l = I]FII. We can now give a geometric version of Corollary 1 of Theorem 2. T h e o r e m 3. Suppose F is holomorphic in O and (a, F(a)) is a minimum point for #x. Then r(a, F ) m ( X ) >_ v ~ / 4 and t(a, F) > s(a, F ) _> 2 artanh 1 / v ~ . Both of these inequalities are sharp. Proof. We m a y assume that a = 0 and F'(0) > 0. If not, then replace F by AF o S, where A is a unimodular constant and S(z) = (z + a)/(1 + az). Notice that we have m ( X ( F ) ) = m ( X ( A F o S)). Thus, we are assuming F'(0) = ][F]]. Then the function F/I]F]] satisfies the hypotheses of Corollary 1 to Theorem 2, and so the conclusions of Theorem 3 follow immediately. Note that equality holds in all the inequalities for the function/;'1. • E x a m p l e 2. It is interesting to look at this geometric interpretation in the special case of the function F1. The e×tremal function F1 has an intimate connection with the ultrahyperbolie metric used in the proof that B > V~/4 via Ahlfors' method. This sheds light upon why Ahlfors' method and Bonk's distortion theorem yield the same lower bound for the Bloeh constant. Actually, it is simpler to consider the function F0 = FI - v ~ / 4 . Then F0 is a twosheeted branched covering of D onto D(3v/3/4) and we let X0 denote the Riemann image surface of F0. The associated function is Go(z) = Fo o S(z) = - ( 3 v ~ / 4 ) z 2. The function # satisfies 2

2

#(w, Fo(~)) = (1 -f~P)IF,~(w)I = (1 -t=l~)lC~,(z)t If Fo(wx) = Fo(w2), then ao(T(w,)) = Co(T(w2)), so T(w,) = -T(w2). This implies that G'o(T(wl)) = -G'o(T(w2)), so #(w~, Fo(Wl)) = #(w2, Fo(w2)). In brief, the function # is invariant under the sheet-interchange function for X0, so # induces a function a on D(3vf3/4). We explicitely determine a. If ; = Go(z) = Fo(w), then

o(C0(z)) = ~(F0(w)) = o ( ~ ( w , F0(w))) = , ( ~ , Fo(w)) = For ( = Go(z) = - ( 3 v ~ / 4 ) z ~ this gives

o-(C) =

4

i¢1,;~ (3-~-~ - tCI)

(~ - I z P ) l a ~ , ( z ) l

The Bloch and Marden Constants

141

a(C)IdC[ is a conformal metric on the punctured disk D(av~/4)\{0} with constant Gaussian curvature -1. Now, the connection with Ahlfors' method becomes apparent. We employ the notation of [M1, §5]. In Ahlfors' method the appropiate ultrahyperbolic metric is constructed from the conformal metric

v TIdq ,,,,4QId¢l

= i 11/2(3 T

_

I¢1)

(Actually, this metric is twice the metric that appears in [M1, §5]; the reason for this factor of two is that ;~c(~)ld~l was taken to be twice the euclidean metric in [5].) This is a conformal metric on D(3r)\{0} which has constant Gaussian curvature -1. The proof that B > v ~ / 4 via Ahlfors' method uses the value r = v~/4. But for this choice of r, O'1, r =

G.

References [A1] L.V. Ahlfors, An extension of Schwarz's lemma, Trans. Amer. Math. Soc. 43 (1938), 359-364. [A2] L.V. Ahlfors, Conformal invariants: Topics in geometric function theory, McGraw Hill, New York, 1973.

[AG] L.V. Ahlfors, H. Grunsky, ~fber die Blochsche Konstante, Math. Z. 42 (1937), 671-673. [B] M. Bonk, On Bloch's constant, Proc. Amer. Math. Soc. (to appear). [C] C. Carathdodory, Theory of functions of a complex variable, vol. II, 2nd English ed., Chelsea Publishing Co., New York, 1960. [H1] M. Heins, On a class of conforraal metrics, Nagoya Math. J. 21 (1962), 1-60. [H2] M. Heins, Selected topics in the classical theory of functions of a complex variable, Holt, Rinehart and Winston, New York, 1962. ILl] E. Landau, Der Picard-Schottkysche Satz und die Blochsche Konstante, Sitzungsber. Preuss, Akad. Wiss. Berlin, Phys.-Math. K1.32 (1926), 467-474. [L2] E. Landau, Uber die Blochsche Konstante und zwei verwandte Weltkonstanten, Math. Z. 30 (1929), 608-634. [M1] D. Minda, Bloch constants, J. Analyse Math. 41 (1982), 54-84. [M2] D. Minda, Domain Bloch constants, Trans. Amer. Math. Soc. 276 (1983), 645-655. [M3] D. Minda, Marden constants for Bloch and normal functions. J. Analyse Math.

42 (1982/s3), n7-127.

D. Minda

142

[Pel] E. Peschl, Uber die Verwendun9 yon Differentialinvarianten bei gewissen Funktionenfamilien und die Ubertragung einer darauf gegrgndeten Methode auf partielte Differentialgleichungen yore elIiptischen Typus, Ann. Acad. Sci. Fenn. Ser. AI

3a6/6 (1963), 23 pp. [Pe2] E. Peschl, Uber unverzweigte konforme Abbildungen, Osterreich. Akad. Wiss. Math.-Naturwiss. K1. S.-B. II 185 (1976), 55-78. [PS] G. P61ya, G. SzegS, Aufgaben und Lehrs~tze aus der Analysis, Bd. 1, Die Grundlehren der math. Wissenschaften in Einzeldarstellungen, Bd. 19, Springer-Verlag, New York, 1964. [Po] Ch. Pommerenke, On Bloch functions, J. London Math. Soc. (2) 2 (1970), 689-695. [R] R.M. Robinson, Bloch functions, Duke Math. J. 2 (1936), 453-459. Received: March 18, 1989

Computational Methods and Function Theory Proceedings, Valpara/so 1989 St. R,uscheweyh, E.B. SalT, L. C. Salinas, R.S. Varga (eds.) Lecture Notes in Mathematics 1435, pp. 143-154 (~) Springer Berlin Heidelberg 1990

On some analytic and computational aspects of two dimensional vortex sheet evolution O. F. O r e l l a n a 1 D e p a r t a m e n t o de Matemgticas Universidad T6cnica Federico Santa Maria Valpara/so, Chile

A b s t r a c t . This survey paper gives an account of recent analytic and numerical results of the initial value problem:

~(~,t) = Z(7,0)

=

'

~

z(.~,O-z(.~,,t)

,

7 + S(7),

which is the Birkhoff-Rott equation for the evolution of a sligthly perturbed fiat vortex sheet. We will indicate some open problems of current research and propose a new physically desingularized Vortex sheet equation, which agrees with the finite thickness vortex layer equations in the localized approximation.

Introduction T h e r e exist two main motivations to study the problem we are concerned a b o u t in these notes: in three dimensions, it is an important unsolved problem of m a t h e m a t i c a l fluid dynamics to determine whether solutions of the Euler equations develop singularities in finite time, with smooth initial data. This problem is i m p o r t a n t for two reasons: from the physical point of view because the existence of such singularities is connected with the onset of turbulence in a high Reynolds number regime, and, from the numerical point of view because the existence of such singularities will generate instabilities in the calculations. Since, this seems to be a hard problem that has been studied through different numerical methods by several authors it is not a bad idea to try a simpler problem namely: show that initially singular solutions can become more singular after a finite 1The author was economically supported by FONDECYT (grant number 235, 1987-88) and Universidad T$cniea Federico Santa Maria (grants 88.12.08, 1988 and 89.12.08) 1989).

144

O . F . Ore//ana

time. Specifically, suppose the velocity field has a symmetry with respect to one of the axes so that the velocity field is two dimensional and has a discontinuity along a smooth curve (for instance a vortex sheet). Does the initially analytic curve stay analytic for all time or develop a singularity in finite time? A number of authors have shown through analytic a n d / o r numerical techniques the appearance of a singularity in finite time (see [l], [21, [31, [41, [51, [71). The second motivation to study the Birkhoff-Rott equation as a mathematical model for the evolution of a vortex sheet is because of its applications to aerodynamics and consequently to the design of more secure and economic airplanes through the calculation of lift of airfoils, the design of airfoils and knowledge and control of the region of turbulence (see [8]). A mathematically rigorous deduction of the Birkhoff-Rott equation can be found in [9] and [10]. This review article describes mathematical analysis results about the evolution of a slightly perturbed flat vortex sheet, because of its importance to the numerics and computation of vortex sheet evolution. Hence we will also comment on the numerical methods to solve Birkhoff-Rott equation proposed by R. Krasny (see [5] and [6]) and the proof of convergence of such methods given by R. Cafiisch and J. Lowengrub in [17]. We also mention open questions of current research in the appropiate place. In the last section of this paper we comment on a new physically desingularized vortex sheet equation proposed by Caflisch, Orellana and Siegel [14] as an alternative to the desingularized equation proposed by R. Krasny [6].

1. Instability analysis and singularity formation for vortex sheets. We will call a 2-dimensional vortex sheet, a discontinuity curve in a fluid domain, which moves with velocity equal to the average of the velocities on its two sides, and across which the tangential velocity, but not the normal velocity, is discontinuous. The jump in tangential velocity across the sheet in a given point is called vortex sheet strength. A good example is provided by the flow field with velocity components: (1)

(u,v)(x,y):=

/ (-½,0)

if y > 0

t

ify<0

(+½,0)

which is a steady weak solution of Euler equations. The flat interface F := {(x,y) E R 2 : y = 0} defines a vortex sheet of uniform strength equal to one, which moves with velocity equal to zero. If we represent the vortex sheet and its corresponding curve in the complex plane by Z ( % t) = X ( 7 , t) + i Y ( 7 , t), where t is the time variable and 3' is the Lagrangian parameter which measure the circulation between a reference point (3' = O) and an arbitrary point on the sheet, then the evolution of the vortex sheet is governed by the singular-integrodifferential equation:

On Some Analytic and ComputationM Aspects ... (2)

O----~(%t) = ~

145

I f_~~ Z(%t)---Z(.y',t) d7'

where the bar on the left side denotes complex conjugate and the slash on the integral sign denotes Cauchy principal value due to the singularity of the integraad at 7 = 7'. Moreover the vortex sheet strength a(7, t) is the jump in tangential velocity across the sheet at Z(7 , t) and it is determined up to sign by:

0Z -1 a(7, t) = Or

(3)

Moreover Z = 7 defines a flat vortex sheet of uniform strength equal to one which is the steady solution of (2) corresponding to the weak steady solution (1) of Euler equations. Equation (2) was derived using the Biot-Savart law by Birkhoff [9] and in a more mathematically rigorous way by Sulem, Sulem, Bardos and Frisch [10]. Therefore the nonlinear evolution of a vortex sheet with a small initial perturbation is given by the initial value problem:

OZ (4)

t

1 /~o

Ot ( % )

=

2~ri

z(7,0)

=

7+s(%0)

dT,

oo Z(v,t) - Z ( f ' , t )

It is well known that the fiat 2-d vortex sheet Z(7, t) = 3, is linearly unstable to small, analytic disturbances. When a slight disturbance preserving the irrotationality of the flow outside the interfase is considered, a linear analysis gives as general modes

z(-y,t)

=

7 + (1 - i)ee~t{alcosn 7 - fl~ sinnT}

z(%t)

=

7 + (1 + i)~e-~{fl2 cosn 7 -- c~2 sinnT}

(5) This shows that the amplitude of disturbances of wavenumber n may grow exponentially in time at the rate I~l; this phenomenon is known by the name of Kelvin-Helmholtz instability (see Batchelor [lid and since the growth rate is unbounded the linear problem is ill-posed in the sense of Hadamard (see Garabedian [12]). From the physical point of view the flow is made well posed by the inclusion of viscosity or non-zero thickness for the Vortex sheet. Mathematically the problem may be made well-posed by considering the solution in the class of analytic function of 7. The initial value problem (4) and (5) has been analyzed by Moore [1] and [2]. He uses a novel formal perturbation analysis to obtain a simpler equation which approximates the singular integral equation (4). Then in [2] performing changes of variables and transformation, Moore obtains a 2 x 2 nonlinear hyperbolic system in the independent variable t and y, where y is the real part of i7 after the complexification of 7 (i.e. Moore's nonlinear hyperbolic system lives on a plane perpendicular to the physical plane). Finally, Moore performs an asymptotic analysis on his nonlinear hyperbolic

146

O . F . Ore//ana

system to describe the formation of a singularity at a critical time to. For an initial perturbation S(% 0) = is sin 7, he showed formally that one of the family of characteristics of his conservation laws has an envelope in the domain D = {(y, t) : t > 0} on which the solution has a singularity. This envelope has the equation

e(t+e-'-l) -~-1, Therefore for small t and y > > 1, the singularity is far from the physical plane. As t increases it moves towards the physical plane and reaches it at the critical time tc = 2l log e I + 0(log l log ¢1) At the singularity the curvature of the vortex sheet is infinite, although the slope of the sheet remains finite and continuous. Moore also predicted that the singularity would be of type S = 7 a/2. For t near tc it is not clear whether Moore's approximate solution is asymptotically correct. But his results have been partially confirmed numerically by Krasny [5] and Meiron et al [3]. Moreover, numerical eomputations by Krasny [6] using the vortex blob method show that roll up occurs immediately (or at least very soon) after the first singularity time and experiments indicate that the vortex sheet will roll-up into a tightly wound spiral with two branches. Under the assumption of analyticity, Caflisch and Orellana [4] using a localized approximation method (see Caflisch, Orellana and Siegel [14]) prove existence almost up to the time of expected singularity formation of the solution of (4)-(5) for a small amplitude, odd, periodic perturbation of the flat vortex sheet. Considering Z(% t) = 7 + S(7, t), extending S(% t) analytically to the complex 3~-plane, define S*(7) = S(~))and write S(%t) = S_(% 0 + S+(7,t) where S_(%t)

=

- ~ A , ( t ) e -i~" re=0

S+(%t)

= ~ An(t)e '~" r~----1

because of the oddness of S(~/), and under suitable assumptions on 5'(7, t). We were able to approximately localize the equation (4) as follows:

(6)

OS'.. at (%t)

=

1 L "~ B[S](% t) = -2r---ij_ (~ + s(v + ~) - s(v))-'d{

=

B[S+](% t) + B[S_](% t) + D[S+, S_]

This equation is just a definition of D. It is shown in section 4 of [4] that D is small since it depends essentially on product terms S+S_. The first two terms of (6) can be evaluated explicity by contour integration. Since the integral of {-1 vanishes,

On Some Analytic and Computational Aspects ...

1 ./~

147

s+(r + ¢) - s+(r)

d~

_ 1{ O-yS+ } 2 1 + OrS +

l{a s }

Analogously

B[S_](r) = - ~

1 + OrS_

'

and ~-~{S_~(7)+S*_(7)}=~

1+07S

+

-2

1+07S_

+D[S+'S-]'

which implies that:

(7)

(8)

1{ &s+ }

o,

= ~ 1 +OTS+ +H+D'

~S_(7) OS. ~" +(7) -

1{

2

07S-

1 +OTS_

}

+H-D'

where Hi[S] = S+. Now consider the change of variables y = i7 and the function f(y) = S+(7). From the oddness of S(7, t) it follows that (7) and (8) are equivalent and ,9+(-7) = - S - ( 7 ) . Moreover since ~ = i(-ff) it follows that

f*(Y) = f(Y) = ( S* +)(-5') = -(S:)(7) Hence neglecting H+D in (8) we can rewrite it as:

(9)

O. l{iOyf} -~f (y,t) = --~ l + iOyf

'

which is exactly the equation derived by Moore [1]. Now denoting ¢ = 1 + iOyf and ¢ = ¢* = 1 - iOyf* and applying derivation and • to (9) we obtain:

(10)

Og, _

i 0(1

at

2oy 7 )

which is a system of two conservation laws. Setting

148

O . F . Ore//ana

and asking that g, h be analytic in y and real for y real, i.e. g'(y) = g(y), h*(y) = h(y), we get Moore's conservation laws [2]: 0

0

gh

= Ng,

0 Ng

=

(11) h

h.

As an example of initial data, suppose that S(7, t = O) = ie sin 3' which is Moore's initial data. Then f ( y , O ) = 2 eu

and therefore: ¢(v,0)

= 1+2

¢(y,0)

=

Z~ y l-~-e

is the initial data for (10) and h(y,0)

=

2+

C2

-e

= -2tan- 1

C

is the initial data for (11). Considering the system of conservation laws (10) and its corresponding initial data, we were able to show that the solution developed a singularity after a finite time (using P. Lax's a priory estimates for the time of blow up of a 2 by 2 system of conservation law, see P. Lax [13]). Finally considering (10) with the error terms that came from H + D and, using the abstract Cauchy Kowalewsky theorem established by L. Nieremberg and improved by Nishida [15] we were able to establish a long time existence theorem for the solution of the Birkhoff-Rott equation (see Caflisch and Oretlana [4]). Also under the assumption of analyticity, Sulem, Sulem, Bardos and Frisch [10] were able to prove short time existence for solutions of the initial value problem (4)-(5). Borgers [16] and Caflisch and Lowengrub [17] proved global existence for the solution of a de~ingularized version of the initial value problem (4)-(5) as proposed by R. Krasny

[6]. The need for the analytic function space setting for the vortex sheet problem, has been demostrated by Caflisch and Orellana [7], who showed that the problem is not well-posed in the Sobolev space H ~ for any n > 3/2. We constructed exact nonlinear solutions with small initial norm for which 0~S become infinite in arbitrary short time for any a > 1. Such singular solutions where also constructed by Duchon and Robert [18], and an alternative proof of ill-posedness was given by Ebin [19]. The singular

On Some Analytic and ComputationM Aspects ...

149

solutions produced in [7] are not believed to be typical (i.e. the type of singularity formation at the initial time is still on open question). Moore [20] has derived corrections to the Birkhoff-Rott equation, for a thin vortex layer approximating a vortex sheet. Numerical solutions and asymptotic analysis for thin layers have been carried out by Baker and Shelley [21] and Shelley and Baker [22]. There are other important analytic aspects about the Birkhoff-Rott equation and its solutions, like similarity solutions of the form =t

z

=

which has been analysed by a number of authors. So far, we learn from the analysis that computation of the evolution of vortex sheet in two-dimensional, incompressible inviscid flow is delicate because of two reasons: a) Kelvin-Helmholtz instability, and b) Singularity formation on the interface after finite time, infinite curvature of the interface. Hence any numerical method must take into account this problem, because numerical roundoff error can excite the physical instability to produce irregular results well before the physically correct singularity formation and roll-up of the vortex sheet.

2. T h e point v o r t e x m e t h o d , the v o r t e x blob m e t h o d and c o n v e r g e n c e of this m e t h o d s for v o r t e x s h e e t The difficulties mentioned above to compute the evolution of vortex sheets were overcame by R. Krasny [5] and [6] by two methods. a) A point vortex methods with filtering to eliminate spurious high wavenumber components, and b) A vortex blob method. From a formal point of view both methods are similar. The former consists of replacing the continuous curve representing the interface or vortex sheet Z(7 ,t) at a fixed time by a finite number of point vortex, corresponding to a uniform "),-mesh. This is achieved by discretizing the integral part of Birkhoff-Rott equation by Simpson's rule, and then using a Runge-Kutta 4th order method to solve numerically the system of O.D.E. that was obtained after discretization of the integral part of equation (4). In the process of iteration there is a filter that at each iteration filters the discrete solution to eliminate spurious high wave number components stabilizing the numerical process in time and space which otherwise will produce irregular results (chaotic motion of the point vortices). Using this numerical method R. Krasny simulated singularity formation for the vortex sheet evolution. Moreover, the jump in tangential velocity developed a cusp at the singular points. Krasny's numerical results are in good agreement with Moore's asymptotic results.

