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

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Author: Edited by W. E. Kirwan and L. Zalcman

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

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

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

|

I

I

Springer-Verlag Berlin.Heidelberg. New York 1976

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

Library of Congress Cataloging in Publication Data

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

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

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

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

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

have been settled.

At the same time,

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

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

These methods have

already yielded a great deal; their promise,

if anything,

their past success.

exceeds

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

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

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

indi-

of which

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

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

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

The present volume

special year.

well over thirty

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

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

Altogether,

lecture courses,

in a

and informal discus-

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

Partly a (very incomplete)

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

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

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

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

functions,

conformal mapping, disparate topics

Fuchsian groups,

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

automorphic

functions,

functions of several variables.

is a certain emphasis,

sub-

quasi-

Uniting these

in point of view or in method,

on problems having concrete geometric content.

This seems only natur-

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

to be published

in the Springer Lecture Note

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

Lec-

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

For help in editing the

IV

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

Thanks also are due Pat Berg and Paula Verdun for an

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

Complex analysis

is indeed alive and well.

W,E, KIRWAN LAWRENCE ZALCMAN

TABLE OF CONTENTS

C.A.

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

1

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

17

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

24

ALBERT MARDEN Isomorphisms

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

. . . . . . . . .

56

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

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

....

79

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

92

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

i01

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

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

138

URI SREBRO Quasiregular Mappings T.J.

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

148

SUFFRIDGE Starlike

Functions

as Limits

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

......

184

PARTICI PANTS IN THE SPECIAL YEAR

Lars V. Ahlfors

(Harvard University)

Albert Baernstein II

(Washington University,

Carlos A. Berenstein

(University of Maryland)

Douglis M, Campbell

(Brigham Young University)

David Drasin

(Purdue University)

Peter L. Duren

(University of Michigan)

F.W. Gehring


Leon Greenberg


Walter Hayman

(Imperial College, London)

Mauriee Heins


J.A. Hummel


James A. Jenkins

(Washington University,

W.E. Kirwan


Jan Krzyz

(Maria Curie-Sklodowska University, Lublin)

011i Lehto

(University of Helsinki)

Albert E. Livingston

(University of Delaware)

Thomas H. MacGregor

(SUNY, Albany)

Albert Marden

(University of Minnesota)

Petru Mocanu

(University of Cluj)

Raimo N~kki

(University of Helsinki)

Bruoe Palka

(Brown University)

John A. Pfaltzgraff

(University of North Carolina)

Albert Pfluger

(ETH, Zurich)

George Piranian


Christian Pommerenke

(Technische Universitat Berlin)

Edgar Reich

(University of Minnesota)

M.M, Schiffer

(Stanford University)

Glenn Schober

(Indiana University)

Uri Srebro

(Technion, Haifa)

St. Louis)

St. Louis)

Viii

Kurt Strebel

(University

of Zurich)

Ted J. Suffridge

(University

of Kentucky)

Jussi V~is~l~

(University

of Helsinki)

Lawrence

(University

of Maryland)

(University

of Maryland)

Mishael

Zalcman Zedek

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

!.

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

in

Cn

and the area of its zero-variety,

and more g e n e r a l l y

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

(see,

e.g.,

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

Griffiths,

Lelong,

[12] for references).

Stoll,

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

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

in the case of a single

variable. M a n y problems

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

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

C I.

(from one

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

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

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

the r e q u i r e d estimates

(Theorem 2 of

w

below).

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

2.

PRELIMINARIES Let us recall

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

~n

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

(cf.

[7]).

d = B + ~ ,

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

~[.

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

The exterior

and w i t h

de =

In

~m

we indicate

so it makes

B(0,r)

~

~n =

= B r = {llzll

A2n

E C n,

=< r } ,

B(a,r)

the Laplace

operator,

to functions

Ag =

of n-complex

x_~, j=l ax. ] variables

~2n.

z = (Zl,...,Zn)

generally, r

A : Am

sense to apply

by identifying If

by

Sr

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

=

:llzll

(z

= r}

= {z : llz-all ~ r}.

for

we w-rite

0 < r < ~.

We can define

two

More

(l,l)-forms

by n

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

= dd c

Then

Cn = r ^ "'" ^ r

logll zll 2 ,

(n times)

more generally

the restriction

linear variety

is the euclidean

other hand,

~n-I

under unitary

transformations

(i)

~2n-I

O.

is the volume

of

is a measure

z~

Ck

form of

~n ,

to any k-dimensional

area form of the variety. of "projective"

and complex

: dc l~

area:

dilations,

and

(complex) On the

it is invariant and

^ ~n-l'

is the area form in the unit sphere

S 1 = {Ilzll = I},

~

~2n-i

i.

1 If i.e.

f

is an analytic A2n ioglf(z) I

the analytic

(2)

defines

variety

~(r)

a positive

measure,

= 0}.

of (2) defines

we can define : ~

loglf(z) I

is subharmonic, whose

support

is

Moreover,

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

that the l.h.s,

As usual,

then

V = {z : f(z)

ddC l~

it follows V.

function,

the countin~

ddC l~

^r

the euclidean function

area form in

by

0 < r < ~.

r

More usually, and

if

D r = D n B r,

D

is a cone

then

in

~n

(having vertex at the origin)

(3)

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

Similarly,

we have the projective

9(r)

: ~

area of

V,

defined

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

If we assume formula

further

that

in Nevaniinna

f(0)

~ 0,

we have the following

theory

(4)

~(r) : ~(r)

r2n-2

Sketch of the proof. one sees easily

9(r)

that

: I

Clearly

~n-i

dCn_l

ddCl~

: I

I.

2.

Furthermore,

by Stokes theorem,

dCl~

^~n-I

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

= ~ 1

r This

For

fails when

of additional

n = I,

a(r)

a(r).

r

simple relation

due to the appearance Remark

: 0.

hence,

= [

Sr Remark

= d~n_l

= IIzll-2n+2r

Br

~D

crucial

boundary

= 9(r)

~

is replaced

by

terms.

= number of zeroes of

f

in

Br 9 The next important ducing

it to the one variable

(5)

~(r)

where the operator inner

formula

integral

allows

us to compute

It is Crofton's

case.

~(r)

by re-

formula

[ii]

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

acts on the complex variable

just counts

the number of zeros of

g(k)

l,

so the in

: f(kz)

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

and finite

that a function

f

is said to be of order

type if there exist constants

A, B

> 0

P such

If(z)l For such functions,

it is known

r§

and therefore

lim r~

Similarly,

Crofton's

degree

then

complex

half-plane g(0)

# 0.

{~ :

I~

-

THEOREM

Denote

= lim 9(r) r§ rp

shows that if

two theorems Let

g

by

< =

f

ePB.

is a polynomial

from the theory of functions

be an analytic of order

9g(r)

If

[9, p.185]

Sg(8)

Cp

=< epB

of

p

function

and finite

the number

defined

type,

of zeroes

of

of one

in the

such that g

in the disk

r/21 ~ r / 2 } .

lim r+~

where

that

< m.

{Re ~ ~ 0},

I.

function

o(r) r p+2n-2

formula

9(r)

variable.

[9, p.44])

(5)

by (4) and

We now recall

(cf.

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

(6)

m,

=< A exp {BllzllP}.

p > 1

then there exists an increasing

such that ~g(r) rp

< =

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

2~(p-l)

cosPe dSg(e)

is a positive constant independent of

< C B = P

g

and

B

is the

constant involved in the definition of finite type. Remark theorem

3.

By using conformal

for functions

larg I I ~ e/2.

of order

This possibility

since by a theorem of Liouville

mappings,

p > T/e,

we can obtain a similar

defined

does not exist

in the angle

in

the only conformal

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

transformations. The generalization

of theorem

of this paper and appears

in w

I to cones

in

Cn

is the objective

Suppose open cone

f

D

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

in

C n.

p

and finite type,

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

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

(7)

f

in an by

z # 0.

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

For

n = i,

h

is even continuous. We say

all

z

the outer

f

lim

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

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

if for almost

we have

E D n SI ,

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

(8)

r§

r p

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

T H E O R E M II.

p > 0

Let

[9, p.182]

and completely

g

be an analytic

regular growth

there exists an increasing

lim r+ ~

The m e a n i n g of of formula

3.

9

g

g

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

function

(r) (9)

function of order Then

such that

Sg(e)

712 _

rP

1

[

2~p ~-~/2

cosP8 dSg(8)

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

(9) to several variables

2. N,

If

N r Sl,

we

i.e.

t > 0},

r"

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

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

PROPOSITION

i.

C SI,

f

and

K c N,

Let

p

a f u n c t i o n analytic in

f(z)

uniformly

in

K .

an open set

such that for every compact

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

Then

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

(12)

Proof.

f

N

N

we have

(ii)

Pm

be a non-zero polynomial,

If

z E SI,

is a homogeneous

polynomial

are of completely

pm(Z)

# 0

t > 0,

regular

p(tz)

: tmpm(Z)

of degree

growth

m.

+ o(tm-l),

Clearly both

in the sense that if

and

z ( N

and

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

Take any such

z E N

and pick

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

s > 0

I

formula

e,

Let

we have

the Jensen

(14)

0 < s < I,

such that

f(sz)

# 0,

so from Crofton's

in n-variables

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

loglf(sz) I

s

:

Gsz(t)

= I

i~

ddC l~

dt asz(t) ~

'

r

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

i~ kl(e)

dt ~

is a positive

~

D' =

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

S1

where

p

then

(13)

where

where

~ kl(e)

constant.