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O . F . Ore//ana

The vortex blob method consists of replacing the continuous curve representing the interface or vortex sheet Z(7, t) at a fixed time by a finite number of circular vortex patches of radius 6 corresponding to a uniform 7-mesh. This is achived by doing exactly the same type of discretization described above, but this time applied to the desingularized integro-differential equation

(12)

0Z

t

1

~

jzZ(% t)

- Z ( e ' , t)d~'

(with no filter). Using this method R. Krasny, was able to simulate roll-up of the vortex sheet, giving numerical evidence of the convergence of the m e t h o d with respect to both discretization parameters (i.e. for the discretization parameters corresponding to the time and space variable respectively) holding 6 equal to const. He also gives numerical evidence of the convergence of the vortex blob method to the vortex sheet computing the solution of equation (12) for several values of the parameter 6 going to zero and holding the other two parameters constant. However, no analytic proof of the convergence had been given until Caflisch and Lowengrub [17] proved convergence of both the point vortex m e t h o d and the vortex blob method with both spatial and temporal discretization and simulated roundoff error. For the vortex point method they proved convergence in the case of a vortex sheet that is initially a small analytic perturbation of a flat, uniform sheet and the perturbation is chosen periodic for simplicity. They proved convergence for a short time. The roundoff error er must satisfy er < max {e -1/h , e-116} This condition on the roundoff error is quite strict, but it is consistent with the numerical results of Krasny. This condition is the numerical interpretation of the requirement of analyticity for vortex sheet solutions. For the spatial discretization size h the m a x i m u m wavenumber is km= 1/h. Analyticity for a function f is (roughly speaking) equivalent to requiring that ] ( k ) < exp(-c[k[). This condition can be verified for all k with Ik[ < k m only if the round off error is sufficiently small; i.e. er < exp(-c/h). First of all Caflisch and Lowengrub prove global existance for analytic solutions of Krasny's desingularized vortex sheet equation (12). This result is established without any reference of closeness to a flat vortex sheet and does not use the Cauchy-Kowalewski Theorem, but it is established for arbitrary analytic initial data. The abstract Cauchy-Kowalewski Theorem as established by Nierenberg and improved by Nishida and a discretized version of it is the basic tool they used to construct the solutions and to prove convergence of the vortex blob method. In the limit 6 -- 0 they get a convergence proof for the point vortex method and for a short time interval, certainly less than the critical time. Hence there is no convergence proof of the point vortex m e t h o d after to. For this result it was necessary to set the analysis in an analytic function space because of the use of Cauchy-Kowalewsky theorem, but more i m p o r t a n t because under this assumption the stability of the point vortex m e t h o d is mantained, otherwise the problem is ill posed as proved in [7]. Hence one of the main contributions of their paper is the clarification of the meaning of analy-ticity for numerical analysis because of its possible application to many other ill-posed problems.

On Some Analytic and Computational Aspects ...

151

3. A n e w p h y s i c a l l y d e s i n g u l a r i z e d v o r t e x s h e e t e q u a t i o n In the previous section we mentioned that R. Krasny [6] formulated a desingularized approximation equation of Birkhoff-Rott equation for the evolution of a vortex sheet and he used it to compute roll-up of a vortex sheet after the critical time (i.e. after singularity formation). Krasny's desingularized equation is:

--

(la)

o,z(%0

= K,[z,-~]

=

~1

T)(z(%~)-- ~!~'' t)d~' ]_'oo00IZ--(~,

After the analytic extension of Z(% t) to the complex 7 plane (13) can be rewritten:

- Z*(~',~)d~' OtZ*(7,*) = K,[Z, Z*] = ~1 f ~oo I z*(%,) Z - - ~ , 7 S 7 # - ( ~ 1 ~ 7 ~ ='

(14)

where z*(%,) = z ( v , 0 .

Even though, from a mathematical point of view, when 6 goes to zero (13) approaches the Birkhoff-Rott equation, the desingularization proposed by R. Krasny seems rather arbitrary. Moreover, from a physical point of view, 6 does not represent the viscosity because the desingularized equation (13) conserves energy, whereas viscosity would dissipate energy. Thus the effect of ~ is dispersive rather dissipative. Here we give an argument that illustrates that ~ does not represent a vortex layer thickness either. If we apply the localization method illustrated in the first section of this paper to equation (14) and the equation of motion of a vortex layer of small thickness derived by Moore [20], namely:

OZ" 1 Ot ( f f ' t ) = M ~ [ Z ' Z * l = ~ i f

dr'

¢ 0 ( OZ -' OZ*~

z(.~,t)-z(~,,t) +-CL-iwiNklO.YI

o'r ]

where the terms of size O(¢ 2) have been ignored; ~ = H/p, H =the dimensional thickness and p the radius of curvature, and w := e~ is the vortex sheet strength with ~ the constant vorticity in the layer; we get:

a,s:

=

. v~[s_,s+]= i5 (

{ a,s+ 17-E~+~j 2 * O~S_

aa~s+

}

+ 8- / (1 + O.rS+)312(1 + O~S*_)31; + (1 + a~s+),/=(1 + a~s:),/= '

1{ 1 +o~s: } O,S_"

o~s; = M,[S_, S;]* = -~

3a{s_ s and

(1 + o,s'_p/~(1 + os+)~n + (1 + ~.,S'_)s/2(1 + oq.~S+)t/2

'

O. F. Orellana

152 o~s:

=

M~[S+,

S:]

1{ 1+c3.yS+ O~S+ } + ~(6wi)_10~{(1+0~8+)-2(1 + 0~S*_)-1 }

2 o,s+

= Mo{S+,S:]"

-

21{ I+0"~S:0~S_*} -e(6wi)-lO~{(l+O'~S*-)-2(l+O'~S+)-'}

respectively. Comparison of these equations shows them to be quite different, hence the desingularization parameter 6 cannot be interpreted as vortex layer thickness. This difference can be understood by noting that Moore's expansion parameter is e/w which has units of (length/velocity). A more physically meaningful desingularization equation is found by replacing 6~ in (13) or (14) by (c/w) 2. Then a factor with units of velocity must be put into the first term of the denominator. Hence we proposed the following physically desingularized vortex sheet equation: --

z

--

z-:

i

t

aZ

2

t

~Z

I

, t

2

p

- ;-:

which agrees up to size O(~): with the vortex layer equation in the localized approximate equation. Analytic and numerical analysis of this last equation is presently being performed.

Conclusion From what has been presented here the importance of taking into account all the aspects involved in the resolution of a given problem is quite obvious. In particular, for this problem, it is useful to notice how the analytic, numerical and physical aspects of the problem play an important role in its resolution and how they relate and complement one other to validate the different solutions and finally the model in question (i.e. Birkhoff-Rott equation as a mathematical model to describe the evolution and dynamics of a vortex sheet).

References [1] D.W. Moore, The spontaneous appearance of a singularity in the shape of an evolving vortex sheet, Proc. Roy. Soc. London, Ser. A, 365 (1979), 105-119.

On Some Analytic and Computational Aspects ...

153

[2] D.W. Moore,

Numerical and Analytical aspects of Helmholtz instability in Theoretical and Applied Mechanics Proe. XVI Internat. Congr. Theoret. Appl. Mech., F.I. Niordson and N. Olhoff, eds., North-Holland, Amsterdam, 1984, 629-633.

[3] D.I. Meiron, R.G. Baker, S.A. Orszag, Analytic structure of vortex sheet dynamics, Part 1, Kelvin-Helmholtz instability, J. Fluid Mech. 114 (1982) 283-298. [4] R.E. Caflisch, O. Orellana. Long Time Existence for a Slightly Perturbed Vortex Sheet, Comm. Pure Appl. Math. 39 (1986) 807-838. [5] R. Krasny, On singularity formation in a vortex sheet and the point vortex approximation, J. Fluid Mech. 167, (1986) 65-93. [6] Krasny, R., Desingularization of periodic vortex sheet roll-up, J. Comp. Phys. 65, (1986) 292-313. [7] R.E. Caflisch, O. Orellana, Singular Solutions and Ill-Posedness for the Evolution of Vortex Sheets, SIAM J. Math. Anal. 20, (1989) 293-307. [8] H.W. Hoeigmakers, W. Vaatstra, W., A higher order panel method applied to vortex sheet roll-up, J. AIAA 21, (1983) 516-523. [9] G. Birkhoff, Helmholtz and Taylor instability in "Hydrodynamic Instability", Proc. Syrup. in Appl. Math. XII, AMS (1962), 55-76

[10] C. Sulem, P.L. Sulem, C. Bardos, U. Frisch, Finite time

analyticity for the two and three dimensional Kelvin-Helmholtz instability, Comm. Math. Phys. 80, (1981) 485-516.

[11] G.K. Batchelor, An introduction to Fluid Dynamics, Cambridge University Press (1967) 511-517. [12] P. Garabedian, Partial differential equations, John Wiley and Sons, 1964. [13] P.D. Lax, Development of singularities of solutions of nonlinear hyperbolic partial differential equations, J. Math. Phys. 5, (1964), 611-613. [14] R. E. Caflisch, O. Orellana, M. Siegel, A Localized Approximation Method for Vortical Flows, SIAM Journal on Applied Mathematics, (to appear). [15] T. Nishida, A note on a theorem of Niremberg, J. Diff. Geom. 12 (1977) 629-633. [16] C. Borges, On the numerical solution of the regularized Birkhoff equation, preprint 1988. [17] R.E. Caflisch, J. Lowengrub, Convergence of the vortex method for vortex sheets, SIAM J. Num., Anal. (to appear). [18] J. Duchon, R. Robert, Solutions globales avec nape tourbillionaire pour les equations d'Euler dans le plan, C.R. Acad. Sci., Paris 302 (1986) 183-186.

154

O.F. Ore//ana

[19] D. Ebin,

Ill-posedness of the Rayleigh-Taylor and Helmholtz problems for incompressible fluids, Comm. P.D.E. 13 (1985) 1265-1295.

[201 D.W. Moore,

The equation of motion of a vortex layer of small thickness, Stud. in Appl. Math. 58, (1978) 119-140.

[21] G.R. Baker, M.J. Shelley, On the connection

between thin vortex layers and vortex

sheets, Part I: J. Fluid Mech. (to appear).

[22] M.J. Shelley, Baker, G.R.,

On the connection between thin vortex layer and vortex sheets, Part II: Numerical Study, J. Fluid Mech. (to appear).

Received: July 30, 1989

Computational Methods and Function Theory Proceedings, Valparaiso 1989 St. Ruscheweyh, E.B. Saff, L. C. Salinaz, R.S. Varga (eds.) Lecture Notes in Mathematics 1435, pp. 155-169 (~) Springer Berlin Heidelberg 1990

On the

Numerical

Decomposition

Performance

Method

N. Papamichael

and

of a Domain

for Conformal

Mapping

N.S. Stylianopoulos

Department of Mathematics and Statistics, Brunel University Uxbridge, Middlesex UB8 3PH, U.K.

1. I n t r o d u c t i o n This paper is a sequel to a recent paper [14], concerning a domain decomposition m e t h o d (hereafter referred to as D D M ) for the conformal mapping of a certain class of quadrilaterals. For the description of the D D M we proceed exactly as in [14:§1], by introducing the following terminology and notations. Let G be a simply-connected Jordan domain in the complex z-plane (z = x + i y ) , and consider a system consisting of G and four distinct points Zl, z2, z3, z4 in counterclockwise order on its boundary OG. Such a system is said to be a quadrilateral Q and is denoted by Q -- {G; zl, z2, zs, z4}. The conformal module re(Q) of Q is defined as follows: Let R be a rectangle of the form (1.1)

n:=

{(~,,):a < ~ < b,c< ~ < d},

in the w-plane (w = ~ + it/) , and let h denote its aspect ratio, i.e. h := (d - c)/(b - a). T h e n rn(Q) is the unique value of h for which Q is conformally equivalent to a rectangle of the form (1.1), in the sense that for h = ra(Q) and for this value only there exists a unique conformal map R -* G which takes the four corners a + ic, b + ic, b + id, and a + id, of R respectively onto the four points zl, z~, z3, z4. In particular, h = r e ( Q ) is the only value of h for which Q is conformally equivalent to a rectangle of the form

(1.2)

Rh{c~} := { ( ( , ~ ) : O < ( < l , a < 7/ < a + h}.

T h e D D M is a method for computing approximations to the conformal modules and associated conformal maps of quadrilaterals of the form illustrated in Figure 1.1(b). T h a t is, the method is concerned with the mapping of quadrilaterals

(1.3a)

Q := {a; z~, z:, z~, z~},

N. Papamichael, N.S. Stylianopoulos

156

where: • The domain G is bounded by the straight lines x = 0 and x = 1 and two Jordan arcs with cartesian equations y = -vl(x) and y = r2(x), where rj; j = 1, 2, are positive in [0, 1], i.e. (1.3b)

a := { ( x , y ) : 0 < x < 1,--TI(X ) < y < T2(X)}.

* The points zl, z2, z3, z4 are the corners where the arcs intersect the straight lines, i.e.

(1.3c)

zl = --irl(O),

z2 = 1 -- it1(1),

z3 = 1 + ir2(1),

z4 = iv:(O).

Let Q be of the form (1.3) and let (1.4a)

Gi := { ( x , y ) : 0 • x .~ 1,--TI(X ) < y < 0},

and (1.45)

G~ := {(z, u ) : 0 < z < 1,0 < u < ~ ( x ) } ,

so that G = G1 U G2. Also, let Q1 and Q2 denote the quadrilaterals (1.4c)

Q, := {G1;z~,z~,l,O}

and

Q~ := {G2;O,l,z3,z4},

and let h := re(Q) and hj := m(Q~); j = 1,2; see Figures 1.2(5) and 1.3(5). Finally, let g and gj; j ---- 1, 2, denote the conformed maps (t.5) (1.6)

g: n h { - h l } -~ G, g l : R h l { - h l } --} G1 and

g2: Rh2{0} ---* G2,

where, with the notation (1.2), R h { - h ~ } := { ( C ~ ) : 0 < ~ < 1 , - h , < ~ < h - hi},

Rhl { - h i }

:= {(~,r]): 0 <: ~ < 1 , - h l < 77 < 0},

Rh~{0} := { ( ~ , ~ ) : 0 < ~ < 1,0 < ~ < h2};

see Figures 1.1-1.3. Then, the DDM consists of the following: (a) Subdividing the quadrilateral Q, given by (1.3), into the two smaller quadrilaterals Q1 and Q2, given by (1.4). (b) Approximating the conformed module of Q by the sum of the conformed modules of Q1 and Q2, i.e. approximating h by (1.7a)

h := hi + h2.

(c) Approximating the rectangle Rh{-h~) and the conformed map g : R h { - h x ) ---+G respectively by

On the Numerical Performance of a Domain Decomposition...

(1.7~)

Ra{-h~} :=

157

{ ( ~ , ~ ) : o < ~ < 1 , - h i < ~ < h=},

and for w e Rh2 {0},

(1.7c)

O(w)

I gl(w) : R < { - h l }

- , GI,

for w e Rh, { - h i } .

The initial motivation for considering the above method came from: (a) The intuitive observation that if the constituent quadrilaterals Q1 and Q2 are both "long" then tz is close to h. (b) Experimental evidence indicating that h is close to h even when QI and Q2 are only moderately long; see [12:§5] and [14:§1]. (It is important to note that h >_ hi + h2 and equality occurs only in the trivial cases where G is a rectangle or rl(X) = 7"2(x), x e [0, 1]; see e.g. [9: p. 437].) The treatment of the DDM contained in [14] is a theoretical investigation leading to estimates of the errors in the approximations (1.7). These error estimates are derived by assuming that the functions rj; j = 1,2, satisfy the following: (i) rj; j = 1,2, are absolutely continuous in [0,11, and (1.8)

dj := ess

[rj(x){ < ce.

sup

0~x
(ii) If (1.9a)

mj := m a x { e x p ( - - ~ ( x ) ) } ;

j := 1,2,

then (1.9b)

ej := d j { ( l + mj)/(1

-

mj)} < 1; j = 1,2.

In addition to the theory, [14] also contains two numerical examples comparing the actual errors in the D D M approximations with those predicted by the theoretical estimates. The present paper is concerned with the numericM performance of the D D M and, in particular, with the performance of the method in cases where the functions rj; j = 1, 2, do not fulfil the rather restrictive condition (1.9) needed for the theory of [14]. More specifically, the main purpose of this paper is to s h o w b y means of numerical examples that some of the theoretical results of [14] remain valid even when the condition (1.9) is violated, and thus to provide experimental support for certain conjectures made in [141. We end this introductory section by making the following remarks concerning the D D M and related matters: • A survey of available methods for computing approximations to the conformal modules and the associated conformal maps of general quadrilaterals is given in [12], where also several areas of application of the conformal maps are discussed; see also [4]-[71 and [9:§16.111. • Although this is not considered here, the D D M can also be applied to quadrilaterals of the form illustrated in Figure 1.4, provided that the crosscut c of subdivision is taken

158

N. Papamichael, N.S. Stylianopoulos

l+i(h-hl)

Z3

Z4

g ....•R h { - h x } . . ,

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

1-ihl

(~)

Z1

g~

(a)

,

G2

(b)

Figure 1.2

1

0 Rhx {-hx }

Z2 Z3

Z4

n h ~ { O } ......

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

"~ (b)

Figure 1.1 l+ih2

V

gl 1-ihl

(~)

Z1 ~

(b)

Figure 1.3

Z2

z4

1

Figure 1.4

zl

Z2

On the Numerical Performance of a Domain Decomposition...

159

to be a circular arc; see [14: Remark 4.7]. In other words, the apphcation of the D D M is restricted to quadrilaterals that have one of the two special forms illustrated in Figures 1.1 and 1.4. We note however that the mapping of such quadrilaterals has received considerable attention recently; see e.g. [2], [8], [11], [12], [15] and [17]. • A more general form of the DDM involves subdividing the original quadrilateral Q into two quadrilaterals Q~ and Q2 of the form (1.4) at the lower and upper ends, and a rectangle in the middle. This can be described more precisely as follows: Let G := { ( z , y ) : 0 < z < 1,--Tl(X) < y < r2(x) + c } , where c > O, let G, := { ( x , y ) : 0 < x < 1 , - r l ( x ) < y < 0}, and

as := {(x, y): 0 < x < 1, c < y < ,2(z) + c}, so that G = G1 U Rc{0} U ~ , and let Zl = --i'rl(0),

Z2

=

I

-- i r l ( 1 ) ,

Z3 = i + i(v2(1 ) -4- c), z4 = i(v2(0) + c).

Then the general form of the DDM consists of the following: (a) Subdividing the quadrilateral Q := {G; zl, z2, z3, z4} into three smaller quadrilaterals, i.e. the quadrilaterals Q I : = {Gi;zl,z~,l,O}

and

Q 2 : = { G 2 ; i c , l + i c , za, z4},

at the lower and top ends, and the rectangular quadrilateral

{Re{0}; 0,1, 1 + ic, i¢}, in the middle. (b) Approximating the conformal module h := re(Q) by (1.10)

h := hi + h2 + c,

where hj := m(Qj); j = 1, 2. (c) Approximating the rectangle R h { - h l } and the conformal map g : R h { - h l } ---* G respectively by R ~ { - h l } and

{ 9~(w): Rh~{c} --, as, (1.11)

#(w) :=

for w • R~{e},

w,

for w • Re{0},

gi(w) : Rha{-hl} --* G,,

for w • Rh,{--h,}.

• The D D M is of practical interest for the following reasons: (i) Given the conformal modules and associated conformal maps of two quadrilaterals Q1 and Q2 of the form (1.4), the method provides approximations to the conformal module and associated conformal map of any quadrilateral consisiting of Q1 and Q2, at the lower and top ends, and a rectangle of any height in the middle.

N. Papamichael, N.S. Stylianopoulos

160

(ii) The method can be used to overcome the "crowding" difficulties associated with the numerical conformal mapping of "long" quadrilaterals of the form (1.3). (Full details of the crowding phenomenon and its damaging effects on numerical procedures for the mapping of "long" quadrilaterals can be found in [12], [13], and [10]; see also [4:p.179], [9:p.428] and [16:p.4].) (iii) Numerical methods for approximating the conformal maps of quadrilaterals of the form (1.4) are often substantially simpler than those for quadrilaterals of the more general form (1.3); see e.g. [S], and [12,§3.4]. • Of the two conditions involved in the assumptions (1.8)-(1.9), only (1.9) is restrictive from the parctical point of view. This condition is more or less equivalent to requiring that the slopes of the two curves y = ri(x); j = 1, 2, are numerically less than unity in [0,1]. This is so because the values mj;j = 1,2, given by (1.9a) are "small", even when the quadrilaterals Qj; j = 1, 2, are only moderately "long".

2. T h e o r e t i c a l

error estimates

As in Section 1, let Q and Qj; j = 1, 2, denote the three quadrilaterals defined by (1.3) and (1.4), let h := rn(Q), hj := rn(Qj); j = 1,2, and let g,gj; j = 1,2, be the associated conformal maps (1.5) and (1.6). Also, let :=

- ihl),

: = Reg(

+ i(h - h i ) ) ,

and X,(~) :-- Reg,(~ - ih,),

22(~) := Reg2(~ + ih2),

and let Eh and Eg{j}, E x { j } , E~{j}; j = 1,2, denote the following domain decomposition errors: (2.1)

Eh := h - (hi -{- h2),

(2.2a)

Eg{1} := m a x { ] g ( w ) - gl(w)] : w G R h l { - h l } ) ,

(2,2b)

E9{2 } := max{[g(w + iEh) - g2(w)t : w C Rh2{0}}, Ex{1}

:=

max]X(()-Xl(~)[, 0<~<1

Ex{2}

:=

max I_~[(~) -- X2(~)[, 0<~_<1

(2.3)

and (2.4)

E~{1} := max ITx(X(~)) -- Tl(Xl(~))l, 0_<~_<1

E~{2} := max [T2(2(~)) - v2(22(~))]. 0<~<1

Finally, assume that the functions Tj; j = 1, 2, satisfy the assumptions (1.8)-(1.9), and let

On

the Numerical Performance of

a

Domain Decomposition...

161

a(ej,e) := 2 / { ( 1 - e j ) ( 1 - e2)½}; j = 1,2,

(2.5a) and

(2.5b)

-e)}~; j=1,2,

-

where the ei; j = 1,2, are given by (1.9), and e := max(el,e2). Then, the main results of [14] are the following estimates of the errors Eh and Eg{j}; j = 1, 2, in the D D M approximations (1.7):

(2.6)

Eh <_ r~-ldla(el,e){ele -2,~h~ + e2e -'~h } + r~-ld2a(e2,e){e2e -2'~h2 + ele -'~h}

and (2.7a)

Eg{j} < max{Mj,Nj}; j = 1,2,

where (2.7b)

2 ~ ! Mj := ~-½(1 + d})a~(¢j,¢){ej + ¢a_je-*h}½{eje -2*hi + ¢a-je -rh }a,

Nj

:=

~rr 2~(O,~){5eje-'~h,+3ea_ie-'~{h-hi)}

(2.7c) Jc-71--ldjol(gj, g){ejg.-2vhj + g3-je.-xh);

see [14:Thms 4.1, 4.4]. Since h > hi + h2, the above estimates show that if the functions rj; j = 1, 2, satisfy the assumptions (1.8)-(1.9), then (2.8)

Eh = O{exp(-2rrh*)},

and (2.9)

Eg{j} = O{exp(-rrh*)};

j = 1,2,

where h* := min(hl, h~). In addition to (2.6) and (2.7), [141 also contains estimates of the errors E x {j } and E~{j}; j = 1,2; see Theorem 4.2 and Remark 4.2 in [14]. These estimates show that under the assump{ions (1.8)-(1.9), (2.10)

E x { j } = O{exp(-rrh*)}

and

where as before h* := min(hl, h~). Finally, shows that if Q1 and Q2 are both "long" from the two sides 7/= - h i and q = h approximated closely by the identity map. (2.11)

Er{j} = O{exp(-rrh*)};

j = 1,2,

[14] contains a theorem (Theorem 4.3), which quadrilaterals, then at points sufficiently far hi of R h { - h l } , the conformal map g can be In particular, this theorem of [14] shows that

max [g(~ + iO) - ~[ = O{exp(-rrh*)}. o_<¢_<1

N. Papamichael, N.S. Stylianopoulos

162

The method of analysis used in [14] for deriving the above results makes extensive use of the theory given in [3: Kap.V,§3], in connection with the integral equation method of Garrick, for the confomal mapping of doubly-connected domains. This involves expressing the three problems for the conformal maps g and gj; j = 1,2, as equivalent problems for the conformal maps of three symmetric doubly-connected domains; see [8], [12:§3.2,3.4] and [13:§3]. (With reference to the above comment, readers who are familiar with the method of Garrick will recognize the very close similarity between the condition (1.9), which is needed for the analysis used in [14], and the so-called cb-condition needed for the theory of the Garrick method.) We end this section by observing that the results of [14] simplify considerably in the case where one of the two subdomains G1 of G2 is a rectangle. For example, let rl(Z) = c > 0, x e [0, 1], i.e. let (2.12)

a l := { ( x , y ) : 0 < z < 1 , - c < y < o} = R c { - ¢ ) .

Then, gl(w) = w, hi = c, dl = cl = 0 and, for any value c > 0, the results (2.8)-(2.10) simplify as follows: (2.13)

Eh : = h - (c + h2) = O { e x p ( - 2 ~ h 2 ) } ,

Eg{j} = O{exp(-~rh2)};

(2.14) (2.15)

Ex{j} = O{exp(-vh2)},

j = 1,2,

E,{j} = O{exp(-~rh2)};

j = 1,2.

Also, in place of (2.11) we now have for any point w := ~ + iq E Rc{-c}, (2.16)

I t ( w ) - wl = O { e x p ( - ~ ( h ~

- 7))}.

Furthermore, M1 the above simplified results hold under the less restrictive assumptions obtained by replacing the inequalities (1.9) by

~2 :=d2{(l+m~)/(1 - m ~ ) } < 1;

(2.17) see [14:Remark 4.5].

3. N u m e r i c a l results and d i s c u s s i o n In addition to the theoretical estimates summarized in Section 2, [14:§5] contains two numerical examples in which the quadrilaterals are chosen so that the functions ~-j; j = 1, 2, satisfy the assumptions (1.8)-(1.9). The numerical results of these examples confirm the theory of [14], and indicate that the DDM is capable of producing approximations of high accuracy, even when the quadrilaterals under consideration are only moderately long. In this section we study further the numerical performance of the DDM, but here we consider its application to quadrilaterals that do not satisfy the conditions (1.9). That

On the Numericed Performance of a Domain Decomposition...

163

is, we are concerned with cases for which the theory of [14] does not apply. Our main purpose is to provide experimental evidence supporting the following two conjectures made in [14]: C o n j e c t u r e 3.1. The results (2.8)-(2.9) hold even when the condition (1.9) is not fuhqfled; see [14:Remark 5.4]. More speci~qcally, the claim here is that

Eh = O{exp(-2zrh*)},

(3.1) and (3.2)

Eg{j} = O{exp(-zrh*)};

j = 1,2,

with h* := rain(hi, h2) even when dj >_ 1; j = 1, 2, where dj are the values given by

(1.8). C o n j e c t u r e 3.2. The errors E x { j } and E~{j}; j = 1,2, are O{exp(-27rh*)}, rather than O{exp(-~rh*)) as predicted by the theory of [14]; see (2.10) and [14:Remark 5.2].

That is, the claim here is that (3.3)

E x { j } = O{exp(-2~rh*)}

and

E , { j } = O{exp(-2zrh*)};

j = 1,2,

and that the above resu/ts hold even when the condition (1.9) is violated. Each of the three examples considered below involves the mapping of a quadrilateral Q of the form (1.3) and, in each case, the decomposition is performed by subdividing Q into two quadrilaterals Qj; j = 1,2, of the form (1.4). In presenting the numerical results we use the following notations: • Eh and E,{j}, E x { j } , E , { j } ; j = 1,2: As before these denote the actual DDMerrots (2.1)-(2.4). More precisely, the values listed in the examples are reliable estimates of the errors (2.1)-(2.4). They are determined, as in [14:§5], from accurate approximations to h, hj, j = 1,2, and g, g5, J = 1,2, which are computed by using the iterative algorithms of [8]. In particular, Eg{j}; j = 1,2, are the maxima of two sets of values which are obtained by sampling respectively the approximations to the functions g(w) - g~(w) and g(w + iEh) -- g2(w) at a number of test points on the boundary segments r / = - h i , 0 of R h l { - h l } and 77 = 0, h2 of Rh2{0}. The values E x { j } and E , { j } are determined in a similar manner, by sampling the approximations to the functions X ( ( ) - Xz((), 2 ( ( ) - 2 2 ( ( ) , etc. at a number of test points in 0 _< ( _< 1. (The only exception to the above are the values of Eh given in Example 3.1, in which Q is a trapezium and the subdivision consists of a smaller trapezium Q2 and a rectangle Q1 := Rc{-c}. In this case, h := re(Q) and h2 := rn(Q2) are known exactly in terms of elliptic integrals. Hence Eh := h - (c + h2) is also known exactly.) • ~h and ~e{J}, ~x{j}, &{j}; j = 1,2: These denote the values used for testing the validity of (3.1)-(3.3). They are determined from the computed values of the errors Eh and Eg{j}, E x { j } , E , { j } ; j = 1,2, as follows: In each of the Examples 3.2 and 3.3, the functions rj; j = 1,2, are of the form

j(x) := oj(x) + l;

j = 1, 2,

N. Papamichael, N.S. Stylianopoulos

164

where l > 0, and in each case the values of the conformal modules and the errors in the DDM approximations are computed for several values of the parameter l. Let hi(1) and h2(1) denote the conformal modules of Q1 and Q2 corresponding to the value l, and let h'(1) := min(hl(/), h2(I)). Also, let E denote any of the errors Eh, Eg{j}, Ex{j} or E~-{j}, and let E(l) be the value of E corresponding to I. Then, the validity of (3.1)-(3.3) is checked by assuming that

E = O{exp(-&rh*)};

h* := min(hl,h2),

and computing various values of 6 (i.e. of 6h, (~g{j }, ~x {j } or (~,{j }) by means of the formula 6 = - {log[E(/1 )/E(12)]}/ {~r[h*(/1 ) -

h*(I2)]}.

(In the examples, ll and 12 are taken to be successive values of the paremeter I for which numerical results are listed.) In the first example, i.e. in Example 3.1, we consider only the error Eh and, because of the form of Q, we check the validity of (2.13) rather than (3.1). That is we assume that Eh =

and determine ~h from the listed values of procedure described above.

Eh, by modifying in an obvious manner the

E x a m p l e 3.1 (See also [12:§5]) Q is the trapezium illustrated in Figure 3.1. That is, Q is defined by the functions

rffx)=c

and

r2(x)=x-l+t,

where c > 0 and l > 1. Here, d2 = 1 and, because of this, the theory of [14] does not apply. As was previously remarked, in this case the conformal modules h := re(Q) and h2 := m(Q2) are known in terms of elliptic integrals. Thus, Table 3.1 contains the exact values of h and h2 corresponding to the parameters t =1.25, 2.00, 2.50, 4.00, 5.00, and c =0.75, 0.50, 1.50, 1.00, 5.00. (These were determined correct to twelve decimal places, by using the formulae of Bowman [1:p.104].) The table also contains the values of the error En := h - (h2 + c) and, where possible, tile corresponding values of ~h- These values of ~h indicate clearly that, for any c > 0, Eh = O{exp(-27rh2)}.

On the Numerical Performance of a Domain Decomposition...

165

Z3

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

i C

1

Z1

Z2

F i g u r e 3.1

tic 1.25

0.75

2.00 o.50 2.50 1.50 4.OO 1.00 5.00 5.00

0.516 1.279 1.779 3.279 4.279

h2 810 878 261 571 359 959 364 399 364 399

029 171 478 489 847

1.297 1.779 3.279 4.279 9.279

h 261 359 364 364 364

571 959 399 399 399

171 478 489 847 847

Eh r 6h 1.2 E-02 9.8 E-05 2.021 4.4 E-06 1.972 3.6 E-10 1.999

T a b l e 3.1

E x a m p l e 3.2. Q and Qj; j = 1, 2, are defined by (1.3) and (1.4) with

rl(X) = 1 + 0.25COS(2~X) + l and

r2(x) = 0.25x 4 - 0.5x; + 1 + I.

Since dl = ~/2 > 1, the above two functions do not satisfy the condition (1.9) needed for the theory of [14]. The numerical results corresponding to the values t = 0.0(0.5)2.5 are listed in Tables 3.2(a)-3.2(c). Table 3.2(a) contains the computed values of the conformal modules, together with the estimates of Eh := h - (hi + h2) and the corresponding values of 6a. (The values of h and hj listed in the table are expected to be correct to all the figures quoted. The algorithms of [8] achieve this remarkable accuracy, because the two curves y = rj(x); j = 1, 2, intersect the straight lines x = 0 and x = 1 at right angles; see [8:§6]) Tables 3.2(b) and 3.2(c) contain respectively the estimates of the errors Eg{j}; j = 1, 2, and E x { j } , E,{j}; j = 1,2, together with the values 5g{j}; j = 1,2, and 5x{j}, 5,{j}; j = 1,2.

N. Papamichael, N.S. Stylianopoulos

166 l 0.0 0.5 1.0 1.5 2.0 2.5

I

hi 0.864 1.364 1.864 2.364 2.864 3.364

086 089 089 089 089 089

] 763 626 632 632 632 632

083 994 342 352 352 352

h2 0.859 360 1.359 560 1.859 568 2.359 569 2.859 569 3.359 569

h 128 944 1.723 659 053 306 2.723 658 647 619 3.723 658 0i8 929 4.723 658 034 974 5.723 658 035'668 6.723 658

Eh 400 858 669 419 6B8 053 668 050 668 050 668 050

2.1 9.0 3.9 1.7 7.2 3.1

E-04 E-06 E-07 E-08 E-10 E-11

I

5h 2.013 2.001 2.OO0 2.000 2.000

Table 3.2 a)

l 0.0 0.5 1.0 1.5 2.0 2.5

Eg{1} 1.2 E-02 2.4 E-03 5.0 E-04 1.0E-04 2.1 E-05 4.5 E-06

6g{1}

Es{2} 1.2 E-02 1.022 2.4 E-03 1.004 5.0 E-04 1.000 I.oE:04 1.000 2.i E-05 1.000 4.5 E-06

•{2} 1.018 1.003 1.000 1.000 1.000

Table 3.2(b)

l 0.0 0.5 1.0 1.5 2.0 2.5

Ex{1} 4.9 E-04 2.1 E-05 9.1 E-07 3.9 E-08 1.7 E-09 7.3 E-f1

6x{1} 1.999 2.000 2.000 2.000 2.000

Ex{2} 1.1 E-03 4.6 E-05 2.0 E-06 8.6 E-08 3.7 E-09 1.6 E-10

6x{2} 2.001 2.000 2.000 2.000 2.000

Er{1} ........ 6~{1} 5.2 E-04 2.2 E-05' 2.005 9.6 E-07 2.000 4.2 E-08 2.000 1.8 E-09 2.000 7.8 E-f1 2.000

EH2} 4.1 1.7 7.6 3.3 1.4 6.1

E-04 E-05 E-07 E-08 E-09 E-11

a{2} 2.002 2.000 2.000 2.000 2.000

Table 3.2(c)

E x a m p l e 3.3 Q and Qj; j = 1,2, are defined by (1.3) and (1.4) with n(x)=0.75+0.2sech2(2.5x)+l

and

v2(x)=x(1-x)+l+l.

In this case, the condition (1.9) is not fulfilled because d2 = 1. The numerical results corresponding to the values l = 0.00(0.25)1.25 are listed in Tables 3.3(a)-3.3(c). (In this example, the values of h and hi; j = 1,2, listed in Table 3.3(a) are expected to be correct to eight significant figures.)

On the NumericM Performance of a Domain Decomposition...

0.00 0.25 0.50 0.75 1.00 1.25

hi 0.815 399 1.065 491 1.315 510 1.565 514 1.815 515 2.065 515

h2 73 1.121 813 26 74 1.371 813"33 77 1.621 813 33 72 1.871 813 33 54 2.121 813 33 71 2.371 813 33

h 1.937 329 02 2.437 329 08 2.937 329 08 3.437 329 08 3.937 329 08 4.437 329 08

167

Eh 1.2 2.4 5.0 1.0 2.1 4.5

] 5h

E-04 E-05 E-06 E-06 E-07 E-08

2.005 2.001 2.000 2.000 2.000

Table 3,3(a)

t 0.00 0.25 0.50 0.75 1.00 1.25

EA1)

EA2}

8.9 E-03 4.0 E-03 1.8 E-03 8.1E-04 3.7 E-04 1.7 E-04

1.027 1.011 1.005 1.002 1.000

8.8 3.9 1.8 8.1 3.7 1.7

E-03 E-03 E-03 E-04 E-04 E-04

1.017 1.007 1.003 1.001 1.000

Table 3.3(b)

l 0.00 0.25 0.50 0.75 1.00 1.25

Ex{1} 8.2 E-04 1.7 E-04 3.5 E-05 7.3 E-06 1.5 E-06 3.2 E:07

~x{1} 2.003 2.000 2.000 2.000 2.000

Ex{2} 6.8 E-04 1.4 E-04 2.9 E-05 6.1 E-06 1.3 E-06 2.6 E-07

5x{2} 1.999 2.000 2.000 2.000 2.000

E~{1} 2.4 E-04 5.0 E-05 1.0 E-05 2.2 E-06 4.5 E-07 9.4 E-08

5~{1} 2.007 2.001 2.000 2.000 2.000

E~{2} 2.2 E-04 4.5 E-05 9.4 E-06 1.9 E-06 4.0 E-07 8.4 E-08

2.000 2.000 2.000 2.000 2.000

Table 3,3(c) The numerical results of Tables 3.1-3.3 indicate clearly that in the three examples considered above 5h=2,

5g{j}=l; j = 1 , 2 ,

and

5x{J}, 6 ~ { j } = 2 ;

j=1,2.

Thus, the numerical results provide experimental support for the Conjectures 3.1 and 3.2, which were made in Remarks 5.2 and 5.4 of [14]. A c k n o w l e d g e m e n t . One of us (NSS) wishes to thank the State Scholarships Foundation of Greece for their financial support.

N. PapamicAael, N.S. Stylianopoulos

168

References [1] F. Bowman, Introduction to Elliptic Functions, English University Press, London, 1953.

[2] N.V. Challis and D.M. Burley, A numerical method for conformal mapping, IMA J. Numer. Anal. 2 (1982), 169-181.