~sz(3/4 s2n-2

es)

For

formula

one

In other words, (i5)

for any

kl(e)

r > 1

asz(

1

es)

I

< ~2n_l(~) -- S1 Since

r > i,

(16)

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

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

such

find

P

function

u(x)

for any

such

aside

u Z -%

that

w = u - Bv

~ BrP+c

for

- C,

v

P

R ~

p~

which

some

we

i8

positive

can f i n d

a

2

: max(p,pl).t

for

the m o m e n t

the

set

{u

exceptional

cases

has m e a s u r e

zero

as c l o s e

Applying

as d e f i n e d

the

= -~}

a 6 K(e)

~ -~.

K(~),

for

0 < 8 < ~,

that

a point u(a)

in

as we w a n t

Green's

in (33),

formula

we h a v e

I

where

n

GR(x,a)Au

denotes

p = pI

+ w(a)

KR(~) the

we have

:

I

for

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

inner

normal.

to take

Clearly,

P* = P I + e,

w ~ 0 e > O.

on

to

to the

2

(41)

$for

result.

Au < MR p*+m- 2 ' =

Let us l e a v e

P = Pl,P2,'''.

pr,i n c i p a l

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

C,

M,

our

3K(e);

14

therefore,

setting

I

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

~GR
Pl'

P ~ PI'

the Phragmgn-LindelSf

shows that

in inequality

Remark 6. for

the interesting case of theorem 1 occurs

w ~ 0

everywhere

in

theorem

(cf. [5,6]

K(e);

the dominant

(42) just gives the integrability condi-

The constant

M3

in (42) is proportional

to

B.

Thus,

theorem I assets that

R-P-m+2 I

Au

< C(6)B.

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

slightly to a bound analogous to that

(w

Because of its importance, analytic functions.

we restate Theorem i in terms of

15

THEOREM 2. Suppose Cn

such that

type. If

as(R)

f

f(0) # 0

is an analytic function in the cone and

f

has order

p > pl(~)

denotes the area of the variety

K(~)

and finite

V N KR(8)

(8 < ~)j

we have

~BCR) lim R+~ ~ (B

of

< C(8)B

is the constant appearing in the definition of the type of

f)

REFERENCES i.

Bateman Manuscript Project, McGraw Hill, 1953J

2.

C.A. Berenstein and M. Dostal,

3.

G. Bouligand,

4.

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

5.

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

6.

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

7.

L. Gruman,

8.

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

9.

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

10.

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

ii.

B. Schiffman,

to appear.

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

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

Transl.

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

1973.

12.

W. Stoll, Value Distribution Theory,

13.

R. Tijdeman,

part B, M. Dekker,

1973.

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

ASYMPTOTICBEHAVIOROF COEFFICIENTSOF UNIVALENTFUNCTIONS PETER L, DUREN

The a u t h o r Tauberian

remainder

coefficients related

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

of u n i v a l e n t

results,

pointed

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

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

functions.

including

estimation

The present

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

between of the

note d e s c r i b e s

on a t h e o r e m

some

of

Bazilevich. As usual,

a nal y t i c Hayman

S

is the class

f(z)

= z + a2z2

and u n i v a l e n t

[4,5]

asserts

(i)

lim

with

equality

only

k(z)

+ a3 z3 + ...

in the unit

that

!anl n

n~

of all f u n c t i o n s

for each

< i.

A t h e o r e m of

f E S,

= e < i, =

for a r o t a t i o n Z

-

IzI

disk

of the Koebe

function

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

(i - z) 2

The proof c o n s i s t s simple

argument

growth,

that

in the

(2)

of two parts. each

It is first

f E $

has a d i r e c t i o n

complicated considerably recently

is

O

step

of m a x i m a l

difficult

discovered out

= ~,

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

is to deduce

more

As pointed

e i80

sense that

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

and the limit

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

(I) from

for

~ > 0

a sim~er approach in [3],

(2).

The second and more

Hayman's

than for in the case

the d e d u c t i o n

of

argument

~ = 0.

Milin

~ > 0.

(I) f r o m

(2)

is

is [8,9]

18

essentially

a Tauberian

step.

f(z) g(z)

:

To be s p e c i f i c ,

(i - z) 2

~

:

let ~

z

fCz)

n

:

b

n=O

z

n

Then n Sn =

[ k=0

bk

=

an+ 1 - a n

n ~ n T1 1 k=0

Sk

=

an+ ~ n+l

and

_

On

After

a suitable

(8)

rotation

tim

of

IgCr)l

f,

we m a y a s s u m e

that

= ~ ;

r+l in o t h e r of the

words~

series

Consider the u n i t number

8

= 0.

while

n o w the

class

disk onto e

as

in

normalized

by

(8).

application

Thus

0 [ bn ,

the

(2).

(1)

is a s t a t e m e n t

is a s t a t e m e n t

S (~)

exterior Assume

Then

of the

(2)

about

about

of f u n c t i o n s

of an a n a l y t i c

that

~ > 0

a square-root

the A b e l

the C e s ~ r o

f E S

which

arc a n d h a v e

and

that

Schwarz

reflection

principle

show

~z

{i + Cl(l

- z) + c2(i

means. map

Hayman

f 6 S (~)

transformation

means

is

and an that

f

has

the

form

f(z)

(4)

-

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

Ill

= e.

has o b t a i n e d

the

-

of

the p o i n t

For

- z) 2 + "''}

z) 2

(i

each

z = i,

f E S~(e)

where

with

l

m > 0,

is a c o m p l e x

number

Bazilevich

[1,2,9]

estimate

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

where We

the c o n s t a n t s

shall

strengthen

B1 this

and

B2

result

depend by

only

eliminating

on the

a,

c I,

and

logarithmic

c 2. term.

ig

The proof is simpler than that of Bazilevich,

since it depends only

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

w h i c h we state first

as a lemma.

LEMMA.

Let

the f u n c t i o n

g(z)

:

[

bkzk

k=l

be a n a l y t i c

in

the

sums

partial

then

o

]z I < 1 and

and

on

at

the

the p o i n t

Ces~ro

means

z : i.

Let

sn

denote

of

b k.

If

sn

the

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

n

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

h(z)

The f u n c t i o n hypothesis,

:

~

h

[6],

(Theorem I.i, p.47).

(as suggested by Korevaar)

we now proceed to give it.

w h e r e it

Since the direct

is short and elegant,

Let

Is n - g(1)] z n .

:

1 - z

n=O

is also analytic

in

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

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

= 0(1),

Izl < 1

h

and at

are bounded.

(proved,

z : i.

By

H e n c e by a variant

for example,

by obvious

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

pp.

the partial sums n

[ k=O are also bounded.

T H E O R E M I. e

ie o

For

of maximal

Is k - g(1)]

In other words,

each

f E S (~)

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

with

~ > 0

growth, -ie O

an e

where

Ikl

= ~ .

:

1)

~ + 0(~

,

and with

direction

73-

20

COROLLARY.

For

each

lanl Proof e o : 0,

f ~ S (~)

~

a n + C,

of Theorem.

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

-

(i

with

-

n = 2,3,--..

Assume again

Z

z) 2

~ > O,

without

the

loss

of g e n e r a l i t y

that

function

f(z)

=

bn zn

Izl < 1

n=O

Since

f

has

analytic

the f o r m

there

f

: ~{i

+ Cl(l

is u n i v a l e n t

are b o u n d e d ~

z : i,

it f o l l o w s

that

g

is

and

g(z)

Since

(4) n e a r

as

and

shown

in

- z) + C2(i

~ > 0~ [3].

- z~ 2 + "'" } 9

the p a r t i a l

Thus

by the

sums

sn

of

~he

bk

lemma,

a _

n

~n-1

as the

theorem

It

should

which

that

appeal

THEOREM

f

is b o u n d e d

that

this

that

true Here

if

(and hence complex

f E S,

has a pole ~

Under

full

of

f

is m o r e

We w i s h

can obtain direct

to t h a n k

and

its

the a d d i t i o n a l

each neighborhood one

u s e of

of

z = i,

an e v e n

and m a k e s

W.K.

Hayman

no for

of a r g u m e n t .

of finite

constants

a role.

outside

the a r g u m e n t

did not m a k e

the u n i v a l e n c e

f E S (a),

and 8uppose

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

proof

Only

played

theorems.

type

Let

the

f E S*(e).

to T a u b e r i a n

2.

n

z = 1

result.

suggesting

O(l_)

near

is c e r t a i n l y

stronger

+

be o b s e r v e d

structure

hypothesis

~

asserts.

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

_

n

area. o f order

and

~,

that for each

Jz-

at most

f

maps

> ~}

ii

Suppose

e > 0,

f

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

two there).

at

z : 1

Then for some

21

(5)

a n = nk

If

COROLLARY.

Proof

f ~ S (~)

of T h e o r e m .

f(z)

where

k

is the

~

Koebe

is a n a l y t i c

choose

+ Us

has

the

form

+ ~(z) ,

function,

~

and

are

constants,

z

at

z : i.

= ~(z)

the c o n s t a n t

and univalent

f

(5) holds with

1 - z '

r and

then

~ > O,

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

_

and

with

= kk(z)

s

n = 2,3,-..

+ ~ + o(i/~),

in some

r

:

Let

- vz,

v

so that

disk

r

Iz - 11 ~

# 0. e,

E > 0.