[31 D. Gaier, Konstruktive Methoden der konformen AbbiIdung, Springer, Berlin, 1964. [4] D. Gaier, Ermittlung des konformen Moduls yon Vierecken mit Differenzenmethoden, Numer. Math. 19 (1972) 179-194.

[5] D. Gaier, Determination of conformal modules of ring domains and quadrilaterals, Lecture Notes in Mathematics 399, Springer, New York, 1974, 180-188.

[6] D. Gaier, Capacitance and the conformal module of quadrilaterals, J. Math. Anal. Appl. 70 (1979), 236-239.

[7] D. Gaier, On an area problem in con/ormal mapping, Results in Mathematics, 10 (1986), 66-81

[8] D. Gaier and N. Papamichael, On the comparison of two numerical methods for conformal mapping, IMA J. Numer. Anal. 7 (1987), 261-282.

[9] P.Henrici, Applied and Computational Complex Analysis, Vol. III, Wiley, New York, 1986.

[10] L.H. Howell and L.N. Trefethen, A modified Schwarz-Christoffel transformation for elongated regions, Numerical Analysis Report 88-5, Dept. of Maths, Massachusetts Institute of Technology, Cambridge, Mass., 1988.

[11] C.D. Mobley and R.J. Stewart, On the numerical generation of boundary-fitted orthogonal curvilinear coordinate systems, J. Comput. Phys. 34 (1980), 124-135.

[121 N. Papamichael, Numerical conformaI mapping onto a rectangle with applications to the solution of Laplacian problems, J. Comput. Appl. Math. 28 (1989) 63-83.

[13] N. Papamichael, C.A. Kokkinos and M.K. Warby, Numerical techniques for conformal mapping onto a rectangle, J. Comput. Appl. Math. 20 (1987), 349-358.

[14] N. Papamichael and N.S. Stylianopoulos, A domain decomposition method for conformal mapping onto a rectangle, Tech. Report TR/04/89, Dept. of Math. and Stat., Brunel University, 1989, (to appear in: Constr. Approx.).

[15] A. Seidl and H. Klose, Numerical conformal mapping of a towel-shaped region onto a rectangle, SIAM J. Sci. Stat. Comput. 6 (1985), 833-842.

[16] L.N. Trefethen, Ed., Numerical Conformal Mapping, North-Holland, Amsterdam, 1986; reprinted from: J. Comput. Appl. Math. 14 (1986).

On ~he Numerical Performance of a Domain Decomposition...

169

[17] J.J. Wanstrath, R.E. Whitaker, R.O. Reid, A.C. Vastano, Storm surge simulation in transformed co-ordinates, Tech. Report 76-3, U.S. Coastal Engineering Research Center, Fort Belvoir, Va. 1976. Received: August 1, 1989.

Computational Methods and Function Theory

Proceedings, Valpara/so 1989 St. Ruscheweyh, E.B. Sail', L. C. Salinas, R.S. Varga (eds.) Lecture Notes in Mathematics 1435, pp. 171-176 (~) Springer Berlin Heidelberg 1990

Planar Harmonic Mappings 1 Glenn Schober D e p a r t m e n t of Mathematics, Indiana University Bloomington, IN 47405, USA

1. I n t r o d u c t i o n Harmonic mappings occur in various frameworks and generalities. For this article they are simply complex-valued harmonic functions f = u + iv of the complex variable z, that are one-to-one and orientation-preserving. It is our purpose to show where these m a p p i n g s occur, to give a brief survey of some areas of study, and to mention several open problems. We begin with some examples, discuss a connection with minimal surfaces, touch on univalent function theory, and conclude with the m a p p i n g theory.

2. A n e x a m p l e Let f ( z ) = z - 1/5 in A = {z : Izl > 1}. Then f is clearly harmonic, and from the polar representation f ( r e i°) = (r - 1/r)e i° we see that f m a p s concentric circles in A onto concentric circles in C\{0}. It is not possible to m a p A conformally, or even quasiconfomally, onto C\{0}. So a harmonic mapping m a y not preserve conformal type. More generally, it is not much of an exercise to show that the functions

f ( z ) = z - 1/2 + Aloglzl are also harmonic mappings of A onto C\{0} when tAl _< 2. We mention t h e m because they are e x t r e m a l functions for several problems [7, sect. 3]. For example, the diameter of C \ / ( A ) is a minimum!

3. M o r e e x a m p l e s and a c o n j e c t u r e Consider mappings of the annulus Ap -= {z : p < ]z I < 1} given by f , ( z ) = tz + (1 1 t)/5. In polar coordinates this is f , ( r e i°) = (tr + - t ) e i ° . Again concentric circles r

are m a p p e d onto concentric circles. We restrict 1/(1 + p2) < t < 1/(1 - p2) so that aThis work was supported in part by a grant from the National Science Foundation (USA).

G. Schober

172

1--t tr + - is an increasing function of r and nonnegative. Then ft maps Ap onto an r

annulusA~ = {z : a < [ z [ < l } Now suppose that f = u + iv onto an annulus A¢. Then U = u apply Harnack's inequality in the

and0Ka<2p/(l+p2). is an arbitrary harmonic mapping of the annulus Ap + 1 is a positive harmonic function to which we shall following form:

H a r n a c k ' s i n e q u a l i t y . If K is a compact subset of a domain D, then there exists a constant M = M ( K , D) such that U(z) < M U ( ( ) for all z, ( • K and all positive harmonic functions U in D. To be specific, choose for K any circle in Ap about the origin. T h e n for topological reasons f ( K ) surrounds the origin in A, and there are points z, ~ • K such that u(z) is to the right of a and u ( ( ) is to the left of - a . T h a t is, we have a + 1 < u(z) + 1 < / ~ l [ u ( ( ) + 1] < M [ - a + 11, and this implies a < ( M - 1 ) / ( M + 1). In other words, a is bounded away from 1 no m a t t e r what f is. On the other hand, the examples at the beginning of this section show that a can be as small as zero. This is a surprising dichotomy. The image annulus cannot be too thin, but it can degenerate to a punctured disk. These examples and the cute little proof were given by J.C.C. Nitsche [9] in 1962. He conjectures that a < 2p/(1 + p2) for all harmonic mappings of Ap onto A~. It is still an intriguing open problem. Try it!

4. A c o n n e c t i o n w i t h m i n i m a l s u r f a c e s Let S denote a nonparametric minimal surface in R3 that lies above a simplyconnected domain /2 in the complex plane, /2 # C. That is, S = { ( u , v , F ( u , v ) ) : u + iv • / 2 } where F satisfies the minimal surface equation

1+ LOvJ J

-2

1+ L~-~j j ~Tv~ :0.

This equation simply states that the mean curvature (the sum of the two principal curvatures) is zero at each point of S. Although S is a nonparametric surface, that is, the graph of the function F, S admits a reparametrization as a parametric surface ,9 = {(u(z), v(z), F ( u ( z ) , v(z)) : z • D} so that each coordinate function is harmonic. This defines a one-to-one mapping from D onto 8, and its projection down to $2 is a harmonic mapping f ( z ) = u(z) + iv(z) from D onto /2. It is no loss of generality to assume that f is orientation-preserving. Here is the strategy. Use knowledge about harmonic mappings of D o n t o / 2 to gain information about nonparametric minimal surfaces S that lie over/2. In 1952, E. Heinz [4] showed that the Fourier coefficients of a harmonic mapping f 3R 2 of D onto the disk /2n = { w : ]w[ < R} with f(0) = 0 satisfy 2([c_1[ 2 + [El[2) > 4----~--"

Planar Harmonic Mappings

173 47r3

As a consequence, this led him to the estimate [K I < ~ - 7 for the gaussian curvature K (product of principal curvatures) at the point above the origin of any nonparametric minimal surface S that lies over ~2a. Although the curvature estimate is not sharp, it has a very important consequence. Suppose that S lies above the entire plane C. Then we may let R --* oc in the curvature estimate to conclude that K = 0. After a translation we find that the sum and product of the principal curvatures are zero at every point. This implies that S is a plane. In this fashion Heinz gave a new proof of the following: B e r n s t e i n ~ s t h e o r e m . The only nonparametric minimal surfaces that lie over (are parametrized by) the entire plane are ~hemselves planes. 27R 2 More recently, R.R. Hall [3] obtained the sharp estimate 2(Ic_112 + [cll 2) > 2rr----~ for what became known as the Heinz constant. Based on it the estimate above for the 16rr 2 gaussian curvature can be improved to IKI _< 27R2 , which unfortunately is still not sharp. We shall explain why later, but the basic reason is that there are more harmonic mappings under consideration than nonparametric minimal surfaces. T h a t is, there is no such surface S that corresponds to the extremal harmonic mapping for the Heinz constant. The sharp estimate for the gaussian curvature is still an interesting open problem. Various estimates for the gaussian curvature can be found in the book by R. Osserman [10]. Sharp estimates in case ~2 is a half-plane, strip, or slit-plane can be found in [8]. The statement that f is harmonic on D can be written as fz~ = 0 in D. This implies that fz is analytic and f~ is anti-analytic. In other words, we have f = h + .~ where h and g are analytic in D. The function a = -f~/fz = g'/h' has special significance. If f is orientation-preserving, then its Jacobian determinant Jf = If~l 2 - I f , l2 is nonnegative. As a consequence, the function a is analytic and la(z)l _< 1. In fact, we have la(z)l < 1, for otherwise, a would be a constant of modulus one and '1f would vanish identically. On each compact subset a is bounded away from one, and so f is locally quasiconformal. However, a harmonic mapping need not be quasiconformat since its distortion may be unbounded at the boundary. In order for f to derive from a nonparametric minimal surface, the function a must be the square of an analytic function. That is, zeros of a must be of even order. This is not the case for the extremal function for the Heinz constant. When it is defined, the function x/~ is called the Weierstrass-Enneper function, and i / v ~ turns out to be the stereographic projection of the Gauss map of the surface 8.

5. U n i v a l e n t h a r m o n i c f u n c t i o n s This is an area pioneered by J.G. Clunie and T. Sheil-Small [2]. Let S 0 denote the class of one-to-one harmonic orientation-preserving mappings f = h + ~ where h and g are analytic in D and h(z) = z + E,~=2 ~ a,~z '~, g(z) = z.,n=2x-"°° b,~z'*. We shall mention only a few results and compare them with the familiar schlicht class S = { f C S ° : g - 0}.

G. Schober

174

If f E S, then the Koebe one-quarter theorem asserts that the disk {w : Iwl < 1/4} is contained in the image f(D). For f E S ° , Clunie and Sheil-Small have the comparable result that the disk {w : ]w I < 1/16} is contained in f(D). They conjecture that 1/16 should be replaced by 1/6. If f E S, then the de Branges theorem asserts that the coefficients satisfy Janl < n. For f E S~/ the coefficient conjectures are la,~] <: ~(n + 1)(2n q- 1) and Ib~] _< l ( n - 1)(2n - 1), n >_ 2. There is a function in S ° that has precisely these numbers as coefficients. It maps D onto the whole plane with a radial slit from - 1 / 6 to infinity. Here, too, most of these inequalities are still conjectures. For f E S } , the corresponding function a = g'/h' is bounded by one and vanishes at the origin. Therefore Schwarz's lemma implies 1252I = la'(0)l < 1. In this way the coefficient conjecture Ib21 <_ 1/2 is verified. What about the conjecture la~l _< 5/2? In [21 they prove that tael < 12,173 (!). More recently, Sheil-Small has reduced this estimate to lael < 57. There is a lot of room for improvement, and very little is known about the higher coefficients. Clunie and Sheil-SmaU [2] have many sharp results in case f E S } is convex, convex in one direction, or close-to-convex.

6. A mapping theorem Here is the problem: M a p p i n g p r o b l e m . Let (2 be a simply-connected domain in C, ~2 ~ C, and let a be analytic in D and satisly la(z)l < 1 for z E D. Does there exist a one-to-one harmonic mapping f of D onto ~2 that satisfies the differential equation f~ = a f t ? The ease a = 0 is taken care of by the Riemann mapping theorem, and so we assume a is not identically zero. First we note that a smooth solution of the differential equation f~ = a f , will necessarily be harmonic. To see this, take the E-derivatives of both sides to obtain I/~zl = lallf~l. Since lal < 1, we must h a v e fz~ = 0. The following example shows that some caution is in order. E x a m p l e . Let J2 = D and a(z) = z. If we write f = h + .q where h(z) = z~n=0v'°°anz" and g(z) = ~n~=l b,z n, then the differential equation becomes g' = zh'. It implies that (n + 1)bn+l = nan for n _> 0. Therefore the area of f ( D ) is A =

2 _ lSle)dxdy

n[la~l 2 -tb~121 = ~r

= ~r

D

n=l

la~l 2 n=0

and the mean M2 =

If(e~°)12dO =

[lanl ~ + Ibn+1151 = ~ n:O

Since2[n---~l] < 1 +

1+ ~

la~l ~-

n=O

[ n - ~ ] 2, we find that A < ~7rM.2. I l l ( D ) C D ,

thenM2 < land

the area of f ( D ) is less than r / 2 . Since the area of D is ~r, the function f cannot map D onto D.

Planar Harmonic Mappings

175

This example shows that the answer to the mapping problem as stated can be no. The following is the best positive statement that we know at present. (cf. [6]).

Mapping Theorem [6]. Let t9 be a bounded simply-connected domain with locally connected boundary. Fix wo E t9, and let a be analytic in D and satisfy [a(z)t < 1. Then there exists a univalent, harmonic, orientation-preserving mapping f with the following properties: (a) f maps D into t9 and f(O) = Wo, fz(O) > O.

(b) f satisfies f~ = afz. (c) There exists a countable set E C cOD such that

(i) the unrestricted limit f ( e ~°) exists, is continuous, and belongs to cOt9 for e i° E cOD\E; (ii) the one-sided limits f(eqS+O)) and f(e~(°-°)) exist, are different, and belong to cOt9 for e~° 6 E; (iii) the cluster set o f f at e ~° 6 E is the straight-line segment joining f ( e i(°+°)) to f ( e i(°-°)). The theorem does not and cannot claim that f(D) = t9. However, the boundary values of f exist and lie on cOt9 with only countably many exceptions. We emphasize that the function [a] is permitted to tend to one at the boundary of D. In that case the mapping f is only locally quasiconformal, the corresponding differential equation is no longer uniformly elliptic, and one may expect pathology at the boundary. In general, the uniqueness of f is still an open question. W h a t happens in the case of our example t9 = D, a(z) = z? The function f(z)=

g(de)ae{(e 'e + z ) / ( e 'e - z))dO

with g(e ~°) = e 2~ki/3 for (2k - 1)~r/3 < 0 < (2k + 1)7r/3 and k = -1, 0, 1 satisfies f~ --- zfz and all other conclusions of the mapping theorem. It maps D onto a triangle with vertices at e -2~i/3, 1, e 2~I/3. Except for three points on cOD, these are the boundary values of f . The cluster sets of f at those three points account for the edges of the triangle. In this case we know that the mapping is unique [5, sect. 6].

7. Constructive approximation It is of interest actually to construct the mappings from the theorem in the previous section. This has been carried out by D. Bshouty, N. Hengartner, and W. Hengartner [1] in case t9 is strictly starlike and Ha[[~ < 1. Under these hypotheses they show also that the mapping is unique.

G. Schober

176

References

[1] D. Bshouty, N. Hengartner, W. Hengartner, A constructive method for starlike harmonic mappings, Numer. Math. 54 (1988), 167-178. [2] J. Clunie, T. Sheil-Small, Harmonic univalent functions, Ann. Acad. Sci. Fenn. Ser. AI 9 (1984), 3-25. [3] R.R. Hall, On an inequality of E. Heinz, J. Analyse Math. 42 (1982/83), 185-198. [4] E. Heinz, ~Tber die LSsungen der Minimalflgchengleichung, Nachr. Akad. Wiss. GSttingen, Math.-Phys. KI., (1952), 51-56. [5] W. Hengartner, G. Schober, On the boundary behavior of orientation-preserving harmonic mappings, Complex Variables Theory Appl. 5 (1985), 181-192. [6] W. Hengartner, G. Schober, Harmonic mappings with given dilatation, J. London Math. Soc. 33 (1986), 473-483. [7] W. Hengartner, G. Schober, Univalent harmonic functions, Trans. Amer. Math. Soc. 299 (1987), 1-31. [8] W. Hengartner, G. Schober, Curvature estimates for some minimal surfaces, in Complex Analysis, Articles Dedicated to Albert Pfluger on the Occasion of His 80th Birthday, J. Hersch and A. Huber Eds., Birkhguser, 1988, 87-100. [9] J.C.C. Nitsche, On the module of doubly-connected regions under harmonic mappings, Amer. Math. Monthly 69 (1962), 781-782. [10] R. Osserman, A Survey of Minimal Surfaces, Dover, 1986. Received: June 1, 1989.

Computational Methods and Function Theory Proceedings, Valparaiso 1989 St. Ruseheweyh, E.B. Saff, L. C. Salinas, R.S. Varga (eds.) Lecture Notes in Mathematics 1435, pp. 177-190 @ Springer Berlin Heidelberg 1990

E x t r e m a l P r o b l e m s for N o n - v a n i s h i n g H p Functions T.J. Suffridge Department of Mathematics, University of Kentucky Lexington, KY 40506, USA

1. I n t r o d u c t i o n The family P of functions that are analytic in the unit disk D = {Iz] < 1} and satisfy P(0) = 1 and ae[P(z)] > 0 when z e D are well understood ([2], [6]). The family B of bounded analytic functions that never assume the value 0 are related to P as follows. Let f E B. Then f(z) = 7e -tP(") where [~/I = 1, t > 0 and P E P. With this simple relationship between the classes :P and/~, it is surprising that the problem: Find A~ = max/or3 [an[ for fixed n = 1 , 2 , . . . (where f(z) = ao + alz + . . . + anz n +'" ") is so difficult even for relatively small n. The problem is trivial for n = 1, it can be solved rather easily for n = 2 but it is much more difficult for n = 3, [3] and n = 4 [5]. The problem has not been solved for n >__5. The solution conjectured by Krzyz [4] is that An = 2/e and that [an[ < 2/e unless (1)

f(z)

= ce ('Tzn-1)/('rzn+l),

Icl =

= 1.

This conjecture is indeed true for n = 1,2,3,4. A rather natural extension of the above question is to assume

[ 1 [2. f e H p, Mp(f) = sup L2--~Jo r
] lip [f(rei°)l'dO

<_ 1

and f is non-vanishing and to again ask for An(p) = sup la, I the sup being taken over the family described above. Hummel, et. al. [3] conjectured that An(p) = (2/e)1/% 1 + ~ = 1, 1 < p and that the only extremal functions are

(2)

21/p

T.J. Suffridge

178

where IcI = I~1 = 1. Brown [1] has verified this conjecture for n = 1 and for other values of n with the added hypothesis that certain coefficients are 0. Hummel, et. al. [3] observed that for p = 1, it is easy to see that A~(1) = 1. This follows from the fact that

an = 21r Jo so that

1[~-1 lanl _ 2~ Jo

r n ~°f(rei°)dO'

1

II(~e'°)ldO <- ~-~

for all r < 1, assuming f E H 1, II/111 ~ 1. However, la~l = 1 for every polynomial 2n

n

n

E aJ zj = c(I-[(1 + eiaJz)) 2 = c ( E bJzi) 2, j=O

where

c

=

j=l

j=O

1/Ej=I~ Ib~l~.

In this p a p e r we study continuous linear functionals on the family Np of functions in NP, 0 < p, with the properties Ilfllp -< 1, and f ( z ) ¢ 0 when z E D or f = 0. For each p, Np is compact in the topology of uniform convergence on compact sets. We let /2 denote the family of complex valued continuous linear functionals on the family of functions t h a t are analytic in D with the topology of uniform convergence on compact sets. For a given L e g, we study max/e/v, a e L ( f ) which is the same as rnaX]eNp IL(f)l because F C Np if and only i f T f C Np for a l l T , 13'1 = 1. Our main theorem is the following. T h e o r e m 1. Suppose L E f~ is non-trivia1 and f E Np satisfies R e L ( f ) = maxgeNp[ReL(g)], 0 < p < ~ , Then there is a function h E H 2 that extends to the dosed disk D so that - - h - - is continuous, I[hll2 = 1 and having the following properties. + z e ic'

(ii) f ( z ) = ( h ( z ) y P I ( z ) , where I is an inner function with I = e -P, P(z) =

fo 2~r 1 + ze ia

1 -- ze i---~d#(c~)'

where ]1 is a positive measure (or O) with support contained in the zero set of

~(d°)

(iii) I f L depends only on the coet~cients ao, al, ..., a,~ of an analytic function ~ aizi, then h is a polynomiM of degree _< n and # consists of at most n point masses. In case p = oo (i), (ii) and Oil) hold for at least one extremM function f . In order to see how one m a y apply theorem 1, we prove theorem 2 below. T h e o r e m 2. Suppose f E ~ , ao > 0 and a~ >__0 (this can be accomplished by replacing f by 3'f(ze i~) for appropriate a and 3'), then

E x t r e m M Problems for Non-vanishing HP Functions

179

(i) al <_ (2/e) 1-1/p when p > 1 with equaJity if and only if I ( z ) = 2-~/p(1 + z)V,e-Cl-,mci-,)/(,+z) and

(ii) a, <_ p-~/2(2 - p)l/P-~/22~-t/~ when 0 < p < 1 with equMity if and onty if f ( z ) = (1 + p~)-~/,(1 + pz) ~zp where p~ = p / ( 2 - p).

Proof. Choose L ( f ) = a~. By theorem 1, (iii), h is a polynomial of degree zero or 1. If degree h = 0, using (ii) f - 1 so L ( f ) = 0 which is clearly not maximal. Therefore degree h = 1. We may assume f has real coefficients because f C Hv implies g( z ) = ( f ( z ) f ( 2 ) ) 1/2 e H~ while g has real coefficients and g( z ) = ao + ai z + . " , f ( z ) = ao + alz + a2z 2 + . . . , i.e. the first two coefficients agree. Using (i) we see that al + aoe i~' + aoe -i~ = al(c0 + c~e-i~)(~ae i~ + c0). Thus Co and Cl can be assumed to be positive and (3)

ao + alz + aoz 2 = al(coz + cl)(c,z + co).

Now suppose # is not 0. Then h has a zero on Izl = 1 and hence Cl -- Co =- 1/x/~ so (4)

f(z) =

2-l/P(1 -[- Z)2/Pe -t(1-z)/(l+z)

is a possible extremal. Since a0 = aa/2 by (3), while aa/ao = 2 / p + 2 t by (4), we conclude t = 1 - 1/p >_ 0 and this is a possible extremal only if p "_ 1. If # is zero then I(z) - l, cl = p c 0 _ < c 0 a n d c 0 = l / ( l + p 2 . Thus

(5)

:(z) = (1 + p~)-'/~(1 + pz) ~/~,

s°thatala0 - 2P by ( 5 ) and a° =

by ( 3 )" Thus l + p2 = 2p2 and ( ~ - l ) p~ =

Since p < 1, this is only possible when p < 1. The proof is now complete since in this P • case, p2 _ 2-p" Note that the result of theorem 2 was given by Brown [1] by a different m e t h o d for p > 1 while the result for 0 < p < 1 is new. Further, it is easy to see that Brown's f(z)---ao+a,~z m+'''+a,~z '~+'-'wherem> ~, p > 1, then result that i f f E N p , la,d < (2/e)1-~ is true, as follows. Under the above assumption, the function

/ ~

~11~

g(z) = [ I I f j ( z ) ] /

, f j ( z ) = f(ze2~i(J-1)/~),

is n-fold symmetric (i.e. g(z) = k(z n) for some k e Np) and satisfies

g(z) = ao + anz ~ + ' " . It therefore follows that a~ for this restricted class has the same bound as al for the entire family. While it is conjectured that the extreme value for la~[ in the entire family is identical to that for lall when p > 1. This is not true for p < 1: As we shall see below, la2] has the sharp bound 2~-2/P(2 + p)2/p-1/p (the square of the sharp bound for Jail) when p < 1.

T.J. Suffridge

180 2. Proof

of main

theorem

Given a non-trivial function f E Np, we may write

f ( z ) = (h(z))2/VI(z),

(6)

where h E H 2, h(z) 7t 0 when z E D and I(z) = e -P(~) is an inner function, h ( 0 ) > O, P(O) >_ O. Lemma

1. Suppose h and k are outer functions in the space H 2 and that h(O) >

o, k(o) > o. zf h" and k" are the corresponding boundary functions and I h * ( e ° ) l

=

[k*(ei°)[ a.e., then h = k. Proof. By hypothesis,

h(z)

=exp

~

~e it-z

,~ e it -- z

= exp = k(z).

L e m m a 2. K h ( z ) = ~ = o ckz k E H ~, h(z) ¢ 0 in D then for 0 < e < 1, the function

he(z) = (1 + eeiZz~)h(z)

[ck12(1+ e2) + 2eRe ~ ckek+~e iz k=O

has the same properties and [[h~[[2 = 1. Proof. Use the fact that for E~=oakz k = f ( z ) C H 2, llfI[~ = Ek=0~ lakl2 and n-1

oo

(1 + c~%")h(z) = F. ~ z~ + ~ ( c ~ + . + c~'ec~)~+". k=0

Lemma

•

k=0

3. I f h and h~ are related as in L e m m a 2, then /" iz ET=o ckek+~

Proof. A straightforward differentiation.

h

•

L e m m a 4. I f L C £ is a non-trivied//near functional and f E Np has the property R e L ( f ) >_ ReL(g) for M1 g C Np, 0 < p < 0% where f = h2/PI, h C H 2 and I is inner, then [[hII2 = 1 and oo

(7)

L ( z ~ f ) = L ( f ) ~_, ck~k+,~ for n = O, 1 , ' " . k=O

Extremal Problems for Non-vanishing H p Functions

181

Proof Clearly [Ifllv = Ilhll2 and flfl[p < 1 implies pf E Np for some p > 1 so that

ReL(pf) = pReL(I)

> a e L ( / ) (because L is non-trivial and the assumed extremal f satisfies R e L ( f ) > 0). Thus, (7) holds for n = 0. Now consider L = (h~)~/~I, h~ as in L e m m a 2. T h e n ReL(f~) _< R e L ( f ) and hence

a e L ~, d~ L o ]

-

We have

df~

= ~(h~)2/p_ll dh~

de ~=0

= 2 f (ei~z~

de ~=0

Reei~?ock~k+~)

P

=

"

Therefore Re

eiZL(z~f) - Re eiz

ck~k+,L

<_ O.

k=0

T h a t is, Re{eia(L(z~f) - ~°=o ck~k+,L(f))} <_ O. Since/3 is arbitrary, the l e m m a now follows. • Note that the expression

oo Z CkCk-l-n k=O

is

2re Jo This is the coefficient of e -~"e in the Fourier series for lh*(e~°)12 and we write it as (lh12,~ ") = (zL thl~). For 0 < p < 1, 0 < p < oo and I extremM for L (i.e. R e L ( f ) _> maxgeNp ReL(g)), we have

L

+

)--f~ = L(f) + 2

L(z'~f)p'~e 'n~ = L(f) + 2 ~ , L(f)(z '~, [hl2)Pnein%

Using the continuity of L and taking real parts, we obtain

-~f]

= Re(L(/) + 2 ~ L(f){z ~, Ih[2)d~) = i(f)(1+ .=1 ~
is continuous. This is (i) of the main theorem for 0 < p < oo. To obtain (ii), note that everything is clear except that the support of # is contained in the zero set of h(e~°). To see this, note that R e L ( f I ~) has a m a x i m u m at e = 0. Thus, R e L ( f log I ) = 0. T h a t is

T.J. Suffridge

182

ReL

fro

'~

2.1+ze.

2,~

i--~e itd#)= L(f) fo Ih(eit)[2d# = O,

and (ii) follows. In case L depends only on the coefficients ao,aa,'",a,~, then L(zmf) = 0 when m > n. This implies (z m, ]hi 2) = 0 for m > n and by a result of Fejer, (since [h[ 2 is a non-negative real trigonometric polynomial of degree _< n), h is a polynomial of degree n. To see this, note that

Ih(¢~°)t2 = ~ d~e~k°, d_~ = dk. With

K(z) = ~ dkz k+~, k=-n

we have

e-ln°K(e i°) = Ih(d°)l ~. Clearly, the zeros of K on

Izl

= 1 are of even multiplicity and since

(i.e., K is self inversive) we have

I~'(z) = c

(1 + ~'",z) 2

(i + p / ~ , z ) ( i j=~+l

where each Set

+ 2e~Jz) PJ

, J

pj <2 1 and c > 0. H(z) = K(z) ~

(1 +

pjei~z)

n

~:,+ipj(i+~e'~'z) :¢'II(i+PJj:l

e'~'z "2

)'

wherepj=l, l _ < j < s , pj < l , j > s , c ' > 0 . Thus, Q(z) = Vf~/t-Ijn=l(1 + pjeia'Z) has the property, Q is outer and IO(e'°)t 2 = lh(e~°)12. This means Q = h. This completes the proof of (iii) in the main theorem. To prove that (i), (ii) and (iii) hold for some extremat f in case p = 0% let fp be extremal for L in h r , p _> 2. Since N ~ C - ~ C N2, 2 _< p, the family {fp} = {(hp)2/PIp} is locally uniformly bounded. Thus, as p --* c~, there is a sequence {p~} such that pn ~ c~ and hp, --* h, Ip, -~ I uniformly on compact subsets of the disk D. Clearly, ReL(fp) > maxg~N~ ReL(g) and hence R e L ( f ) _> maxgeN~ ReL(g) by continuity of L. The theorem now follows. •

183

E x t r e m a l P r o b l e m s for Non-vanishing HP Functions

3. Applications

Proposition.

I f f E Np then If(r)l <

S(z)-

wi~h equality for

(1 - r2)llp (1 - r2) 1/v

Of course one can prove this directly using the fact that for f E H 2,

If(r), =

tlfll ~ 1,

]N=. f(z)

2 r Jo

pe ie -- r

2~r

.

\1/2

( 1

f2~

)

,0 2dO~ 112

< - -1 -

l ~ - Z ~ _ r 2,

while 9 e g v , ][g[[v < 1, g(z) ~ 0 in D implies 9(z) = ( f ( z ) ) 2/p for some f e g 2, []f[[ _< 1. Using T h e o r e m 1, the proof is as follows. For the extremal function f , r ~ f ( r ) = L ( z " f ) = (z ~, Ihl2>f(r).

This implies

Ih(e~°)[ ~ = 1 + E.%~ ""(e ~"° + e -~"°)

reiO =

I

+

-

-

+

1 - re ~8

re-lo -

-

1 -- re -~°

1 -- r 2

I1 - re~°} 2 We conclude h(z) - x/1 1 - - -- r zr 2 (up to a constant of modulus 1) and since h(z) # 0 when (1 - r2) 1/"

Izl = 1, I --- 1. Therefore, f ( z ) - (1

rz)~/p is extremal and f ( r ) < 1/(1 - r2) lip.

•

T.J. Suffridge

184

Remark 1. Theorem 1 remains true in the restricted class {f E Np : f(0) > 0) provided the functional L has the property that L( f ) > 0 for extremal functions f with max ReL(g ) = ReL( f ) over the restricted family. Theorem 3. I f f E

when (1 + r ) ( l - r)'/p-'

%, and f(0) > 0, then

2 1, where

and

when (1

+ r ) ( l - ~ ) ~ / p - \ 1, where p is a solution of the equation

Remark 2. Note that the result in Theorem 3 is not the correct bound for the entire class N,. This is easily seen in case p = m. In this case, since Ref(r) 1, if we require f(0) > 0, clearly Re(f ( r ) - f(0)) < 1. However, if f (z) = - e - t P ( z ) where P(0) = 1 and t is small and positive then f(0) can be near -1 but with r near 1, we may choose P so that tP(z) = u vi, u small, v = a. Then f (r) - f (0) will be nearly 2. The problem maxj~,v,Re( f (r) - f (0)) is more difficult.

<

+

Proof of Theorem 3. If f E N,, f (0) > 0 then the function g(z) = (f (z) . fo)1/2 E NP, g(0) = f (0) > 0 and g(r) = If(r)l 2 Ref(r). Thus, by Remark 1, Theorem 1 applies to the functional L(f) = f(r) - f(0). Further, the extremal function can be assumed to have real coefficients. We have rnf (r) = L(znf ) = [f(r) - f (0)](zn, 1 hi2), so that

ExtremM Problems for Non-vanishing HP Functions

where p is real, ]p[ < 1, z = e ~°, and K -

185

rf(O) If [p[ = 1, it is easy to check f(r) - f(O)"

t h a t p # - 1 (i.e., the relation 1 -

r2 -

2r 2

f(O)

2rf(O)

_

f(r) - f(O)

f(r) - f(O)

c a n n o t hold), so Ipl = I implies p = 1. In this case, we find t h a t

h(z)

Since

tlfllp

= 1,

= v~

K-

l+z

and

I - rz

f(z)

~

K 1 / p ( I + z ~ 2/p e - iP-* 'gg. \1 - rz/

1 -2 r , s o t h a t f(r) - f(O) = 1 2r_r f (0),

hence

~1 _+ r~ ~ ] 21pe _ t ( l _ r ) / ( l + r )

_

f(r) -- 11+ r f ( o )

and

-i-r~ -'. 1+

nus, e,+, = ( l + r ) ( 1 - r ) 2/p-a a n d we obtain the value of t given in the t h e o r e m . Because t >_ 0, this can be e x t r e m M only for the values of p for which (1 + r)(1 r ) 2]p-1 > 1 (this clearly includes p >_ 2). In case [p[ < 1, we find as before t h a t p > 0, so t h a t W

~

--

lh(z)l -

1 K 11 - ~ z l ~ 7 I(1 + p z ) ( p + z)i -

z=ei°,andweeoncludeh(z)=,/~SI+pz V p 1-rz' A g a i n u s i n g Ilfllp = 1, w e get K -

f ( r ) - f(O) =

T h u s , since f(O) =

I z l < l . Thus, f ( z ) =

l +p(1 2 p r- +rp2)2

r(1 + 2pr + p2)

~(~-~)

1 K [1 + pzl 2, I1 - ~ z p p

( K ) 1/" ( l + p z ~ 2/" \ ~ ] "

' so

. f(O), f(r) = r +pp ( l + p1r ) f tr02a . , ,

we have

(1 + ?<+_

f(r) - f(O) = P \

1 - r~

Using f ( r ) -- r + p (1 + pr~ ( 1 + flr~ 2]p--1 - P \1__--'-S-~] f ( 0 ) we see t h a t p m u s t satisfy \ 1 _ - ~ - ~ ] --

r

3t- p P

T h i s clearly has no solution when p > 2 a n d since p varies continuously with p a n d tends to c~ as p -~ 2- and p = 1 when (1 + r)(1 - r) 2/p-1 = 1, the t h e o r e m now follows. Note t h a t if we consider ( f ( r ) - f(O))/r for the e x t r e m a l f in the t h e o r e m and let r -+ 0 + , t h e result agrees with T h e o r e m 2. • We n o w consider in detail the special case L ( f ) = a~. Since f E Np if a n d only if 7 f ( z e i~) E Np for all 7 and a, [71 = 1, a real, we m a y assume t h a t for the e x t r e m a l , f ( 0 ) > 0 a n d a , > 0. T h e following definition will be useful.

T.J. Suffridge

186 Definition

1. If f ( z ) = ao + alz -4-... and 9(z) = bo + blZ + ' "

Jr" are analytic in a

n

n e i g h b o r h o o d of 0, t h e n f ~ g means a0 = b0, al = b l , - ' - , a~ = bn. R e m a r k 3. N o t e t h a t if f n g, h is analytic in a n e i g h b o r h o o d of 0 a n d F is analytic in a n e i g h b o r h o o d of a0, then /1

(i) (ii)

h'f~h'g Fof.~

n

Fog.

Using T h e o r e m 1 a n d assuming L ( f ) = a,, we see t h a t h is a p o l y n o m i a l of degree < n for the e x t r e m a l I ( z ) = (h(z))~/PI(z) and t h a t we m a y take m

h(0)=c0>0

and

I(z)=e

-tP(~),

tP(z)=Etjl+ze; 1 - zei~J ' j=l

where 0 < m < n, each tj > 0 and each factor (1 - zei%) is a factor of h. Now set

h ( z ) = z=h (3) so t h a t [A(eiC~)12 = e-in~[h(ei~)h(ei~)] a n d note t h a t hh is a self-inversive p o l y n o m i a l of degree n + degree h < 2n. Further, each zero of h t h a t lies Oll lzl --- I is also a zero of A a n d therefore is a zero of hh of even multiplicity. Using T h e o r e m I (i), for the extremM, we see t h a t n-1

a. + ~ ( a k e i(~-k)~ + ~-~e-i(~-k)*) = a . e i " ~ h ( e - i ~ ) h ( e -i~) k=O

a n d thus,

ao + •lz + 62z 2 + . . . + a~z ~ + an-1 zn+l "4-"" q- ao z2n = a n h ( z ) h ( 2 ) , when ]z[ = 1. This equality therefore persists for all z and we see t h a t a0 = a,~coc,, so t h a t c,~ > 0 a n d degree h = n. Further,

ao + a l z + . . . + anz ~ + a ~ - l z T M + . . . + aoz 2n = a~h(z)A(z). W e n o w state the next theorem. Theorem

4. I f f E Np satistles f(O) > O, am > 0 and a~ > [g(~)(0)[ for a1I g E Np -

then

n!

(i) f ( z ) = h2/pI, where h is a p o l y n o m i M of degree n, I = e -tP(z), where m 1 q- ze'% t P ( z ) = ~ tj 1 j=l for s o m e m , o f h,

O <_ rn <_ n, each t d > 0 , t = ta + " " + t m and each (1 - ze iaj) is a factor

(ii) f n a~hh, and (iii) for 1 < k < n,

ExtremM Problems for Non-vanishing HP Functions

m

j=l tie ik~j = ~1

( ~ ) m~ 1-

j=l

eik~~ +

187

~___~fi[ (2)] 1 k -2¥+pj j=m+l P'J .

m n 1-pje'~z), where h(z) = c l-Ij=l(1zei"~z)1-Ij=,~+l(

1-

eik~j,

0 < pj < I for each j, m < j < n.