Then

r

is a n a l y t i c

Let

7. c n z n , n=O

so t h a t

(6)

an

In v i e w of integral

:

kn

+

(6) we n e e d

over

~

only

the u n i t

prove

imply

(7),

D r

n

'

show

that

r

has

=

2~3,

finite

....

Dirichlet

n=l

that

c

n

: o(i/vrn),

we use the hypothesis

I

Thus

n

disk:

I~I 0,

is l i n e a r ?

A complete

question.

v < 0 He h a d

answer

are obtained

is c o n t a i n e d

in

THEOREM

k

Let

i.

v

be the number

nite

of

harmonic

of components

if and only

the degree

be a real

if

v,

v

of the set

is a polynomial.

we have

(i 9 i)

the sharp

If

k : 2,

n = 1

v > 0

in

of

plane

harmonic

given

in

THEOREM

D

D

point

Then

v # 0.

Then

k

In thi8 case,

if

Let

is fin

is

inequalities

so that

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

all

in the plane.

< n < k. ~i k --

COROLLARY.

if

function

2.

entire only

and

from

Let

v--+0

inside

functions

D

function8

D.

v(z) as

has

is linear 9

a domain

approaches

It is n a t u r a l

can have

be a plane

3 cases

the

domain

f = u + iv

the f o l l o w i n g

z

v

such

same

and

any

to ask tract.

let

that

F(D)

v

has

as a tract

D

finite

boundary

in w h a t w a y A full

two

answer

be the class D

is

of

as a tract.

are p o s si b l e

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

25

(i)

F(D)

is empty;

(ii)

F(D)

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

entire function.

fo

is any m e m b e r of

all other members are given by

F(D)

a > 0

(iii)

In this case if

and

b

is real;

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

F(D)

In this case

fo"

where

f = af 0 + b,

consists of all functions

F(D)

where

af 0 + b/f 0 + c,

entire f u n c t i o n

a ~ 0,

ab > 0

b [ 0,

f =

and

c

is

real.

Before

considering

Example

i.

ly c o n n e c t e d not

Take

but

sufficient

Example Zn---+~

v

is n o t

Let n

proof

zn

and

of T h e o r e m

v = 1 - e x cos y .

to c o n s i d e r

2.

with

the

the

an

of

0

examples.

is e v i d e n t -

in T h e o r e m

1 it is

v.

of r e a l

be p o s i t i v e an

set

Thus

tracts

be a s e q u e n c e

let

The

a polynomial. just

1 we give

numbers

numbers

such

such

that

that

~.

n=l l+IZnl Set

f(z)

=

u + iv

an z-- z

:

n-

n:l f(z)

Then

y.

Since

tal, tions

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

it is c l e a r with

functions

in the p l a n e

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

Theorem

poles.

Also,

f(z),

Theorem

since

v

of a r b i t r a r y

1 does there

2 does

and

not

not

has degree

extend

is a v e r y extend

the

same

either.

as

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

to h a r m o n i c wide

sign

choice

funcfor

the

26

i.i.

Proof of T h e o r e m

i.

The proof of T h e o r e m

L EMM A

1 is based

Under the hypotheses of Theorem I, let

i.

tracts of

v.

Then the finite boundary

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

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

which go from infinity on

D

~ = 1

consists of

to

p,

to infinity in the

from c o n t i n u i t y

sidering

the b e h a v i o u r

that

has

J o r d a n arcs curves.

r ,

of

be one of the

z

p

where

p

plane.

is

Also,

F.

It is evident

F

r

D

results.

The

of

sectionally going

from

latter

F

that

case

on

F.

near a b r a n c h point

analytic ~

u = 0

to

character.

~

and

is h o w e v e r

of

Thus

(possibly)

excluded

Also, by conv,

F

we see

consists

of closed

of

Jordan

by the m a x i m u m

princi-

ple. It remains finite.

to show that

It is clear

domain

D

where

the total number

that each

v < 0.

F

Further,

Since the total n u m b e r of tracts

of

p

of curves

separates

D

all these

D9

-v

F

is

from at least one are distinct.

is finite,

p

must

be

finite.

L EMM A

ber.

2.

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

Then the equation

where

f

f(z)

has at most

= w0

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

We map the unit

disk

I~I < 1

(1.2)

z

onto

:

w0

v

D

be any c o m p l e x nump

roots in

D,

as imaginary part.

by a c o n f o r m a l

mapping

t(~),

and set

F(~)

Then

U

is p o s i t i v e

=

f{t(~)}

and h a r m o n i c

in

=

U + iV.

I~I < 1

and

V

vanishes

27

continuously sible by

as

~

exception

of

(1.2).

It t h e n

approaches

any p o i n t

p

~i

points

follows

for f u n c t i o n s

of p o s i t i v e

6.13,

that

p.

179)

f r o m the imaginary

F(~)

=

ic ~=i

where

the

Thus

c

are r e a l

F(~)

evidently tion

f(z)

Lemma

2.

has

= w0

It f o l l o w s only

have

it f o l l o w s tial

from

from

Lemma

=

the

is f i n i t e Set

and

f = u + iv

number

and

of c o m p o n e n t s

show

each

the

total

(i.i).

let

F0

of the

e.g.,

0, the

I~I z

[6],

~ = 1

to theorem

Theorem

< 1.

the

to

equation

v

equation

in the p l a n e f(z)

the

This

that

= w0

equa-

proves

and f(z)

for a n y

cannot

p. F(~)

Thus

= t(~).

have

-v = w0

w 0.

Now

an e s s e n -

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

To see this be the

correspond

set

complement

of

k

to s h o w that

of t r a c t s

we a r g u e

v = 0. F0

of

if

Ivl

as follows.

Let

in the

k

be the

closed

plane.

that

k

s

is the n u m b e r

counted

with

To p r o v e F0

and

in

f(z)

of T h e o r e m

then

(1.3)

where

so

the p o s -

+ ic 0,

c~ ~

that

with

representation

the h y p o t h e s i s

of r o o t s

and

satisfies

and

= i

which

(see,

~+~ -9 ~-~

roots

theorem

proof

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

part

of tracts, that

number

at

p

I~I

classical

~v1

2, and

number

~p

function

roots

Picard's

singularity

f(z)

p

at m o s t

a finite

To c o m p l e t e

We

has

a finite

has at m o s t

constants

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

to

on

considered

(1.3)

n + s + i,

of f i n i t e

correct we

=

branch

points

of

f

on

F

0'

multiplicity.

study

as a n e t w o r k .

the

disection

As v e r t i c e s

of the we t a k e

closed the

plane

finite

by branch

28

points

Vg,

9 = 1

multiplicity of order Let

to

s ,

q

of

together

f

on

r0,

which

with

~,

which

we

acts

supposed

as a b r a n c h

to have point

n - i. e

be t h e

total

number

of edges,

to a v e r t e x ,

possibly

the

same

2e

arcs

r0

coming

analytic

branch

point

arcs.

Summing

of

of order

s

of

over all the

one

(namely

out

F0

each ~).

are

points

are

thus

vertices.

2s+2

of

from a vertex

There

of all t h e

there

branch

going

At

just a

such analytic

F0

including

~,

we

see t h a t

2e

q [ ~=I

:

(2s~+2)

+ 2n

:

2s + 2n + 2q,

i.e.,

e

On t h e

other hand

it f o l l o w s

k

which

is

finite,

We note

k

also

must since

in t h e

f

plane

from

is

ples

f = zn ,

show that

(i.i).

is

degree

n- 1

+

q.

Euler's formula

:

is sharp.

n

and

completes

f : zn + i

(i.i)

n

that

s + n + 1

that

since

q

and

so

e

is

be f i n i t e . has

This

+

incidentally

n + 1

which

s

e - q + 1

(1.3).

Finally points

:

:

i,

v : I m ( x + iy) n =

polynomials

8 + i sin e)

k.

k

1 ~ ~ ~ k.

inequality

to odd

even

and has

satisfies

the

we r e p l a c e and

R2

the h a r m o n i c

Iv I s h o w that

everywhere, at t h e s e

u

z = r(cos

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

of

in

However,

isin(}kS) u

at p o i n t s

these is evi-

I

Also,

satisfied,

1 n = [ k

and

I

is un-

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

so t h a t

u = 0~ u

is still

s.h. However,

Heins'

theorem

lies m u c h

deeper

than

the

corresponding

32

result for harmonic polynomials. Denjoy

conjecture

proved by Ahlfors

totic values of an entire

k

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

Since the values ferent

9

a9

in the open plane

as

z-~

F

are disjoint paths going from

zV

lies in

from

0

to

z

We complete

in common.

D

for

from a classical each domain

9 = 1

to

k,

a~ ~ ag+ I.

from dif-

F

~

in

Izl ~ R,

by straight from

0

so that

y9

Yk+l

that

~

having only

arranged

and

= Yl"

f(z)

line segments

to Yv

where

Y~+I

in antibound a

Also it follows

must be unbounded

in

since f(z)--~a9

and

Julia

F

Thus we may assume that the to

where

values

F .

We may assume the

theorem of

D ,

~.

f(z)

such that

distinct the paths

y9

clockwise order around the origin, domain

along

z9

and obtain paths

these endpoints

finite asymptotic

F

will not intersect near

Izl = R.

To see this, suppose that

distinct

are supposed

the famous

on the number of distinct asymp-

function.

is an entire function with a .

In fact, it contains

Thus

as

z --~

If(z) I < M, yg,

along

Yv"

say on the union of the paths while

If(z)l

in each domain

> M

somewhere

D v-

Consider now u(z) u

: u(z)

the plane and yg,

E : u(z) E

> 0

while

u(z)

> 0

somewhere

in each

has at least one component

has at least

k

If(z)l1

{----~I

: Yk+l

Then

paths

l~

components

and

u(z)

is clearly u(z) D .

= 0

s.h.

in

on the

Thus the set

in each domain has at least

D9 k

and so

33

tracts.

4,

u(z)

has

and since the lower order of

u(z)

coincides

definition

of

if

Thus,

by T h e o r e m

of the lower order

k ~ 2.

in

k

we deduce

Ahlfors'

1 P ~ 7

also yields

and

which have at least

Rm

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

in

any o r d e r

if

theorem

k = i.

m

both greater than one,

let

Rm 1

k > 2

k

to

What is

is e s s e n t i a l

has at least

subject

tracts.

since

one tract and

0 < I < ~.

s

any u n b o u n d e d

such a f u n c t i o n

In view of T h e o r e m

s.h. m a y have

4 we have

1 = ~ k.

s

It turns m > 2 when

out that

into

One reason

k

congruent

divides

into

k

AN UPPER BOUND

moni c

Heins' m e t h o d

though the results k = 2.

plane

3.

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

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

s

Rm

1 ~ k,

at least

the f o l l o w i n g

Given integers

PROBLEM.

f(z),

In fact the a r g u m e n t

We can now formulate

lower order

polynomials

examples

sharp w h e n

or

in all cases

shall

see,

sible

in certain

THEOREM

5.

good bounds

k = 2

m = 2

Let

for and

for h a r m o n i c

for general

except

fact that we cannot

divide

s.h.

which is even.

polynomials

some harThese

are c e r t a i n l y

They may even be although,

as we are pos-

cases.

m, n

be integers,

k

is defined as follows.

k

of degree

m ~ 2,

v

where

u,

2~/k.

of tracts.

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

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

opening

to c o n s t r u c t

large n u m b e r

s k

in the way that the

of a n g u l a r

We proceed a relatively

to the case sharp,

cones

sections

s

quite well

are no longer

is the

circular

congruent

having

sharp

can o b t a i n

for this

right

FOR

[3] yield

~

extends

n

in

R m,

n ~ i.

There exists a

such that Suppose

that

Ivl

has

34

n + m - 2

(3.1)

where

p, q

are integers

(3.2)

and

k

We n o t e

that

:

k : 2n

(3.3)

:

(m-l)p

+ q,

0 ~ q < m-l.

Then

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

if

m : 2.

k = 2n,

If

if

m > 2,

we h a v e

n < m

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

(3.4)

2,

m > 2

m

rl~ k] + 1 Llog 2J

2m

y. We

shall

need

we

shall

quote

certain

properties

f r o m the

b o o k in

[13].

k kj

and satisfying

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

k < 2m

'

i8 the integral part of

We p r o c e e d

orthogonal

O, k

having

there exists a distinct

the differential

+ k(k+2k)y

:

0.

~ero8

equation

35

pkk)(t)c

Also k

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

t

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

is even or odd.

The polynomials nomials.

P~l)(t)

The property

and the equation statement

are the so-called

of the zeros

(3.6) is given

is formula

(4.7.4),

ultraspherical

is discussed

poly-

on p. 117 of [14]

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

p. 80.

We deduce LEMMA

5.

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

Let

nomial of degree

xI

in

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

and set

v v

~

=

be a homogeneous

x m.

to

Let

t = Xm+i/R.

be a positive

integer

Then if

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

is a homogeneous

k

harmonic poly-

1

1 ~ + ~ (m-l]

=

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

s + k

in

xI

to

Xm+ 1 9 Suppose

first that

~i k

degree

in

k

is even.

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

is clearly Similarly

a homogeneous if

k

Rk_(~ r k )(t) where

~

homogeneous

is a polynomial polynomial

It remains x = Xm+ I,

:

=

is a polynomial

of

Thus

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

k

in

xI

to

Xm+ I.

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

of degree

to show that

p2 = [mv:l xg, 2

P~I)

by Lemma 4.

polynomial

is odd,

Then

v

k

=

Xm+iR2Pr

p,

so again

in

xI

is harmonic.

t : x/R, Q

=

Rkp(x/R)

to

Rkp~l)(t)

is a

Xm+ I.

To see this,

we write

36

where

P = P~k).

(3.7)

~-~ P(t)

and for

9 : !

~x This

P(t)

leads

We note :

to

that for

p2 P'(t) ' ~-~

~Q ~x

kRk-2xp + Rk-302p '

:

m xx9 R3 P'(t),

:

x : Xm+ I

3Q

kRk_2x

:

~x

p

-

Rk_3xx 9

P'

to

(3.8)

82Q 3x 2

:

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

(3.9)

~2Q Bx2

=

(kRk_2 + k(k_2)Rk_4x~)p

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

~ 9

Then V2(Qu)

=

QV2u + uV2Q

=

uV2Q + 2

+ 2

~ ~=i

8u 3xv

9 8__q__ 8x9 3u

m

~

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

~=i :

by Euler's

theorem,

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

on

k-2P

=

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

since xI

u

to

_ Rk-3xp,)}

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

xm

only.

From

of degree (3.8)

and

s

harmon-

(3.9) we have

_ Rk-3xp,)

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

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

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

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

_ 2s k-3)

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

: in v i e w of

Rk-2{(l-t2)p"

(3.6).

We now p r o c e e d

Thus

- (m+2Z)tP'

v = Qu

to p r o v e

+ k(k+m+2s

is harmonic.

Theorem

5.

Let

This n

= proves

0 Lemma

be a p o s i t i v e

5.

37

integer.

Let

k2

to

km

be non-negative

integers

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

such that

n.

We define

u2(x) This function that

Up(X)

=

is harmonic

Rg(xl+ix2)k2.

and

has been defined

UP is a harmonic kp+ 1 = 0

lu2(x) I

polynomial

we define

v

m = p.

:

of degree

k 2 + --- + kp

Up+ 1 = Up.

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

where

p = 2.

any values of

xI

of

Vp+ 1

tp

fixed,

where tp

Up+ l(tp)

to

has

Xp

-I

largest value of

u2

Um(X)

We wish to maximize Then

to

to

as Up+ 1

in

p.

Xp.

If

s = k2+'''+k p,

Xp+ 1

are

goes from

-~

to

+~.

p + 1 = m,

km

tracts.

of degree

s 2 = k2,

we see that k2 + k3 +

s9 = k

s~ : n+m-2. s~

Since

sign, we see that

and having

if we choose the

intervals Xp

xm

m ~:2

different to

to

Write

Then clearly for

are independent.

Taking

This

xI

xI

km"

components.

Also, when

of constant

tracts.

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

(kv+l)

kp+ 1 + 1

sign.

+i,

intervals

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

3 ~ ~

2 m.

We s u p p o s e

we h a v e

q = I.

Thus

~1 (n+l) 2

integer,

first

1 p : ~ (n+l),

that

m = 3.

q : 0,

while

Then if

n

we h a v e

k

:

~1

we d e f i n e

n

to be the

or

{(n+l)2_l}

smallest

integer

that

(n+l)

T h e n we can

find

(n+l) 2 = 2k I

a function

2

>

2k.

of o r d e r

or

2kl+l ,

so t h a t

suppose

m > 3.

Let

n

having

k I ~ k.

kI

domains,

Thus

~(k,3)

largest

integer

~ n ~

where (2k) 89

as r e q u i r e d . Next

2

n

be the

(n+m m-I

r

0 < S < i,

where

r

is convex

then

(4.2)

s

~

r

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

whenever (4.1).

[ S

In fact, since

= I, and this yields

This technique

lest results

~(S)

is convex we have

s

is due to Dinghas

in this direction.

from Jensen's

~ r

in view of

[4] who obtained

the earl-

42

Before of

m.

going

For, a n y

Rm+l

of t h e

B(r))

when

lower

bound

s

we

for

Rm

if

t = cos

stead of mial

choose mined

that

be c h o s e n cos ~ related

in

becomes

Rm

order

with

(in f a c t

respect

to

yields

a s.h. the

function function

same

value

X m + I.

Thus

any

a lower

bound

for

4 shows

that

the

satisfies

function

6.

s : 0,

For we may However,

now

is an i n t e g e r , which

the cone

largest

y

will

but

for

z e r o of

has

=

k = ~

not

at

solid

y

(-i,i)

in

S

y = 0

If

and

e,

(4.4)

S

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

=

is d e t e r B

has

8 = e. S

are

dx~

where

F(lm) (4.5)

fm =

1 (m-2) 2

p.

82] on

u = (sin

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

m > 2.

8)Vy,

where

v =

that

(4.6) and

r(89 setting

vanishes (For

The proof

at

l

sin 2 8

8 = 0, e

m = 2, u = 0 of Theorem

at

and

=

8 = ~'

6 is r a t h e r

0

J

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

in b e t w e e n at

long and will

in-

We must

e

then for

m

be a polyno-

function.

angle

and

equation

and write

t = i.