Proof. (i) and (ii) follow from the discussion preceding the statement of the theorem. To obtain (iii), use the fact that ½log f ~ ½log(a, hh). • Actually, if the extreme value for an = a~(p) is known to occur when h given in Theorem 4 has all its zeros on lzl = 1, 1 < p, then a,~(p) = (a~(oa)) ~-Vp. Clearly, this supports the conjecture that an(p) = (2/e) I-1/p since a~(o~) is known to be 2/e, n = 2, 3, 4, with the extremal for p = c~ occuring only when h(z) = ~1( 1 + z ~) or a rotation. Incidentally, Theorem 4, (ii) can hold for h(z) = c(1 + z)(1 + a z cos a + z 2) when n = 3 and h 7~ ~ ( 1 + z a) (see [3]). That is there are are other "local" maxima for a3 than the one that yields the absolute maximum. 1 < p, then (ii) of 11 Theorem 4 holds if and only if f ~ ( z ) ~ an(oc)hh and a~(p) = (an((X))) 1-1/p. Theorem

5. If all zeros of h lie on Izl = 1 and fp(z) = h~/vI,

Proof. Given h E H 2 of degree n with all zeros of h on I~l = 1, h(0) > 0, h(z) h(z), we have f ~ ( z ) e -tP(") satisfies f ~ ( z ) ~ a~(cc)hh if and only if e -t(x-Vp)P(~) (an(~))l-a/ph 2-2/p if and only if h2/Pe-t(~-l/P) P(z) .~ (an(~))l-1/Phh using Remark The theorem now follows.

= n 3. •

Further, the inner part of f cannot be trivial for the extremal when p > 1. We believe it is trivial when p < 1. T h e o r e m 6. If f E Np is extremal /'or the problem L ( f ) = a~, n > O, and 1 < p, then the inner part I in the representation f ( z ) = (h)2/PI is non-trivial O.e. 1 7~ 1). Proof. We know that if I is trivial then (h) 2/p ~ a,,h]~. Hence, with h(z) = Co + ' . . + c~z '~, we have cg/p = a,~coc,, However, since h(z) 7~ 0 in D and Ilhll= = 1, 0 < ic~l _< Ic01 < 1. Thus, cg/p < a,~c~ so that c~/p-2 < a,~. However p > 1 implies 2/p - 2 < 0 so a~ > 1. We know this last inequality is false and it follows that I cannot be trivial. • We have a complete solution for n = 2. T h e o r e m 7. If f E Np, then

la~l _< (2/e) 1-V',

(s)

la l < (_L_2

(9)

- \2-p)

1 < p,

_2 p'

p
with equality if and only if 2-1/P(1 + z2)2lpe -O-lIp) 1-"2 1+~2, (10)

f(z) =

P)z+

Pz21

/

1 < p,

,

p
T.J. Suffridge

188

or a rotation. Proof. Using T h e o r e m s 4 and 5, (8) clearly holds when m = 2 with equality as s t a t e d when p > 1. F u r t h e r , f e i"V'pimplies (f(z)fi~.)) i/2 E Np. However, g(z) =

ao

l+2Realz+(2 ao

)

a2 + l a l l ~ z 2 \ ao

= a o + 2 R e a l z + (a2+

2ao~ ]

i/2

+""

[alt2-(Real)2) 2~o

z2+''"

so clearly al is real for the e x t r e m a l and in fact we m a y a s s u m e f has real coefficients. In case rn = 0, f(z) = (Co + clz + c2z2)2/p, Co > 0, c2 > 0, we t h e n have either 0 = ( ~ + p(1 - ~ ) ) c o s ~ , (11) 0=(p~+p2(1-~))

cos2~,

or I o = i + i + (p,+ p~)(i- 2), Pl

P2

P

(12) I I 2). 0 = p-~ + P-~2+ (p~ + p~)(1 - P

In the second case,

0 ~---(pl-l'-p2)(p@p2"~(]-- ~)): (p~.~_p2)(pllp22..[_(1 - ~)) . 1 2~ -

Thus 2 / p - 1 =

PiP2 done

1

so pip2 = 1 and thus pl = p2 = 4-1, p = 1. T h i s case is

PiP2

o

--p2'

In the first case, cos a ~ 0 yields (2/p - 1) = 1

0=

2

~+

p

-

~/-~. p = V Z 2p ....P a n d either

1=--2,p

1 - 2 -Pp or cos 2 a = 0. T h u s cos c~ ¢ 0 implies cos 2~ = 0. T h e n p2 we m a y take cos ~ = ~

p=l ' cos (~ = 4-1._}__ v~' and

(i.e. replace f(z) by f ( - z ) if necessary). Thus,

)

,

Extremal Problems for Non-vanishing H v Functions

f(z) =

+

189

+ 2 22

Of course this is only a possible extremal for p < 1 and 1 _ 2 - p > 1. In fact this p2 p y i e l d s the b o u n d given in the theorem. If c o s a = 0, then f(z) = g(z 2) where g(z) = [c(1 + pz)] 2/p and this implies p < 1 for such an extremal. It is easy to see from Theorem 2 and the above result that g(z 2) cannot be extremal for a2. It remains to check m = 1. In this case, -

h(z) = c(1 + Z)(1 + pz) = c(1 + (1 + p)z + pz2), 1

where f i = 2(i + p + p2) while = 1 - -i + ~ ( ~ + p(l P

t

))

(13)

_1

1

Therefore, (14)

1-1= p

(l+p)(1-p)3 >0 2p~(1 + 2p - p2)

(hence p > 1).

W h e n p = 1, p = 1 and p increases as p decreases and p = oc when

(using (13)). Thus, p assumes all values from 1 to ec as p decreases from 1 to (1 + x/5- 2~

2 v ~ ) / 2 . Using (13) and (14) together with T h e o r e m 4, an extremal satisfies

c0~/"e-~ = a2c0c~, c0 = (2(1 + p + p~))-'/~,

a2 = ![2(1 + p + p )] P

c2 = pc0 = d hence ~

Thus, to complete the proof of the theorem, we require

On taking logs and using (14) we require

(1 + p ) ( 1 - p)3 [~rl°g'l + p2) ~-(-i T-2,; Z T ) +P

-

p~] < -1 1- - - ;~ - 2 p

+ l o g p.

T.J. Suffridge

190 Now log p = log0

=

-

(1

- ( 1 - p)

-

p)) >__-(1

l+p 1 + 2 p - p2

(1 - p)3 log(1 + p + / )

p)

(1 - p)~ 2

(1 =3 p)3 o~ ~(1 k=O

- p)k

p)2 (1 - p)3

(1 2

Further

-

3p

l+p

< 1. Thus, it is sufficient to show 1 + p + ( p - p2) _< p [1 - / - 2p(1 - p) - p(1 - p)~ - ~(1 - p)3 + (1 - p)3] _ 4(1 _ p ) 3

3

p.

Finally, 1 + p + p2 < e3!p = 1 + -~p + Sp2 ~ + .-- is sufficient. The last inequality clearly 1 1 2 holds since 0 < ~p - uP , 0 < p < 1. This completes the proof. •

References [1] J. E. Brown, On a Coe~ieient Problem for nonvanishing Hp functions, Complex Variables, Theory and Applications 4 (1985), 253-265. [2] C. Carath4odory, Uber den Variabilitdtsbereich der Fourier3chen Konstanten yon positiven harmonisehen Funktionen, Rend. Circ. Mat. Palermo 32 (1911), 193-217. [3] J. A. Hummel, S. Scheinberg and L. Zaleman, A eoej~cient problem/or bounded nonvanishing functions, J. Analyse Math. 31 (1977), 169-190. [4] J. Krzy~, Coej~cient problems for bounded nonvanishing functions, Ann. Polon. Math. 20 (1968), 314. [5] Delin Tan, Coejficient estimates for bounded nonvanishing functions, Chinese Ann. Math. Ser. A4 (1983), 97-104 (Chinese). [6] O. Toeplitz, (/bet die Fouriersche Entwieklung positiver Funktionen, Rend. Circ. Math. Palermo 32 (1911), 191-192. Received: September 3, 1989.

ComputationalMethods and Function Proceedings, Valpara/so 1989

Theory

St. Ruscheweyh, E.B. SalT, L. C. Salinas, R.S. Varga (eds.) Lecture Notes in Mathematics 1435, pp. 191-200 ~) Springer Berlin Heidelberg 1990

Some

results on separate of continued

convergence

fractions

W.J. Thron 1 Department of Mathematics, University of Colorado Campus Box 426, Boulder, CO 80309, U.S.A.

1. Introduction The term separate convergence was introduced recently to describe the phenomena which result when conditions - - stronger than needed for convergence - - are imposed on continued fractions or similar algorithms. The term separate is motivated by the first result of this type due to Sleszyfiski in 1888. He proved that if the continued fraction K(anz/1) satisfies the condition ~ lant < c~ then not only does the sequence of approximants {A~(z)/Bn(z)} converge but {An(z)} and {Bn(z)} converge separately to entire functions A(z) and B(z). In later investigation the conclusions frequently are less sweeping. One settles for the existence of an "easily described" sequence {Fn(z)} such that {A~(z)/F~(z)} and {Bn(z)/F~(z)} converge separately for z E A. Usually the convergence will be uniform on compact subsets of A, which may be a proper subset of the complex plane C. The restriction that the F,~ can be "easily described" is essential because for convergent continued fractions one can always choose In(z) = Bn(z) and thus the distinction between ordinary and separate convergence would become meaningless. Most, if not all, instances of separate convergence occur for limit periodic continued fractions with elements that are functions of a complex variable. Sometimes separate convergence is a tool in the derivation of results on analytic continuation or behavior on the boundary of the function to which the continued fraction converges. In other cases it may only be a by-product of such investigations. Since orthogonM polynomials can be obtained as denominators of the approximants of certain continued fractions, there is an overlap between our work and known results about the asymptotic behavior of 1This research was supported in part by the U.S. National Science Foundation under grant No. DMS8700498.

W.J. Thron

192

orthogonal polynomials (see, for example [2]). Our results can also be used to obtain information on the asymptotic behavior of orthogonal L-polynomials. Instead of describing at length the results obtained on separate convergence, we give in the References a list of all articles on the subject we know'about. This includes some papers where separate convergence is provable, but is not explicitly established. We shall concentrate here on the proof of a basic theorem from which some, but not all, of the known results on separate convergence can be derived. The theorem also yields new results, in particular on general T-fractions. We shall also discuss briefly the impact of equivalence transformations on separate convergence. An account concerned mainly with separate convergence of PC fractions, Schur fractions and Schur algorithms is being prepared by O. Nj£stad. In a subsequent article we hope to return to the subject and explore some other general approaches which yield results on separate convergence.

2. A s y m p t o t i c behavior of numerators and denominators of approximants of limit periodic continued fractions. In this section we shall prove a quite general and rather involved theorem. In the next section corollaries of a much simpler nature will be given. T h e o r e m 2.1. Let an(z), b~(z), a(z) and b(z) be hotomorphic functions for z E A. Further assume that a,~(z) ¢ 0 and

~im a~(z) = a(z),

j i m b~(z) = b(z)

fo~. z e z~

Then

(ao(z) ~=, \ b~(z) ]

is a limit periodic continued fraction for z E A. Set (2.2)

am(z) = a(z) + 6~(z), b,~(z) = b(z) + ,7,~(z).

Note that ~ ( z ) and ,~(z) are holomorphic and lim 6,~(z) = 0,

?I--+OO

lim rl~(z) = 0

for z e A. Let x f f z ) and x2(z) be the solutions of

~,~ + b( z )w - a( z ) = o and assume that the solutions have been so numbered that (2.3)

xxl(z) -~

Further assume that the series

<1

for

z E A* c A .

Some Results on Separate Convergence...

(2.4)

193

~ JSk(z)l and

~ trtk(z)l

k=l

k=l

converge uniformly on compact subsets of Ao C ,4 and that (2.5)

Ix~(z)l > 0 for

z e ,40.

Finally, let ,4(~) be such that I ~ ( z ) - ~(z)l > 2~ for

(2.6)

z e `4(<

Then A~(z)

lira (_x2(z))~+ ~

(2.7)

and

B~(z) lim (_x2(z)),~+l

both exist and are holomorphic in At : U z~* N Ao N ,4 (~). e>0

Here A,~(z) and B~(z) are the numerator and denominator, respectively, of the nth approximant of the continued fraction (2.1). Of course the theorem is of interest only if a'¢O. Proof. From now on we shall usually not indicate the dependence of the various functions under consideration on z. We shall simply write an, b~, a, b, 5n, qn, x,, x2, An, B~, C~, Dn. We note that

(2.s)

--a =

XlX2,

-b

=

Xl + x 2 .

As was shown in [16] one then has

B~ + xlBn-1

= (b + xl)Bn-1 + aBn-~ + rl,~Bn-1 + 5,~B~-2 =

(-x2)Bn-, + (-z~x2)Bn-2 + ~,~B,~_, + 6,,B,~_~.

Iterating these equations one arrives at n-1

(2.9)

n-1

Bn + x l m _ , = (-x~) n + E ( - x ~ ) n - ~ - % ÷ , B ~ + F_,(-x~)"-I-%÷~B~-~. k=0

k=0

n-1

n-1

An analogous derivation leads to

(2.1o)

B. + x~m_, = (-~,)~ + ~..(-x,)~-~-%÷,B~ + F , ( - Z l ) " - ~ - % ÷ I B ~ - . k=O

k=O

If Xl ~ x2, one can solve the system of equations (2.9), (2.10) for B,~. The result is

W.J. Thron

194 n-1 - - ( - - X l ) n + l -~- ~-'~.((--Z2) n - k -k=0

(xl - *=)Bn = (_x=)n+,

(--Xl)n-k)rlk+lBk

n-1

+ E ( ( - x ~ ) n-~ _ (-~,)'~-k)6~+~B~_,. k=O

Introduce (2.11)

On

Bn -

(_x~)n+,

In terms of Dn we then have (xl

-

x2)Dn = 1 -(x-'Z'~ n+l+ E

\X2]

1-

77

,k+,Dk

k--0

(2.12) 1 n-'(1 (X__Al~n-kI q--------~2k~=O -- \X2/ ] 5k+lDk-1. Similarly one obtains for (2.13)

Ca

An -

(--Z~)n+l

the formula (Xl

-x2)Cn ~ Xl -

(Xl~n+l n-1 ( ( X l ~ n - - k ~ -x2+~ 1- ~ ~k+lCk \X2/ k=0 \X2/ 1

(2.14) + --X2 k----O

KX2/

]

From (2.12) one can deduce for z E A* the inequality n-1

(2.15)

Ix~ - x211Dnl _< 2 + 2 ~

2

I~k+~l IDkl + ~

k=O

n--1

~

16k+~lIDk-~t .

k=0

We would like to prove that there exists a constant M > 0 such that for z in certain subsets of A0 n A* ] D n I < M for all n > 1. To do this we prove the following lemma. A similar result can be found in [2, p. 455]. Lemma2.2.

Leta>0,%>0,

n>_ 1 and Sn k O, n >__O. If

n-1 (2.16)

S~ < a + ~ %+lSk,

k=O

~hen

n >_ 1, So <_a,

Some Results on Separate Convergence...

195

it

(2.17)

Sit < a r I ( 1

n>l.

+7,Q =: P,.,,

k=l

Proof. We have $1 _< a + 71So _< a(1 +71)Set Po = a and assume t h a t Sk 5 Pk for 0 < k < n - 1. T h e n sit

5 ~ + E kn=- 1o ( (

1

. . . . l i p ~+~ - Pk) + 7k) - 1)Pk = a -~ z~,=oV

=a-Po+P.=P~. T h e l e m m a is thus proved by induction. • We apply the l e m m a to (2.15) by setting [Dit I = Sit, a = m a x ( 2 / I z , - a2[, 1/[a2[), %

_

I11

_2 x=l (It/k[ + 17)~k+1 "21/

For K a compact subset of A0 N A* we then have [x2] > dK > 0, [x1/x2[ < rg < 1. In view of (2.4) the sequence {Dit} has a uniform b o u n d M ( K , e) satisfying (2.18)

IDd_< max(1/e, 1 / d g ) f i

(1 + I~kl+ ~ ) <

M(K,e).

k=l

for z E K VI A (~). Let z E K N A(~), set Ix~/x21 = r < rK < 1. F r o m (2.12) one deduces It--1

Ixl -- x2]lDit+m - Dit[ <_ r It+'~ + rit + ~_,(r "~+m-k + rn-k)Jqk+lllDkl k=0 1

n--1

~-~( r~+m-k + r~-k)[~Sk+l IlDk-1 I + ~ - [ k=o n+m-1

+

(1 + r'~+m-k)lrlk+lrlDk I kmit

(2.19) 1

n+ra-1

+~--~ ~ (1 +,'~+m-k)iak+,llDk_ll k=it

< 2r~ + 2Mr~K/2 ~

]~k+ll +

d~ J

k=0

k=[it/2]

One thus can write for z E K gl A (*), where K is a compact subset of Al0 N Al*,

W.J. Thron

t96

\

k=[n/2]

It follows t h a t {D,~} converges uniformly on compact subsets of A(e) := A0 N A* M A (~), since every compact subset of A(e) is a compact subset of A0 N A*. Hence D = l i m D n is a holomorphic function on A(e). Analytic continuation can then be invoked to conclude that D(z) is hotomorphic for all z E A* = U A(e). e>0

Similarly one shows that An

lim

- C

exists and is holomorphic for all z E z2t.

•

3. Applications of T h e o r e m 2.1. We begin with the theorem of Sleszyfiski of 1888. Sleszyfiski's approach also allowed one to conclude that A and B are of order at most one. T h a t A and B have no common zero was first proved by Maillet [4], (see also Perron [5, p. 150]). By further refining Sleszyfiski's m e t h o d one can get stronger results on the order and type of the two entire functions A and B. We plan to present results of this kind in a future article [15]. C o r o l l a r y 3.1. If the regular C-fraction (3.1)

K n=l

satis~es

~la~I < c~, then lirn A,~(z) = A(z)

and

lirn B,~(z) = S ( z )

for all z E C. A(z) and B(z) are entire functions.

Proof. This is the special case of Theorem 2.1 where a = 0, b = 1, ~,~ = 0, xl = 0, x2 = - 1 and hence A=A*=A

0=C,

A(~)=C,

0<e<

C o r o l l a r y 3 . 2 . / f the regular C-fraction (3.1) satist~es

(3.2) then

~lan-a I<ec, a#0,

1.

•

Some Results on Separate Convergence...

197

2~+lA~(z)

2~+lB~(z) lirnoo(1 + ~ ) ~ + 1

nli~lTl~(1 + ~ y1/ ~ ' 4 ~ ) n+l = C(z),

for all z ~ ~o = [ z : o z e C ~ [s:~ e R , - 1 < ~ < - ~ ] 1 R e x / - - > 0 for z E ¢~.

= D(z)

Here ~ - - is chosen so tha~

Proof. In this ease A = C, A* = A 0 = ¢2 and

[

1 A(~) = Lz : - - ~ a -- z > ~2/4]a[] T h e p r o o f of the convergence of (3.1) under the a s s u m p t i o n (3.2) is also due to Sleszyfiski [13]. O u r conclusion a b o u t s e p a r a t e convergence is not explicitly s t a t e d in his article b u t can easily be deduced. Corollary

3.3 If the g e n e r a / T - f r a c t i o n

F,z +GozJ satist~es E~°°=I [F~[ < cc and E,~=~ [G~ - G[ < cx~, G ~ O, then lira A,~(z) - C*(z), ~-oo (1 + az)T M for all z 6 C ~ [ - l / G ] .

lira

B~(z) - Dr(z) (1 -~- az)T M

The functions C t and D t are holomorphic in C ,~ [ - l / G ] .

Proof. x~ = O, x2 = - 1 - Gz, A = C, A" = C ~ [ - l / G ] = A0, A (~) = [z : [1 + 2c]. Corollary

azl

> •

3.4. In the general T-fraction (3.3) iet

(3.4)

lira F~ = 1,

lim G~ = - 1 .