8 < e

is h a r -

by

[14,

of

0

in g e n e r a l

a Gegenbauer

is a n a l y t i c

8 < e

y > 0

set

r~f(e)

the differential

+ B(B+m-2)y

so t h a t

We find

in

< m.

solution

is t h e

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

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

~

so t h a t

lower

~(k,m)

automatically

y = f(8)

m + i.

unless

u

constant

of Lemma

(4.3)

where

u

and

s m'

that

function

order

set

The method in

s.h.

same

when

monic

on w e n o t e

for

8 = 0. ) be published

to Thus

43

elsewhere

[3].

I should like to indicate

it depends and then to deduce method is due to Huber

only some ideas on which

some numerical

to Lemma 7 have been proved by Bandle

let

8 are the following

The first and third are due to Huber

LEMMA 6. y

Let

D

[9].

g r a d i e n t of

y

Results related

Rm

be a smooth domain on the unit sphere ~n

D

and i8 p o s i t i v e

three

[2].

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

the boundary of

The

[9].

The main tools of the proof of Theorem lemmas.

consequences.

in

D.

Let

D

and

which vanishes on

Vy

denote

the

along the surface of the unit sphere and set

I (Vy) 2 do (4.7)

k(D,y)

~

, f y2 do

where

do

fixed

D

~(D)

is

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

y,

and varying

when the function

subtended by

D

~(D,y)

v = IxlPy(x/Ixl)

at the origin.

(4.8)

attains

l(D)

D.

Then for

its m i n i m u m value

~ =

i8 harmonic in the cone

Here

=

U(p+m-2).

Lemma 6 means that (4.9)

in

Ay + Xy

D,

where

of the Laplace

A

is the Laplace

value of the equation rems on expansions

operator,

~

that part

along the

is thus the lowest eigen-

The result follows

of functions

i.e.,

with differentiating

The quantity (4.9).

0

Beltrami

operator concerned

surface of the sphere.

=

from standard theo-

in series of eigenfunctions

of

elliptic partial differential operators. The next result contains

the main new idea of our proof.

It is

44

based

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

of the

isoperimetric

result

due

LEMMA S

7.

to E.

Among

times

cap

are

by

related

satisfying

suppose

and

final

8. D(r)

D(r)

be

together

be d e f i n e d in terms

of

sphere,

be s.h.,

the

support

xI =

boundary

the

B(r)

=

C

is a

The result problems give

a well-known inequality earlier bound

when

Suppose fine

ug(x)

R TM

a

in

u

on the of

u(x)

whose

8.

when

Here of

is

D

is

S

and

8

only

u = 0,

1 ~ = ~ (m-2) p

area

when

and

we

= ~/2a

= 1/2 S.

Rm

sphere

D(r)

of

>

u > 0

somewhere.

of radius

on the u n i t

D(r)

C exp

and

and

r,

let

sphere.

Let

p = ~(r)

{Irle

as in

(4.8)

}

ro

t

constant.

extends

q.

to

case

which

was

without

of

direction

Dg,

it was

for

is due k u

to

= 0

additional by

Heins

extension

in p r o v i n g

Theorem

Talpur D~,

which

[8]

and r e d u c e s

It is the

m = 3,

tracts (x)

proved

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

obstacle

is small,

in

any e s s e n t i a l

Wirtinger.

now that we have = u(x)

where

ease, L e m m a

our m a i n

in this

the a r e a

Rm

m = 2,

In this

inequality

result

in

use

is

u > 0 of

lemma

conditions

y = u = cos pS,

in terms

max

f r o m the

Theorem

of a s p h e r e

is m i n i m a l

Also

Ixl=r where

essential

is a f u n c t i o n

0 < e < ~.

Then

~.

makes

Ixl cos

with

projection

(4.7)

l(D)

8)~y

result

u

as in

surface

u = (sin

m = 2,

radial

itself

in the p r e v i o u s

where

for

subsidiary

be the

and

If

on the

D

of the unit

u > 0

Let

Let

domains

(4.4)

which

[13].

0 ~ 8 < ~,

m > 2.

The

LEMMA

all

(4.6)

e = o, ~,

inequality

Schmidt

the area

a spherical

argument,

gives

to to

of t h i s 6.

An

a good

[15]. 9 = 1

elsewhere.

to This

k.

We de-

function

is

45

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

denotes the m a x i m u m of

By(r)

>

i

C exp

k

Ck B (r) v

Now the quantities

> --

~v(t)

tities for spherical

values

exp

is at least

, r0

t

r r/e

]ro

_>

max By(r) l_ Also

in. t h i s

theorem

since

]0 , -~-

case

_

if

J0

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

(4.4)

cosec28

i 4S

that

is t h e

: 2.4048

then

S

Also

shows

C2

.2 30 ~2

>

gives =

! - cos 2

- 8 -2

~

increases

i 4 sin 2 ~

< --

i

:

8,

with

"

we

W

+ i 2

~

. 2 sln 7

deduce

0 [15]

30

2

m=

'

3

result.

m > 4

S

=

fm

is0 (sin x)m_ 2 dx

--> fm

=

fm ~

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

(sin

e)m-i

r(89 Thus

if

dm

: fm/(m-l)

= 2F( ~i) F {

d ~)m-i (

[i( m + l ) }

,

we have

1
4, ,

and

K.

+ ~(i-~) sin 2 8

9(l-u)

let

JK

!

0,

1 v : ~ (m-2)

where

be t h e

first

zero of Bessel's

We have

,

[C%m) } m-i

>

j

K

2 +~-

--

1

and

C

=

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

[{~ + ~

i

Thus

(~.12) where = ~

JK

is t h e

(m-3).

first

I + i - ~-m,

zero of Bessel's

It is k n o w n

(see,

e.g.,

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

purpose

we

~ +

some results

function

[1],

as

m >

p.

371)

= 2

which

are

h V~''

~ 2

t

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

As

m

§ m

we have

of order that

independent

and

-

4 ,

m -~=.

set

e,

89 (m-2).

}

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

>

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

m.

T a k i n g for instance

[15]

as

k --+~.

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

Dv

each h a v i n g

choose a point

x

small diameter.

and project

to obtain a plane region and

A

A

r

having area

range to have

Q

r2

Av

such hexagons

H

p

If

N

we xv

is large

will be close to

Dv

4~.

We

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

Thus if first

is small compared with

Q

By joining each

Av.

[ Av

These h e x a g o n s have area

chosen large and then

Dv

N

which will be allowed to tend to zero and

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

In each region

into

onto the tangent plane at

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

now choose a q u a n t i t y

side

Dv

R3

>

Hp,

(l-c)

I/N

N

is

we can ar-

where 8~

r2/(27)

to the origin we obtain

Q

non-overlapping

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

C in

P

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

Cp,

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

Ixl Pyf

Up where

1

P

~p,

Ip

l

D

of

C

P

are related as in (4.8) with

i.e~,

Up(Up+l) Also

1

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

with the unit sphere and m = 3,

Cp

u (x) which P and of the form

D

=

Xp.

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

r

and so

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

P for a plane hexagon of side

r.

This is

g i v e n by

Ir -2,

where

5~

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

I,

such a

is V(w)

=

w

=

i

c r w + r--~T w

+

...

.

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

is a s i m p l e

criterion

mapping

as a s o l u t i o n .

LEMMA.

Let

to

(3.1)

It is g i v e n

r < n < 2r-2,

having

at

0

for

the

V(w)

and

(3.1) by the

let

V(w)

to h a v e

only

the

identity

following

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

solution

expansion

:

w + c wr +

....

r

Then

V

The p r o o f

must

be

of t h e

the

identity

lemma

mapping.

is m o d e l e d

after

the p r o o f

of L e m m a

XXIV

in

[4]

83

as

far as that p r o o f

deals with uniqueness.

Let

(3.2)

where,

~

=

V(w)

for the m o m e n t ,

Q(w)

k

:

w + c_+~wK •

k+l

may be any p o s i t i v e

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

=

+

W

9

m

m

~

integer,

An-i

and lei

~ 0

W

be such that (3.3)

Q(z)dz 2

Clemrly, branch

we may a s s u m e

of

(Q(w))i/2

(3.4)

1~

first

An_ 1 = I.

=

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

that

n

where

G0 :

ml

= ~' m

tion.

Hence,

taking

square

~)W ~

expansion:

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

w

I (Q(w))i/ 2 dw

W -m

Q ( w ) d w 2.

has the f o l l o w i n g

Q(w) I/2

We a s s u m e

=

is odd:

n = 2m + I.

Then

=

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

of

(3.4)

roots

+ O

and

in

Z -m

:

0

c

is a c o n s t a n t

(3.3)

and

~ (~) z ~

integrating

+

log

of i n t e g r a we obtain

Z

0

Or

z TM

(3.5)

+ cwmz m

=

wm ~ a

0

But

(3.2)

z(w) ~

zv

+ awmz m log

w ~

'

0

implies

=

w~(l

= 1,2,...,

+ ek+lWk

+ ...)~

=

w ~ + ~ek+lWk+~

+ o(wk+B+l),

and

log z(W)w

=

log(l

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

=

o(wk).

8~

Substituting

~ wm+U(l

in

(3.5),

we o b t a i n

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

+ cw2m(l

+ o(wk))

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

and

1 s0 = - ~

[0= a w m+9

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

2 r - 2,

so t h a t

for all

k > 0;

2~

n

Let

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

of the

k > m.

V(w)

n o w be even:

e0

=

fact

implies

c = 0

: w,

thus

n = 2m.

Then

=

Taking

1 1 m - -2

O.

k + 1 ~ r

This

hence

:

we h a v e

w -m+l~

proving

square

and

and

the

n : 2m + 1

then

lemma

Ck+ 1 = 0

for odd

n.

[ e w~ + c 0

we

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

lemma

Q(w) I/2 dw

with

sides,

gives -Ck+lWm+k

Under

on b o t h

u

roots

in

(3.3)

and

integrating,

see t h a t w -m+l~

~ euwU 0

+ c

=

z-m+~2

~ e zv 0

Or

0 Since

all

terms

that

c = 0;

but

cz m-~2

hence,

Z

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

squaring

2m-i ~

in the

tions

previous

z = V(w)

solution

V(w)

case

~wU

:

~

= w.

This

it f o l l o w s

W

2m-I ~

~ ~

~Z

9

0

of odd

= w + c2w2

w = 0,

yields

0 As

at

+

D

it t u r n s

n, 9

O

proves

~

this the

out

equation

lemma

that

among

admits

for e v e n

n.

only

all the

func-

85

Now,

if

n < 2r- 2

from

the

lemma.

the

If

uniqueness

n = 2r- 1

part

and

c

in T h e o r e m

C follows

= 0,

(3.2)

then

at once

holds

with

r

k + 1 = r + i. r + i

4.

Since

instead

Consider

of

now

n : 2r - I,

corresponds mizes

and

function

of m a x i m i z i n g

the and

f~

~4.1)

within

- an)

- 21ar(a

real .

are

reduces

Yr

One

occurs

with real.

and

Let

of

Then

S f

f

which maxi-

from Theorem

A it

the o t h e r

part

If the

F the

(4.1)

of if

f

and

this

the

the

inequal-

an

and

l

ar : ar .

:

A it

follows

Hence

f : f,

i.e.,

la r2

-

F : -(a n - la~)

and

gets

i.

"'"

is r e p l a c e d

+ Irar

all real,

analogous

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

if

maximizes

(minimizes)

coefficients

a r*

one maximizes

+

are

implies

all real.

~ > r/2,

an + l l a n - i

real.

because

" a]. = x 3. + lyj,

and

from Theorem

result

the

~,

2

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

I. ]

is

0

~)Yr I,

be an e x t r e m a l

n < 2(r+l)

i,

by

n = 2r - i,

- la$, the

i.e.,

same if

method

of p r o o f

I < r/2

(l > r/2)

Re F

among

those

functions

a2,...,ar_ 1

real,

then

f

has

gives

all

in

S,

a

and which

its c o e f f i -

86

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

Re{a 3 - la~}

Thus the e x t r e m a l set

functions Using points

differential

of this r e g i o n ,

is r e d u c e d

+ a3z3

r = 2) in

Proposition S

for

i

~ < 1

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

in the real

= z + a2z2

Schiffer's

(hence

( minimized)

functions

problem

{(a3,a2)}

f(z)

n = 3

is m a x i m i z e d

only for s c h l i c h t

the p o i n t

made

c a s e of

xy-plane

+ ...

equation

one finds that

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

in

corresponding

S

of

to the

w i t h real coefficients.

corresponding it is b o u n d e d

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

up by the two arcs:

for

0 < t ~ I

AI:

a3

=

2 a 2 - i,

A2:

a3

:

i + t2(l

a2

:

•

and

{a 3 - l a ~ } ,

m(1) for

0 < I < ~,

A2

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

outside

points For

and

(3,-2),

which

and

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

straight line

through

it c o n t a i n s point

Let

( A I U A 2.

if

0 < I < i,

I < 0,

To

through

~ > i,

the same p o i n t s

the are

region

through

If

~ < i,

for

I < i

(3,2)

and

point (3,2)

=

z (l-z) 2 the

(3,-2).

a 3 = la$ + m(l)

(3,-2),

it t o u c h e s

k(z)

corresponds

the p a r a b o l a

(3,2) and

A I.

~ = 0

to

(it t o u c h e s

the p o i n t s

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

passing if

is t a n g e n t

and at the s i n g l e

it p a s s e s

respectively.

a 3 = M(0)

a 3 = la~ + M(1)

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

correspond

On the o t h e r hand, passes

(a3,a 2)

the p a r a b o l a

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

if

t = 0.

min

A2

(i,0),

for

max :

Then,

+ (i - log t) 2)

- log t)

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

M(1)

-2 ~ a 2 ~ 2

while AI

for

~ = 1

at the single

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

(minimized)

in

that S

by e x a c t l y

two

(I > i) (real)

Re{a 3 - la~} functions.

is m a x i m i z e d

87

9

The

case

minimized

(in

f(z) and that

~ : I S)

by the

h(z) l-ah(z)

:

-

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

equation

As

in 1936

functions

shown

The

I~

Let

Theorem which the

in

are

A.

Then,

maximizes

had used

h*(z)

section,

we

LSwner's

{(la2r_ll,larl)}

-

z l-z 2 "

should

remark

differential

for

(r-l)-

r = 2,3, . . . .

4, T h e o r e m

A gives

n = 2r - i, other

good

within

information suitable

if o n e

subclasses

of

as d e f i n e d

in

examples.

be a n o n - e m p t y

I > r/2

a

is

z = l+z2

h(z)

where

Basilewitsch

some

if

R e { a S - a~}

functions

of this

(minimizes)

coefficient

b y the

the material

S,

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

that

where

-2 < a < 2, ---

'

R e { a n - laS},

following

S)

the region

in S e c i i o n

extremizes S.

J.

to d e t e r m i n e

symmetric

5.

with

-2 _< a _< 2, (in

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

One knows

functions

'

it is m a x i m i z e d

f(z)

that

is c l a s s i c a l .

subclass

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

there

of

S

is at m o s t

__in S(a2,

one

..,ar_ I)

f a n d has

real. r

In f a c t ,

(4.1)

Re{(a*n and

if

la .2) r

f*

is a l s o

(I- r/2)(a

- ar)2 ~

a r* = at, < r/2

c a n be w r i t t e n

so t h a t

-

(a n

function

this

inequality

0.

I > r/2

and

* ar

Thus

equality

if

occurs

-Re{a

solution

in

(4.1)

which

reduces

implies

f*

to

then = f.

If

- la2}. r

to the p r o b l e m

a 2 = ...

0,

is r e a l ,

is k n o w n

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

of

(5.1)

< --

a maximizing

one maximizes

A complete

form

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

n

2~

in t h e

-- ar_ I -- 0,

r > 2.

case

88

The

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

Extended

General

Coefficient

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

by J.A.

Jenkins

Theorem

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

to f u n c t i o n s

using variational

extremal

functions

Let

f

maximize

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

i.e.,

(4.1) h o l d s

for the

almost

a r = ar

and

R e { a n - ka~},

an = an.

A it f o l l o w s

used

one to d e t e r m i n e

n = 2r -i,

functions

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

The m e t h o d

the

immediately.

among those

functions

which were guessed

methods.

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

the

f*

: e-lf(ez), Hence

for w h i c h in

lel

equality

within

the

(5.1)

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

holds

If

c r-I

subclass

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

then

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

that

f(z)

This r e l a t i o n

=

implies

2hi e : er-i

e-lf(ez),

a. = 0 ]

if

j - i

is not a m u l t i p l e

of

r - i,

i.e., f(z)

There

=

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

is a f u n c t i o n

F

in

F(z)

+

3(r-l) a3r-2Z

+

) ....

S,

=

z + A2z2

+ A3z3

+ ...,

such that f(z)

(5.2) Indeed,

=

I F ( z r - l ) r-I

=

z

+

=

~A2r

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

~r-2 (A 3

! + ...)r-i

+ A3z2(r-l) 222r-i

z

+

-

A

=

z(l + a r _ i Z

+ a2r_2Z

)z

+

....

define 2 F(z)

. . . )r-l. + 1

F

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

If 1,2,

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

in

D

and n o r m a l i z e d

a n d if

f(z I) = f(z2),

zk

at

is c h o s e n

hence

zI = z2

0,

and

such that and

f(z)

= F(zr-l) r-l.

r-i ~k = Zk '

E1 = ~2"

This

k = shows

89

i that

F

then

f

is in

S.

Conversely,

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

PROPOSITION

If

2.

S(0,...,0),

a2,...,a n

vanish,

maximizes

then

f i

9

9

n = 2r - i,

S

maximizes

within

whose

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

where

the real part of - r/2 + i,

~=

r-i

S.

Conversely, a function

in

is of the form

A 3 - ~A~,

in the class

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

Re{a n - la2},

of functions

+ A3z3 +

F(z) = z + A2z2

and

This proves

f

the class

F E S

if

f

any such function maximizing

similar proposition

F

produces in

Re{a n - la~}

holds for functions

by

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

S(0, .... 0).

minimizing

Of course,

a

in

Re{a n - la~}

S(O,...,O).

3~

Let

a2,...,ar_ 1

aje j-I = aj

be such that

for

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

2wi where

e = e k

and

k

is a divisor of

j - 1

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

k.

f(z)

S(a2,...,ar_ I)

i.e.,

a. = 0 ]

if

By an argument similar to that of

Example 2, one shows that a function this subclass

r - i,

f

maximizing

Re{a n - %a~}

in

satisfies the functional equation

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

Remark.

In the two foregoing examples,

the extremal function

IF

2hi satisfies the equation k > i,

f(z)

the extremal domain

nents in

C,

differential vanishes~

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

e = e k

Consequently,

for

has at least two boundary compo-

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

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

the coefficient

By the work of Jenkins it then follows that

f

A1

is extremal

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

90

meromorphie

in the

zr + z + ar

r+l

unit

r = 2

having

at

0

the

expansion

+

ar+lZ

If

disc

.... (k = I, n =

3) the

situation

is d i f f e r e n t .

have

A1

hence

the a b o v e

=

remark

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

=

applies

I = i.

only

if

2a2(i-

I);

By

(2.2)

we

REFERENCES i.