Further, assume that

~lF,-ll<

n=l

~,

]C~+11< o~. n=l

Then limo~ A , ( z ) = A(z),

limB~(z) = B(z)

and A a n d B are holomorphic for Iz] < 1. Also lira A,~(z) _ C(z),

n...*oo

zn+l

lira Bn(z) z~+l : D(z)

n~oo

azid C and Z) are holomorphic for [z I > 1. Proof. A = C. Xst case: x I --Z, X2 2 n d c a s e : xl = - l , x2=-z,A*=[z:lz[> :

=

- 1 , A* = [z: [z[ < 1], A(0 = [z: [ z - l [ > 2¢]. 1], A(') = [z : [ 1 - z[ > 2 ¢ ] . "

This corollary overlaps some results of W a a d e l a n d [21] and T h r o n a n d W a a d e l a n d

[151.

W.J. Thron

198 C o r o l l a r y 3.5.

In the J-fraction

(3.5)

K

n~--1

A,,(z)

nli~lTl°°(C -- Z) n+l --

and C* and D* are

C*(z), n---*c¢ lim B,~(z) -D*(z) (C : Z) n-'''-+l

holomorphic in C -~ [c].

P~oor. n = a - = c , z , = 0, x~ = z - c, a 0 = c ~ [cl, a (~) = [ z : Iz - cl > 2 d . This is Theorem 5 in Schwartz [12].

4. C h a n g e s

Let

in separate convergence transformations.

under

equivalence

K~=1(a,~/b,,) be a given continued fraction, let

be defined by (4.2)

5~=7~_:7~a~,

b~=7,~b~,

70=1,

%~0,

n > 1.

Then the two continued fractions are equivalent in the sense that their sequences of approximants are identical. However for the numerators and denominators of the approximants one can show, using the recursion relations, that (4.3)

Ao=AoI]~o

,

B~=B~II%,

v=0

,

n>-:,

v=O

provided one uses the convention FIE = 1, for m < k. We use this idea to prove another result for J-fractions (3.4). The continued fraction (4.4)

K(k'(z-c')-:(z-c'~-:)-l) l

,

Co = z - l ,

is equivalent to (3.4). Under the conditions (4.5)

c,~ ---* co,

< oc,

the numerators Am and denominators/~, of (4.4) satisfy, for z E C ,-, [Cl, c 2 " "] l i m ii. = A,

lim/}. = B.

Some Results on Separate Convergence...

199

It follows from (4.3) that for the J-fraction (3.4) one has

An

Bn

l i m l_i~:o(Z _ c,) - B .

lirn I_t~=o(z _ cv) - A, If one strengthens (4.5) to

one obtains (Theorem 2 in Schwartz [10]) ,~-~ I1~=1(_c. ) - A

1- z

,

lim I]~=~(-c.) - B

v----1

1-

.

v=l

References [1] R. J. Arms, A. Edrei, The Padd tables and continued fractions generated by totally positive sequences, Mathematical Essays dedicated to A. J. Macintyre, Athens, Ohio (1970), 1-21. [2] F. W. Atkinson, Discrete and continuous boundary problems, Academic Press, New York, 1964. [3] A. Auric, Recherches sur les fractions continues algdbriques, J. Math pure et applique, (6) 3 (1907), 105-206. [4] A. Edrei, Sur des suite~ de hombres lides h la thdorie des fractions continues, Bull. Sci. Math., (2) 72 (1948), 45-64. [5] Lisa Jacobsen, A note on separate convergence of continued fractions, preprint. [6.] E. Maillet, Sur tes fractions continues aIgdbriques, J. Ec. poI., (2) 12 (1908), 41-62. [7] O. Perron, Die Lehre yon den Kettenbrgchen, 3. Anti., 2. Band, Teubner, Stuttgart, 1957. [8.] H.-J. Runckel, Bounded analytic functions in the unit disk and the behavior of certain analytic continued fractions near the singular line, J. reine angew. Math., 281 (1976), 97-125. [9]

, Continuity on the boundary and analytic continuation of continued fractions, Math. Zeitschr., 148 (1976), 189-205.

[10]

, Meromorphic extension of analytic continued fractions, Rocky Mtn. J. Math., to appear.

200

W.J. Thron

[11] J. Schur, Uber Potenzreihen die im Innern des Einheitskreises besehrdnkt sind (Fortsetzung), J. reine angew. Math., 148 (1918/19), 122-145. [12] H. M. Schwartz, A class of continued fractions, Duke Math. J., 6 (1940), 48-65. [13] I. V. Sleshinskii (J. Sleszyfiski), Convergence of continued fractions (in Russian), Zapiski matematicheskago otodieleniia Novorossiiskago obshchestvoispytatela, 8 (1888), 97-127. [14] T. J. Stieltjes, Recherches sur les fractions continues, Ann. Fac. Sci. Toulouse, 8 (1894), 1-122. [15] W. J. Thron, Order and type of entire functions arising from separately convergent continued fractions, submitted. [16] W. J. Thron and H. Waadeland, Convergence questions for limit periodic continued fractions, Rocky Mtn. J. Math., 11 (1981), 641-657. [17] Walter van Assche, Asymptotics for orthogonal polynomials and three term recurrences, Proceedings of Columbus conference, to appear. [18] H. von Koch, Quelques thdor~mes concernant Ia theorie gdndraIe des fractions continues, Oversigt av kongl. Vetenskaps-Akademiens FSrhandlingar, 52 (1895). [19]

, Sur la convergence des determinant~ d'ordre infini et des fractions continues, Comptes Rendus, 120 (1895), 145.

[20] . . . . . . . Sur un thdor~me de Stieltjes et sur les fonctions dd~nies par des fractions continues, Bull. Soc. Math. France, 23 (1895), 33-41. [21] H. Waadeland, A convergence property of certain T-fraction expansions, Kgl. norske videnskabers selskabs skrifter 1966, No. 9, 3-22. [22] H. S. Wall, Some recent developments in the theory of continued fractions, Bull. Amer. Math. Soe., 47 (1941), 405-423. [23]

, The behavior of certain Stieltje~ continued fractions near the ~ingutar line, Bull. Amer. Math. Soc., 48 (1942), 427-431.

Received: August 7, 1989, in revised form November 28, 1989.

Computational Methods and Function

Theory

Proceedings, Valparaiso 1989 St. Ruscheweyh, E.B. Saff, L. C. Salinas, R.S. Varga (eds.) Lecture Notes in Mathematics 1435, pp. 201-207 (~) Springer Berlin Heidelberg 1990

A s y m p t o t i c s for t h e Z e r o s o f t h e P a r t i a l S u m s o f e ~'. II R.S. Varga* Institute for Computational Mathematics Kent State University, Kent, OH 44242, USA and

A.J. Carpenter D e p a r t m e n t of Mathematical Sciences Butler University, Indianapolis, IN 46208, USA

1. I n t r o d u c t i o n n

With

s,(z)

:=

~--~zJ/j!

(n = 1 , 2 , . . . ) denoting the familiar partial sums of the

j----0

exponential function e ", we continue our investigation here on the location of the zeros of the normalized partial sums, sn(nz), which are known to lie (of. Anderson, Sail', and Varga [1]) for every n > 1 in the open unit disk A := {z e C : Izl < 1}. For notation, let the Szeg5 curve, D ~ , be defined by

(1.1)

D ~ := {z E C : lzel-Zl = 1 and [z t < 1}.

It is known that D ~ is a simple closed curve in the closed unit disk ~ , and t h a t Do~ is star-shaped with respect to the origin, z = 0. If {zk,n}~=l denotes the zeros of sn(nz) (for n = 1,2,---), then it was shown by Szeg5 [7] in 1924 t h a t each accumulation point of all these zeros, {zk,~}k=l,n=1, m u s t lie on D ~ , and, conversely, that each point of D ~ is an accumulation point of the zeros n~oo n,oo {zk,n}k=l,n=l. Subsequently, it was shown by Buckholtz [2] that the zeros {zk,~}k=l,,=l all lie outside the simple closed curve Doo. As for a measure of the rate at which the zeros, {zk,~}~=l, tend to D ~ , we use the quantity *Research supported by the National Science Foundation

R.S. Varga and A.J. Carpenter

202

dist [{Zk.n}~=l'~D¢01 := max (dist [zk,~;Doo]),

(1.2)

l<_k<_n

and a result of Buckholtz [2] gives that (1.3)

dist [{zk,,}L1;D~] _<

2e

5.43656. • •

-

(n = 1 , 2 , . . . )

It was later shown in Carpenter, Varga, and Waldvogel [3] that the result of (1.3) is best

possible, as a function of n, since

(1.4)

lim{v~-dist [{zk,,)k=l, D~]} > 0.63665.-. n

.

> 0.

It was also shown in [3] that there is substantially faster convergence of the subset of the zeros {zk,n}~=l, to the Szeg5 curve Do¢, which stay uniformly away from the point z = 1. More precisely, for the open disk C~ about the point z = 1, defined by (1.5)

C6:={z~C:lz-11<5}

(0<~<1),

it was shown in [3] that, for any fixed 6' with 0
(1.6)

0

(n --* ec),

and, the result of (1.6) is also best possible, as a function of n, since (cf. [3, eq. (2.27)]) ~-~¢¢1im lo--~' dist [{zk,~}k=l\C6, D~o] >_0.10890... > 9,

(1.7)

for any fixed ~ with 0 < $ _< 1. In [3], an arc, D~, was defined for each n = 1, 2 , . . . by Dn:=

1-z

zeC:[zd-~l~=r~

~

(1.8)

, tz] _< 1, and I arg zl _> cos-1 ( ~ - ~ )

},

where from Stirling's formula, n! (1.9)

1

vn.-- n~e_~ 2v/~-~ 1 + ~ + 2 8 8 n

1 2

139 51840n 3 + . - . ,

asn-+oc.

n This arc was introduced to provide a much closer approximation to the zeros { Z k,n}~=l of s,(nz), than does the Szeg5 curve. With the notation of (1.5), it was shown in [3] that, for any fixed ~ with 0 < 5 _< 1,

(1.10)

dist [{zk,,}k=1\C6, ON]

O

(n ~ ~ ) ,

and moreover that (1.10) is best possible, as a function of n, since (cf. [3, eq. (3.18)]) (1.11)

limoo{n2 • dist [{zk,,}~=l\c6;n~]} >_ 0.13326... >

0,

Asymptotics for the zeros of the partial sums of e~. H

203

for any fixed 6 with 0 < 6 ~ 1. It turns out (cf. [3, Prop. 3]) that, for each positive integer n, the arc Dn is starshaped with respect to the origin, z = 0, i.e., for each real number 8 in [-Tr, +~r] with [8[ >_ c o s - l ( - ~ ) , there is a unique positive number r = r,~(8) such that z = re i° lies on the arc D,~ of (1.8). Let :D~ be the closed star-shaped (with respect to z = 0) set defined from the arc D~, i.e.,

L A, Izl _< 1, and

D= := {z E C : [zel-~[ ~ _<

(1.12) [ argz[ > cos-l(rt : 2 ) } ,

(rt ---- 1,2,'.-).

n Recently, R. Barnard and K. Pierce asked if the zeros, {Z k,~}k=l, of s~(nz) all lie outside ~D~ for every n _> 1. (This would be the natural analogue of the result of Buckholtz [2] which established that all the zeros {zk,~}k=l,~=l lie outside of Doo.) This is not at all 16 and "t f z k,2rIk=l 127 obvious from the graphs of [3], since it appeared that the z e r o s {Zk,16}k=l of s16(16z) and s2r(27z) were, to plotting accuracy, respectively on the curves D16 and D27. It turns out that the zeros, {z~,,}~=l of sn(nz) do not all lie outside :D,, for every n > 1. This follows from our first result below (to be established in §2).

P r o p o s i t i o n 1 If {zk,n}~=l denotes the zeros of s~(nz) with increasing arguments,

i.e,, (1.13)

0 < argzl,, _< argz2,n _< --. ~ arg zn,n < 27r,

then (cf. (1.12)) zl,, is an element of D, for all positive n su~ciently large. As a consequence of Proposition 1, there is a least positive integer, no, such that (cL (1.12)) (1.14) {zk,,~}~=l AD,~ ~ 0 for all positive integers n > no, i.e., at least one zero of s~(nz) lies in ~D~ for every n > no. By direct calculation of the zeros of s~(nz), it appears that (1.15) no = 96, and also that (1.16)

{Zk,n}~=lN~)n

=

$

(n = 1 , 2 , . . . , n 0 ) .

The size of no = 96 is somewhat surprising. Because no is so large, it was necessary to calculate the zeros of s,~(nz) with great precision, and for this, Richard Brent's MP package was used with 120 significant digits. As a consequence of Proposition 1, it is natural to ask if there is a simple modification, say ~ , of the definition of the closed set T~,~of (1.12) which would have all the zeros {zk,,,}~=l outside f),~ for all n > 1. To give an affirmative answer to this question, we define, for each n = 1, 2,-.-, the arc

(1.17)

/ ) . :=

.----I1 -Re z z E C : lzel-~l'~ = T,~.d2~'n 1 - - - ~ 1 ' [zl < 1, and ,~,

~ ~o~-~ ( ~ ) } ,

204

R.S. Varga and A.J. Carpenter

1.0

-

1.0

0.5

0.5

0.0

0.0

-0.5

-0.5

-1.0

-1.0 I

I

I

I

-1.0

-0.5

0.0

0.5

1.0

I

I

l

I

I

-1.0

-0.5

0.0

0.5

1.0

F i g u r e 1. D~7

F i g u r e 2./)27

and its associated closed star-shaped (with respect to z = 0) set

(1.t8)

~n :~_ {Z E C : Izel-zln ~ Tn 2V/'2~ n 1 - R-e z 1, Z

Izl_
Unfortunately, this modification does not preserve the accuracy of (1.10). Our main result (which will be sketched in §2) is T h e o r e m 2 With the detinition of the set D~ of(1.18), then (1.19)

{zk,~}~=lN/~= = 0

(n = 1,2,...),

and, with the de~nition of(1.5),

(1.20)

dist [{zk,~}k=l\C~, D=] = 0

(n -* oo),

/'or any/~xed ~ with 0 < ~ <_ 1. We remark that the bound of (1.20) is best possible, as a function of n. We include here Figures 1 and 2 which respectively display tile arcs D2~ and D2r, along with the 27 zeros, { Z k,2r}k=l, of s2r(27z). These zeros are denoted by x's on Figures 1 and 2. Figures 3 and 4 similarly display the arcs D49 and D49, along with the zeros {zk,49}49=i of s49(49z).

2. P r o o f of P r o p o s i t i o n 1 It is easy to verify (by differentiation) that (2.1)

1£

e-Zs~(z) = 1 - -~.

(~e-¢d(

(z e C, n = O, 1,...),

Asymptotics t'or the zeros of the partial sums of e~. H 1.0

-

205

1.0

0.5

0.5

0.0

0.0

-0.5

-0.5

-1.0

-1.0 I

I

t

t

i

I

I

I

t

I

-12

-0.5

0.0

0.5

1.0

-1.0

-0.5

0.0

0.5

1.0

F i g u r e 3. D49

F i g u r e 4./)49

and replacing ~ and z, respectively, by n~ and nz in(2.1) results in 72n+ 1 t z

(2.2)

= 1 - -P-F. J0

With the definition of r~ of (1.9), the above equation becomes (2.3)

z

e-~zs,~(nz) = 1

(~el-~)nd~

(z E C, n = 0 , 1 , . . . ) .

Now, in [3, eq. (2.14)], it is shown that (2.4)

z(zel-~)~ { 1 (~)} e-~Zs~(nz) = 1 - " r ~ - - z) 1 - (n + 1)(1 - z) 2 + 0

,

uniformly on any compact subset 12 of/~\{1}. On fixing i2, then for any zero, zk,,~, of s,~(nz) in i2, we evidently have from (2.4) that (2.5)

r~--z-~-~)

1-

+ 0

= 1,

2+0

=1.

so that

~:-~:~k.~I

1-(n+l)(1-zk,n)

It is nov,- clear that the arc Dn of (1.8), is just the approximation of (2.6), with a continuous variable z, when the quantity in braces in (2.6) is replaced by unity, i.e., (2.7)

~ ( z ) :=

Iz(zel-~)~l ¢~2,/5-~11 - zl"

It is further evident from (1.12) that (2.8)

a zero zk,,~ of s,~(nz) lies in/Pn iff A,(Zk,n) <_ 1.

R.S. Varga and A,J. C a r p e n t e r

206

We now examine the particular zero za,. of s~(nz) which has smallest a r g u m e n t (cf. (1.13)). As discussed in [3, eq. (2.3)1 , we can write

(2.9)

(n --, ~ ) ,

Zl,~ = 1 + }f~(t, + o(1))

e-' dt in the upper half-plane (i.e., Im t~ > 0)

where tl is the zero of erfc(w) :=

which is closest to the origin, w = 0, and it is known numerically (cf. Fettis, Caslin, and C r a m e r [4]) t h a t (2.10) t~ = - 1 . 3 5 4 8 1 0 . . . + i l , 9 9 1 4 6 7 . . . . On evaluating A~(z~,~) from (2.7), (2.9), and (2.10), it can be verified t h a t (2.11)

lim A~(Zl,~) = 0 . 9 8 5 9 6 4 . . - < 1.

n--*OO

Thus, with (2.8), zl,~ is contained in D~ for all positive n sufficiently large, which establishes Proposition 1. • We r e m a r k that it is because the constant, 0.985964.-., of (2.11) is so close to unity, t h a t it is difficult to see, graphically, that there are zeros of s,(nz) which lie interior to :D,~, for all n sufficiently large.

3. P r o o f o f T h e o r e m

2

We consider the integral (cf. (2.3)) (3.1)

L(~¢l-¢)nd~

I,(z) :-~-

(z e C, rt = 0, 1,-.-),

and, with z = re i°, we choose the line segment ~ = pei°(O < p < r) for the p a t h of integration in (3.1). Then,

(3.2)

Ix~(=)l <

(pe'-o¢°'°)~dp--: J~(r; 0).

For 0 = :t:~r/2, we see that J.(r;-t-7r/2) can be expressed as (3.3i)

Jn(r; =i=Tr/ 2 ) -

r(rel . . . . . o)~ n + 1

<

r(r~'

. . . . .

O)n

n(1 - r cos0)

.

W h e n cos 0 < 0, J,~(r; O) can be expressed as

1

fi¢osolo '+r~<°~°'

J~(~; o) = t cos Oln+l Jo

u(~)

v"-' 1 + ,~(v) d,

u~,+~) (~ :=

Because u/(1 + u) is strictly increasing, it can be verified t h a t r ( r e 1. . . . . o),~

(3.3ii)

J,(~; 0) < ~(1 - r cos 0)

(0 < ~ < i, and cos e < 0).

Asymptotics for the zeros of the partial sums of e*. H

207

But this same derivation also shows that the above holds for all 0 < r < I and cos ~ > 0. Thus, with (3.2) and (3.3), we have (3.4)

IzJlzel- l II~(z)[ < n'~---~-e z) (0 < [z[ < 1, n = 1,2,..-),

and from (2.3), we further have that (3.5)

r~(z)

= 1 - e-~%(nz).

Thus, if {zk,~}~=l denotes the set of zeros of s~(nz), then ~-~2 I~(zk,~) = 1, which implies from (3.4) that

(3.6)

> 1.