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

2.

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

3.

J.A. Jenkins,

4.

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

5.

M. Schiffer,

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

Univalent functions whose n 329-349.

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

first coefficients are

ON INCLUSION RELATIONSFOR SPACESOF AUTOMORPHICFORMS CH , POMMERENKE

INTRODUCTION

i.

Let

F

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

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

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

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

and

1 ~ p ~ ~,

analytic

in

let

D

A~(F)

F

D

with area

onto itself,

and let

8F = 0.

q = 1,2,...

For

denote the space of functions

g(z)

that satisfy

(I.I)

g(r162

q : g(z)

(r ( r )

and

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

(1.2)

< |

if

i < p < |

if

p : =.

F

sup

(~

-

i~I~qlg(z>l

i,

p = i.

have

93

J. Lehner

[5] has recently proved

exists a constant

y = y(F) > 0

inf d(z,r z~D where

d

such that

I Y

for all hyperbolic

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

results on universal particular,

properties

it follows that

(1.3) for the case that there

distance.

He uses A. Marden's

of Fuchsian groups

(1.3) holds

r E F,

if

F

[7].

In

is any subgroup of a

finitely generated group. We shall show that on

(1.3) is not true without

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

r.

THEOREM

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

I.

A12(r)

(1.5) and therefore

such that

~ A~(r)

that

(1.6)

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

(1.5)

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

from (1.4).

F

(1 ~ p

implies

Then

(1.6)

|

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

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

and

It is a pleasure to acknowledge

Marden and L.Greenberg

Iz I > i~

= 0

as

o(t).

for all functions

where

t ~ 09

and its Schwarzian

within an error of

ff v(z)g(z) d x d y

0 < t < 1,

lle(',t)ll~ = o(t)

f(t,z),

1

+ e(z,t),

t

In this case derivative

Suppose

g

holomorphic

in

U

for

U

which

Ilgll = f[

Ig(z)l d x d y

< ~.

It turns out [3,10] that

U

II~(',t)ll~ = o(t), same boundary

t § 0;

values

as

i.e., f,

the mapping

5.

1

of Extremal

Mappings

be the boundary

homeomorphism h

of

U

onto

to quasiconformal

smallest maximal

U.

W(Zl) ,

chordal

distance

f E Qh i}.

of

be extended Choose

that

h.J --~ (3 144 Z

§

0 .H 4~

O

4~ 9H

N

~

N

,-,

V

)4

9

N

N

H v

S

~

9,-I

m

~L

O C_)

.,

'-."

8

~ 0

N

0

--

~

..~

-,-I

I

||

~

v

N

vii 9H

N

~

,,., II

,-4

'~

hl

~

"0-

..

~H ii

N V

QI

~4

,-4

'

11

II

II

v

N

~

N

D

0 H

v 7

~

0

i!i

simple zeros.

The general result follows by a simple approximation

procedure. 2.

Trajectories

trajectories r

and orthogonal

Differentials

traOectories

in the Disk.

of holomorphie

play an important role in the interpretation

A trajectory of differential orthogonal i.e.

of Quadratic

r

is an arc in

displacement

on which

functions

and proof of

r

2 ~ 0

in the direction of the arc).

trajectory of

a trajectory of

U

The so-called

~

-~.

is an arc in Conceptually,

U

on which

the simplest

(dz

(M). is a

An r

2 ~ O;

situation

is that in which

r

= f$s

is both single-valued

de

and univalent.

In that ease,

trajectories

orthogonal

trajectories

are merely inverse images under

horizontal

and vertical

lines,

this situation continues point

z0

for which

three trajectories

r

= z

~(z O) ~ O.

for

(3 strip

domains )

of

In the general ease,

to hold locally in the neighborhood

meet at

trajectories

respectively.

9 -I

z0

If

r

of any

has a simple zero at

(under equal angles).

trajectories with

and

for a

2 simple

r

zeroes

(4 strip domains)

Zo,

112

trajectories for where right.

~I(U)

r

: r!

2

i8 region shown at

(5 strip domains).

The heavily drawn trajectories are the

pre-images of the horizontal dashed lines, The following decomposition

r

is possible [17]. Up to a n set of Lebes~ue 2-dimensional measure O, U : U Zk' where {Zk} k-i are disjoint simply connected "strip" domains. Each Zk is swept out by a family of trajectories single-valued

is horizontally

#k(Zk )

the intersection

convex,

many

Zk

U

r

Ck(Z)

Zk

= f/~Tdz.

there exists Each region

i.e. if a horizontal

r

is holomorphic

so that countably many,

can occur.

and in each

line intersects

consists of a single open interval.

it is merely assumed that {Izl ~ i},

of

schlicht branch of

@k(Zk )

of

of

under the mapping

for

{Izl < i},

(In [17] instead

instead of merely finitely

Actually,

in our use of the strip domains the

advantage of limiting ourselves

to finitely many is purely didactic.)

The above sketches

indicate

several possible configurations.

The

strip domains are bounded by the heavily drawn trajectories. 3. case

Proof of (M). S = {Izl = i}.

r = ~,2 ~(U)

We now proceed with the proof of

where

is schlicht and holomorphic

convex.

unnecessary.

for

Izl ~ 1

and

This occurs when there is merely a single

strip domain for the orthogonal the discussion,

for the

To clarify the ideas we will first assume that

#(z)

is vertically

(M)

trajectories

the concept of trajectory

of

r

For this part of

is quite trivial and

113

We introduce and

~

domain of

{I~I
i

and c o n d i t i o n

(z ( ~ )

(22),

w r i t t e n in n o n - n o r m a l i z e d form becomes

ff(fz) (23)

dx dy

sup ~ r holom, ins ffIr

That is,

= i. dy

as a result of our last t h e o r e m we know that (23) is a

n e c e s s a r y c o n d i t i o n on

~

for the affine m a p p i n g

FK(Z)

of

~

to

be extremal for its b o u n d a r y values. Let

fla

be the angular region

0 < arg z < m,

(0 < a < 2w).

124

As a n i l l u s t r a t i o n

of the application

we w i l l p r o v e that

(23) does not h o l d for

that the a f f i n e v a l u es. [16].)

s t r e t c h of

~

(This was o r i g i n a l l y Let

A : ffr n

E = {w dxdy

of the n e c e s s a r y ~ ,

and thus c o n c l u d e

is not e x t r e m a l

for its b o u n d a r y

p r o v e d by an e x t r e m a l

I 0 < Im w < ~}.

condition,

length method

Then f(w)

: ff e - 2 i V f ( w ) d u d v , E

= e2Wr

(w : u + iv).

and

B = fflr

dxdy

fflf(w)ldudv

=

=

E Since

B < ~,

is a n a l y t i c in

f_|

f~dvf 0

du

exists

lf(u§

du.

-~

for a.a.

v,

and since

E, co

f ( u + i v ) du =

c

=

const

c e -iS s i n e ,

B ~

for

a.a.

Hence, A = c~e

- 2 i v dv

=

Icl~.

Therefore,

[ABI
O

- HI9].

431) into (21) and

then (32)

426).

implies that

k*(t)

: o(t).

EXAMPLES

Refering to III.3 and IV.2,

that a__nnaffine stretch of a simply c o n n e c t e d r e g i o n for its b o u n d a r y values if and only i f

423) holds.

we can now state ~

is extremal

W h e t h e r or not

130

the affine

stretch

is extremal

is independent

of the dilatation

K

and of the direction of stretch. We illustrate

the condition by a brief new proof of the fact

[16] that an affine 7 is extremal Proof

!

stretch of = { z ]Re z >

0,

0 < Im z < i}

for its boundary values. Let

9

f (z) = -i- e-Z/n n n

,

z

E

X

One gets

I{? lim n§

: i.

fflfn(Z)Idxdy E,

2.

An Extremal

be a measurable For functions [Ig[[ =

ff

Problem for Functions Analytic complex-valued

g(z)

function of

holomorphic

Ig(z)I d x d y

< ~,

in

z,

{Iz] < i},

consider

in the Disk. ]z] < i,

Let

v(z)

0 < []v[[~ < ~.

with

the question of whether the

Izl R.

We write the real part of (3.13) as

N

(3.15)

If(z) - f(Zn)l

Clearly the limit as z = Zm,

min ~ F negative of that in (3.2).

logIR'

- ZmZnl

the variational Therefore

derivative

in (1.8). is simply the

the second inequality

in (1.8)

in an entirely analogous manner.

Remark.

Since the above proof is based on variational

the inequalities

from (3.13) and inequality

(1.8) are n e c e s s a r i l y

(8.14) that an extremal

in (1.8) is of class

zn

Similarly, is of class behaves

XmXn

This proves the first inequality

For the problem

points

- Z~nl + constant.

exists. We may therefore substitute n N and sum on m to obtain (since 7. x n = 0) n-i

1 IK-L = K ~K + L]m,n.l

r

in the extreme case.

ations,

= K\K + LI n~iXn l~

z § z

m u l t i p l y by Xm,

(3.16)

follows

Z~E - L~ N

n~lXn log Iz _ znll/K

where it behaves an extremal C~

like

C~

like

We also note

function for the first

except on

Izl = R

and at the

An(Z - Zn) IZ - Zn I(I/K)-I + B n.

function for the second inequality

except on

Izl = R

and at the points

in (1.8)

zn

where

A~(z - Zn) IZ - Zn IK-I + B'.n The extremal functions

defined by (3.13) and (3.14) provide interesting homeomorphisms

sharp.

consider-

of

C

onto

C.

examples of

q.c.

it

REFERENCES 1.