From (1.18), this means that all zeros {z~,~}~=~ of s~(nz) lie outside the set 7~, for all n _> 1, which is the desired result of (1.19) of Theorem 2. The remainder of Theorem 2, to establish (1.20), now similarly follows, as in the proof given in [3, Theorem 4], by expressing a zero, zk,~, of s,(nz), as ~ + ~, where is a suitable boundary point of ~ , and where 6 is assumed small. This argument also shows that the result of (1.20) is best possible. •

References

[1] N. Anderson, E.B. Saff, and R.S. Varga, On the Enestr~m-Kakeya Theorem and its sharpne88, Linear Algebra Appl. 28 (1979), 5-16. [2] J.D. Buckholtz, A characterization of the exponential series, Amer. Math. Monthly 73, Part II (1966), 121-123. [3] A.J. Carpenter, R.S. Varga, and J. WMdvogel, Asymptotics for the zeros of the partial sums of e~. I., Rocky Mount. J. of Math. (to appear). [4] H.E. Fettis, J.C. Caslin, and K.R. Cramer, Complex zeros of the error function and of the complementary error function, Math. Comp. 27 (1973), 401-404. [5] E.B. Saff and R.S. Varga, On the zeros and poles of Padd approximant8 to e~, Numer. Math. 25 (1975), 1-14. [6] E.B. Saff and R.S. Varga, Zero-free parabolic regions for 8equence~ of polynomials, SIAM J. Math. Anal. 7 (1976), 344-357. [7] G. Szeg6, Uber eine Eigen~cha~t der Exponentialreihe, Sitzungsber. Berl. Math. Ges. 23 (1924), 50-64. Received: February 28, 1990

L e c t u r e s presented during the conference R. A. Askey, Madison, USA "Polynomial inequalities". R.W. Barnard, Texas Tech, USA "On two conjectures in geometric function theory".

H.-P. Blatt, Eichst£tt, FRG "Erd6s-Tur£n Theorems on Jordan curves and arcs". P. Borwein, Halifax, Canada "A remarkable cube mean iteration". A. Cdrdova, Wiirzburg, FRG "Maximal range problems for polynomials". C. FitzGerald, San Diego, USA "Slit Mapping problems with no corresponding extremal problem". R. Fournier, Montreal, Canada Starlike univalent functions bounded on a diameter". R. Freund, Wfirzburg, FRG "A constrained Chebyshev approximation problem for ellipses". W.H.J. Fuchs, Cornell, USA "On a conjecture of Fischer and Michelli". W.K. Hayman, York, UK "A functional equation arising from the mortality tables". D. Hough, Z/irich, Switzerland "A Symm-Jacobi collocation method for numerical conformal mapping" J.A. Hummel, Maryland, USA "Numerical solutions of the Schiffer differential equation". L. Jacobsen, Trondheim, Norway "Orthogonal polynomials, chain sequences, three-term recurrence relations and continued fractions". W.B. Jones, Boulder, USA "Zeros of Szeg6 polynomials associated with Wiener-Levinson prediction". A. Lewis, Halifax, Canada "On the convergence of moment problems". D. Mej~a, Medellln, Colombia "Hyperbolic geometry in spherically k-convex regions".

210 D. Minda, Cincinatti, USA "An application of hyperbolic geometry". P.D. Miletta, Santiago, Chile "Approximation of special functions to delay differential equations". P. Nevai, Ohio State, USA "Computational aspects of orthogonal polynomials". O. Orellana, Valparaiso, Chile "On the point vortex method and some applications to problems in a~rodynamics". N. Papamichael, Brunel, UK "A domain decomposition method for conformal mapping onto a rectangle". C. Pommerenke, TU Berlin, FRG "Conformal mapping and the computation of bad curves". B. Rodin, San Diego, USA "Circle packing and conformal mapping". F. R0nning, Trondheim, Norway "A result about the sections of univalent functions". St. Ruscheweyh, Valparaiso, Chile, and W/irzburg, FRG "Convexity preserving operators". A. Ruttan, Kent State, USA "Optimal successive overrelaxation iterative methods for p-cyclic matrices". E.B. Saff, Tampa, USA "Distribution of extreme points on best complex polynomial approximation". L. Salinas, Valparaiso, Chile "On abstract conjugation and some engineering applications". G. Schober, Bloomington, USA "Planar harmonic mappings". D.F. Shea, Madison, USA "An extremal property of entire functions with positive zeros". F. Stenger, Salt Lake City, USA "Explicit exponential and rational approximation of continuous functions on R". H.J. Stetter, TH Vienna, Austria "Numerical inversion of multivariate polynomial systems". T.J. Suffridge, Lexington, USA "On nonvanishing H p functions".

211 W.J. Thron, Boulder, USA "Consequences of separate convergence of continued fractions". R.S. Varga, Kent State, USA "Recent results on the Riemann Hypothesis". R.A. Zalik, Auburn, USA "On the nonlinear Jeffcott equations". D. Zwick, Vermont, USA "Recent progress on best harmonic and subharmonic approximation".

Other Chilean participants J. Almanza, Concepci6n J. Bestagno, Concepci6n H. Burgos, Temuco V. Gruenberg, Valparalso S. Martlnez, Concepci6n W. Moscoso, Temuco F. Novoa, Concepci6n H. Pinto, Valpara/so V. Valderrama, Punta Arenas J.C. Vega, Concepci6n V. Vargas, Temuco

L E C T U R E N O T E S E d i t e d b y A. Dold,

IN I~;~TH EI~;/%T B. E c k m a n n and F. T a k e n s

I C S

Some general remarks on the publication of proceedings of congresses and symposia

Lecture Notes aim to r e p o r t new d e v e l o p m e n t s - quickly, i n f o r m a l l y and at a high level. The following describes c r i t e r i a and procedures which apply to proceedings volumes. The editors of a volume are strongly advised to inform contributors about these points at an early stage. §i.

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§2.

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§3.

The c o n t r i b u t i o n s should be of a high m a t h e m a t i c a l standard and of current interest. R e s e a r c h articles s h o u l d p r e s e n t new m a t e rial and not d u p l i c a t e other papers a l r e a d y p u b l i s h e d or due to be published. T h e y s h o u l d contain s u f f i c i e n t i n f o r m a t i o n and m o tivation and they should p r e s e n t proofs, or at least outlines of such, in s u f f i c i e n t detail to enable an e x p e r t to complete them. Thus resumes and mere announcements of p a p e r s a p p e a r i n g elsewhere cannot be included, although m o r e d e t a i l e d versions of a contribution m a y well be p u b l i s h e d in other p l a c e s later. Contributions in n u m e r i c a l m a t h e m a t i c s m a y be a c c e p t a b l e w i t h o u t formal theorems resp. proofs if they p r e s e n t n e w algorithms solving problems (previously u n s o l v e d or less w e l l solved) or develop innovative q u a l i t a t i v e methods, not yet a m e n a b l e to a more formal t r e a t m e n t . . Surveys, if included, should cover a s u f f i c i e n t l y b r o a d topic, and should in g e n e r a l not simply r e v i e w the author's own recent research. In the case of such surveys, exceptionally, proofs of results may not be necessary.

§4.

"Mathematical Reviews" and "Zentralblatt f~r Mathematik" recommend that p a p e r s in proceedings volumes carry an e x p l i c i t statement that they are in final form and that no similar paper has been or is b e i n g s u b m i t t e d elsewhere, if these papers are to be c o n s i d e r e d for a review. Normally, papers that satisfy the criteria of the L e c t u r e Notes in M a t h e m a t i c s series also satisfy

this requirement, ting authors be b e g i n n i n g or end cases where this the paper is still

but we strongly r e c o m m e n d that the c o n t r i b u asked to give this g u a r a n t e e e x p l i c i t l y at the of their paper. There will occasionally be does not apply but where, for special reasons, a c c e p t a b l e for LNM.

55.

Proceedings should appear soon after the meeeting. The p u b l i s h e r should, therefore, receive the complete m a n u s c r i p t (preferably in duplicate) w i t h i n nine m o n t h s of the date of the m e e t i n g at the latest.

§6.

Plans or proposals for p r o c e e d i n g s volumes s h o u l d be sent to one of the editors of the series or to Springer-Verlag Heidelberg. They should give sufficient i n f o r m a t i o n on the conference or symposium, and on the p r o p o s e d proceedings. In particular, they should contain a list of the e x p e c t e d c o n t r i b u t i o n s with their p r o s p e c t i v e length. A b s t r a c t s or early v e r s i o n s (drafts) of some of the c o n t r i b u t i o n s are helpful.

57.

Lecture Notes are printed by p h o t o - o f f s e t from c a m e r a - r e a d y typed copy p r o v i d e d by the editors. For this p u r p o s e S p r i n g e r Verlag provides editors w i t h technical i n s t r u c t i o n s for the preparation of m a n u s c r i p t s and these should be d i s t r i b u t e d to all contributing authors. Springer-Verlag can also, on request, supply stationery on w h i c h the prescribed t y p i n g area is outlined. Some h o m o g e n e i t y in the p r e s e n t a t i o n of the c o n t r i b u t i o n s is desirable. Careful preparation of manuscripts will h e l p k e e p p r o d u c t i o n time short and ensure a s a t i s f a c t o r y a p p e a r a n c e of the f i n i s h e d book. The actual p r o d u c t i o n of a Lecture Notes v o l u m e n o r m a l l y takes 6 -8 weeks. M a n u s c r i p t s should be at least I00 pages should include a table of contents.

§8.

long.

The

final v e r s i o n

Editors receive a total of 50 free copies of their v o l u m e for d i s t r i b u t i o n to the c o n t r i b u t i n g authors, but no royalties. (Unfortunately, no reprints of individual contributions can be supplied.) They are e n t i t l e d to purchase further copies of their book for their personal use at a d i s c o u n t of 33.3 %, other Springer m a t h e m a t i c s books at a discount of 20 % d i r e c t l y from Springer-Verlag. C o n t r i b u t i n g authors may p u r c h a s e the v o l u m e in which their article appears at a discount of 33.3 %. C o m m i t m e n t to publish is made by letter of intent rather than by signing a formal contract. S p r i n g e r - V e r l a g secures the c o p y r i g h t for each volume.

Addresses: Professor A. Dold, Mathematisches Institut, Universit~t Heidelberg, Im Neuenheimer Feld 288, 6900 Heidelberg, Federal Republic of Germany Professor B. Eckmann, Mathematik, ETH-Zentrum 8092 ZUrich, Switzerland Prof. F. Takens, Mathematisch Instituut, Rijksuniversiteit Groningen, Postbus 800, 9700 AV Groningen, The Netherlands Springer-Verlag, Mathematics Editorial, Tiergartenstr. 17, 6900 Heidelberg, Federal Republic of Germany, Tel.: (06221) 487-410 Springer-Verlag, Mathematics Editorial, 175, Fifth Avenue, New York, New York i0010, USA, Tel.: (212) 460-1596

L E C T U R E

N O T E S

ESSENTIALS FOR THE PREPARATION OF C A M E R A - R E A D Y M A N U S C R I P T S

Springer Springcr-Vcrlag Bcrlin Hcidclbcrg Ncw York London Paris Tokyo Hong Kong

The preparation of m a n u s c r i p t s w h i c h are to be r e p r o d u c e d b y p h o t o o f f s e t r e q u i r e s p e c i a l care. M a n u s c r i p t s w h i c h are s u b m i t t e d in technically unsuitable f o r m w i l l be r e t u r n e d to the a u t h o r for retyping. T h e r e is n o r m a l l y no p o s s i b i l i t y of c a r r y i n g out f u r t h e r c o r r e c t i o n s after a manuscript is g i v e n to p r o d u c t i o n . H e n c e it is c r u c i a l that the f o l l o w i n g i n s t r u c t i o n s be a d h e r e d to closely. If in doubt, p l e a s e s e n d us 1 - 2 s a m p l e p a g e s for e x a m i n a t i o n . General. The c h a r a c t e r s m u s t be u n i f o r m l y b l a c k b o t h w i t h i n a s i n g l e c h a r a c t e r and d o w n the page. O r i g i n a l m a n u s c r i p t s are required: phot o c o p i e s are a c c e p t a b l e o n l y if t h e y are s h a r p and w i t h o u t smudges. On request, S p r i n g e r - V e r l a g w i l l s u p p l y s p e c i a l p a p e r w i t h the text a r e a o u t l i n e d . The s t a n d a r d T E X T A R E A (OUTPUT S I Z E if you are u s i n g a 14 p o i n t font) is 18 x 26.5 c m (7.5 x ii inches). This w i l l be s c a l e reduced to 75% in the p r i n t i n g process. If y o u are using computer t y p e s e t t i n g , p l e a s e see also the f o l l o w i n g page. M a k e sure the T E X T A R E A IS C O M P L E T E L Y FILLED. Set the m a r g i n s so that t h e y p r e c i s e l y m a t c h the o u t l i n e and type r i g h t from the top to the b o t t o m line. (Note t h a t the p a g e n u m b e r w i l l lie o u t s i d e this area). L i n e s of text s h o u l d not e n d m o r e t h a n t h r e e s p a c e s i n s i d e or o u t s i d e the r i g h t m a r g i n (see e x a m p l e on page 4). Type on one side of the p a p e r only.

S p a c i n g and H e a d i n g s (Monographs). Use O N E - A N D - A - H A L F line s p a c i n g in the text. P l e a s e l e a v e s u f f i c i e n t space for the title to stand out c l e a r l y and do N O T use a n e w p a g e for the b e g i n n i n g of s u b d i v i s o n s of chapters. L e a v e T H R E E L I N E S b l a n k above and TWO b e l o w h e a d i n g s of such s u b d i v i s i o n s . Spacing and H e a d i n g s ( P r o c e e d i n g s ) . Use O N E - A N D - A - H A L F line s p a c i n g in the text. Do not use a n e w p a g e for the b e g i n n i n g of s u b d i v i s o n s of a single paper. L e a v e T H R E E LINES b l a n k a b o v e a n d TWO b e l o w headings of such s u b d i v i s i o n s . M a k e sure h e a d i n g s o ~ e q u a l importance are in the same form. The first page of e a c h c o n t r i b u t i o n s h o u l d be p r e p a r e d in the same way. The title s h o u l d s t a n d out clearly. We t h e r e f o r e r e c o m m e n d that the e d i t o r prepare a sample page and pass it on to the a u t h o r s together with these instructions. Please take the f o l l o w i n g as an example. B e g i n h e a d i n g 2 c m b e l o w u p p e r e d g e of text area. M A T H E M A T I C A L S T R U C T U R E IN Q U A N T U M F I E L D T H E O R Y J o h n E. R o b e r t M a t h e m a t i s c h e s Institut, U n i v e r s i t ~ t H e i d e l b e r g Im N e u e n h e i m e r F e l d 288, D - 6 9 0 0 H e i d e l b e r g Please author,

leave THREE LINES blank below h e a d i n g and a d d r e s s of the t h e n c o n t i n u e w i t h the a c t u a l text on the same page.

Footnotes. These should preferable be avoided. If n e c e s s a r y , type t h e m in S I N G L E L I N E S P A C I N G to f i n i s h e x a c t l y on the outline, and sep a r a t e t h e m f r o m the p r e c e d i n g m a i n text by a line.

Symbols.

Anything

BLACK

ONLY

cannot be t y p e d m a y be e n t e r e d b y h a n d in f i n e - t i p p e d r a p i d o g r a p h is s u i t a b l e for this p u r p o s e ; a g o o d b l a c k b a l l - p o i n t will do, but a pencil will not). Do not d r a w s t r a i g h t lines by h a n d w i t h o u t a r u l e r (not e v e n in fractions).

AND

which

BLACK

ink.

(A

Literature R e f e r e n c e s . T h e s e s h o u l d be p l a c e d at the e n d of e a c h pap e r or chapter, or at the e n d of the work, as desired. Type t h e m w i t h single line spacing and s t a r t e a c h r e f e r e n c e on a n e w line. F b l l o w " Z e n t r a l b l a t t fur M a t h e m a t i k " / " M a t h e m a t i c a l R e v i e w s " for a b b r e v i a t e d t i t l e s of m a t h e m a t i c a l j o u r n a l s and " B i b l i o g r a p h i c G u i d e for E d i t o r s and Authors (BGEA)" for c h e m i c a l , b i o l o g i c a l , and p h y s i c s journals. P l e a s e e n s u r e that all r e f e r e n c e s are C O M P L E T E and A C C U R A T E .

IMPORTANT P a g i n a t i o n . For t y p e s c r i p t , n u m b e r p a g e s in the u p p e r r i g h t - h a n d c o r ner in L I G H T B L U E OR G R E E N P E N C I L ONLY. The p r i n t e r s w i l l i n s e r t the final page numbers. For c o m p u t e r type, y o u m a y i n s e r t p a g e n u m b e r s (I c m a b o v e o u t e r edge of text area). It is s a f e r to n u m b e r p a g e s ted. Page 1 (Arabic) s h o u l d Roman p a g i n a t i o n (table of ments, b r i e f i n t r o d u c t i o n s ,

A F T E R the text has b e e n t y p e d and c o r r e c be THE F I R S T P A G E OF THE A C T U A L TEXT. The contents, preface, abstract, a c k n o w l e d g e etc.) w i l l be done b y S p r i n g e r - V e r l a g .

If including r u n n i n g heads, t h e s e s h o u l d be a l i g n e d w i t h the i n s i d e e d g e of the text a r e a w h i l e the p a g e n u m b e r is a l i g n e d w i t h the outside e d g e n o t i n g that right-hand pages are odd-numbered. Running h e a d s and p a g e n u m b e r s a p p e a r on the same line. N o r m a l l y , the r u n n i n g h e a d on the l e f t - h a n d p a g e is the c h a p t e r h e a d i n g and that on the r i g h t - h a n d page is the s e c t i o n heading. R u n n i n g h e a d s should not be i n c l u d e d in p r o c e e d i n g s c o n t r i b u t i o n s u n l e s s this is b e i n g done cons i s t e n t l y by all authors. Corrections. When corrections h a v e to be made, cut the n e w text to fit and p a s t e it over the old. W h i t e c o r r e c t i o n f l u i d m a y also be used. Never make

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

in the text by hand.

If the t y p e s c r i p t has to be m a r k e d for any reason, e.g. for p r o v i s i o nal p a g e n u m b e r s or to m a r k c o r r e c t i o n s for the typist, this can be d o n e V E R Y F A I N T L Y w i t h B L U E or G R E E N P E N C I L but N O O T H E R COLOR: t h e s e c o l o r s do n o t a p p e a r a f t e r r e p r o d u c t i o n . C O M P U T E R - T Y P E S E T T I N G . Further, to the a b o v e i n s t r u c t i o n s , p l e a s e n o t e w i t h r e s p e c t to y o u r p r i n t o u t t h a t - the c h a r a c t e r s s h o u l d be s h a r p and s u f f i c i e n t l y black; - it is not strictly necessary to use S p r i n g e r ' s s p e c i a l t y p i n g paper. A n y w h i t e p a p e r of r e a s o n a b l e q u a l i t y is a c c e p t a b l e . If you are using a s i g n i f i c a n t l y d i f f e r e n t font size, y o u s h o u l d modify the output size c o r r e s p o n d i n g l y , k e e p i n g l e n g t h to breadth r a t i o 1 : 0.68, so that s c a l i n g d o w n to i0 p o i n t font size, yields a t e x t a r e a of 13.5 x 20 cm (5 3/8 x 8 in), e.g.

Differential equations. : u s e

output

Differential equations. : u s e

Differential equations.

size

output

: use

13.5

size

output

16

size

x

20 x 18

cm.

23.5 x

cm.

26.5

cm.

I n t e r l i n e spacing: 5.5 m m b a s e - t o - b a s e for 14 p o i n t c h a r a c t e r s stand a r d f o r m a t of 18 x 26.5 cm). If in any doubt, p l e a s e s e n d us 1 - 2 s a m p l e p a g e s for e x a m i n a t i o n . We w i l l be g l a d to g i v e advice.