O. Lehto and K.I. Virtanen, Quasikonforme .Abbildungen, Verlag, Berlin-Heidelberg-New York, 1965.

2.

H. Renelt, Modifizierung und Erweiterung einer Schifferschen Variationsmethode f~r quasikonforme Abbildungen, Math. Nachr. 55 (1973), 3 5 3 - 3 7 9 .

3.

M. Schiffer,

4.

M. Schiffer

Springer

A variational method for univalent quasiconformal mappings, Duke Math. J. 33 (1966), 395-412. and G. Schober, An eztremal problem for the Fredholm eigenvalues, Arch. Rational Mech. Anal. 44 (1971), 83-92, and 46 (1972), 394.

QUASIREGULAR MAPPINGS URI SREBRO

i.

INTRODUCTION

i.i. in

Quasiconformal, R n,

n e 2,

quasiregular and quasimeromorphic mappings

seem to be reasonable generalizations of conformal,

analytic and meromorphic functions, of these mappings

respectively;

and the theory

is in many respects complementary to the geomet-

ric theory of functions in

r

Furthermore,

complex analysis are not applicable

since most methods of

in general for quasiconformal,

quasiregular and quasimeromorphic mappings

in

R n,

the proofS in

the theory of these mappings a~e usually more direct and mostly of a geometrical nature;

in many cases,

they give better insight

into various phenomena connected with these mappings as well as with analytic functions in In this paper,

r

which is partly expository,

I shall survey

some of the elementary properties of quasiregular mappings,

illus-

trate the use of the main methods in this theory by proving several distortion theorems for quasiregular mappings and conclude with the introduction of the concept of conformal measure with an application to a two constant theorem for quasiregular mappings.

Further

properties and applications of the conformal measure will appear in a forthcoming paper. For more information about quasiregular mappings the reader is refered to V~is~l~'s 1972 expository report raphy at the end of that report, this note. know, (9)

IV4],

to the bibliog-

and to the list of references of

Several open problems are listed in [V4].

three of them have been answered: by S. Rickman [Ri 2] and

(13)

(8)

by O. Martio [M2],

by T. Kuusalo

the 1973 Analysis Colloquium in Jyr~skyl~,

As far as I

(announced in

Finland).

149

n

1.2.

N o t a t i o n and terminology.

For

x { Rn

we w r i t e

x =

'

where

el,---,e n

is an o r t h o n o r m a l basis in

R n.

r > 0

we denote

Bn(a,r)

Bn(r)

B n = Bn(1),

sn-l(a,r)

~B n.

The closure

sets

A

Rn.

in

D

2.

the b o u n d a r y

~n = R n U {=}

is a d o m a i n in

QUASIREGULAR

sn-l(r) 8A

x.e. i

a E Rn

I

and

= Bn(0,r),

= ~Bn(r)

and

S n-I =

and the c o m p l e m e n t

CA

of

will always be t a k e n w i t h r e s p e c t to

f: D § R n

continuous and that

I x - a I < r},

= ~Bn(a,r),

A,

By w r i t i n g

that

= {x:

For

[ i=l

or

Rn

f: D + Rn

or in

Rn,

we shall always assume

respectively,

that

f

is

n a 2.

MAPPINGS

2.1. A mapping

f: D + R n

qr,

f(D)

if either

is said to be quasiregular,

is a point in

Rn

or else

f

abbreviated

has the fol-

lowing properties: (i) sets in

f

is open

Rn),

for every (it)

y

(i.e.,

discrete in

f(D))

f E ACL n

f

(i.e.,

maps open sets in f-l(y)

D

onto o p e n

is a d i s c r e t e set in

D

and s e n s e - p r e s e r v i n g ;

(i.e.,

f

is locally a b s o l u t e l y c o n t i n u o u s on

almost all line segments p a r a l l e l to the c o o r d i n a t e axes and its partial d e r i v a t i v e s belong to belongs to the Sobolev space (2.1.1)

(iii)

for some

K E [i,~).

Here

L~oc(D) ,

or in other words

1 Wn,loe) ,

If'(x)l n c KJ(x,f)

f' =

~

\

a.e.

2.2.

D

f,

If'(x)I

,j=l

denotes the s u p r e m u m norm of the linear o p e r a t o r J(x,f)

in

is the formal d e r i v a t i v e of

]

f

f'(x)

and

= det f'(x).

A mapping

bmeviated

qm,

f: D + Rn if either

is said to be q u a s i m e r o m o r p h i c , f(D)

is a point in

Rn

or else

ab(i) -

150

(iii) hold where

(ii) and

(iii) are checked at

~

and at

by means of a u x i l i a r y M~bius t r a n s f o r m a t i o n s w h i c h map

2.3. A m a p p i n g ated

2.4.

qc,

f: D + Rn

if

f

Conditions

is

is said to be q u a s i e o n f o r m a l ,

(i) - (iii) are not independent. see R e s h e t n y a k

is not a constant and satisfies

open,

~

into

R n.

abbrevi-

qm and injective.

(though not so easily, f

f-l(~)

One can show

[Re i] and [Re 2])

(ii) and (iii),

then

that if f

is

discrete and sense-preserving. Condition

(i) says that,

branched covering map.

locally,

Conditions

a bounded d i l a t a t i o n in

D.

f

is a s e n s e - p r e s e r v i n g

(ii) and (iii)

This is why

qm

say that

f

has

m a p p i n g s are some-

times called m a p p i n g s of bounded d i l a t a t i o n or of bounded distortion.

By (ii) the partial

exists

a.e.

in

D.

d e r i v a t i v e s of

Moreover,

is d i f f e r e n t i a b l e

0

onto a set of m e a s u r e zero and by Martio, J(x,f)

> 0

a.e.

is d i f f e r e n t i a b l e and where now every ball

f'(x) B c D

in

in

J(x,f)

mappings

by R e s h e t n y a h

f

[MRV i]

a.e.

qm

D,

D.

f: D § Rn

[Re i] and [Re 2],

and maps every set of m e a s u r e Riekman and V~is~l~

At points

> 0,

f(x+h)

x

in

D

where

f

= f(x) + f'(x)h + o(lh[),

is a n o n - s i n g u l a r linear o p e r a t o r w h i c h maps onto an e l l i p s o i d

E

and

If'(x) In/j(x,f)

is the ratio b e t w e e n the volume of the ball w h i c h c i r c u m s c r i b e s and the volume of 2.5.

Let

dilatation

E

E.

f: D + Rn

be a n o n - c o n s t a n t

K O = Ko(f,D) ,

the maximal d i l a t a t i o n

qm

mapping.

the inner d i l a t a t i o n

K = K(f,D)

K I = KI(f,D)

are defined by If'(x)l n

K0

:

ess sup x(D

KI

=

J(x,f) ess sup xED ~(f'(x)) n

J(x,f)

The outer and

151

K

=

max

(Ko,K I )

w her e

s

(x))

inf If'(x)hl. lhl : 1

K 0 ~ K n-i I ,

By linear a l g e b r a

analytic

and

conformal

functions

C

qr

qc

K = I.

self-evident

3.

are subclasses

These

subclasses

The g e o m e t r i c

by v i r t u e

K-qm,

of

qm,

K-qr

then

The classes of m e r o m o r p h i c ,

and

is

n = 2

to the case.

respectively.

f

if

according

in

we say that

and

If

n = 2

= K

n-i KI < _ K0

K 0 = K I.

pings,

K(f,D)

:

are o b t a i n e d

meaning

of the last r e m a r k

of

K0

or

and

K-qc

map

by letting

and

KI

are

in 2.4.

THE MULTIPLICITY FUNCTIONS, THE LOCAL TOPOLOGICAL INDEX AND THE BRANCH SET OF OPEN DISCRETE MAPPINGS

3.1.

Suppose

serving. c G

f(BD)

nent of

if

of a point

with

x ( G

r > 0

defines

~ c G,

f-l(Bn(f(x),r)).

Then

U(x,r,f)

is a normal

whenever

0 < r ~ r

o"

with

then

D

~ c G

domain

for

[MRV i] there neighborhood

denote exists

and

o

compo-

and

Rn

small-

[MRV i].

such n e i g h b o r h o o d s .

the r

Fur-

neighborhood

f

in

if

d o m a i n and

has a r b i t r a r i l y

way to c o n s t r u c t

f

is a normal

is a c o n n e c t e d

complement

U(x,r,f)

for

D § f(D).

is a normal

x E G

connected

let

D

sense-pre-

domain

is said to be a normal

is a standard x E G

D,

is a normal

with

and

a closed mapping

E v e r y point

and

discrete

is a d o m a i n and

D

= {x}.

there

A domain

D r G

if

neighborhoods

Moreover,

f

A domain

n f-l(f(x))

For

if

is open,

is said to be a normal

= Bf(D).

D' r f(G)

f-l(D')

= D'.

normal

D c G

if and only

thermore,

f(D)

f: G § R n

A domain

and

domain

that

> 0

CU(x,r,f)

x-component such that is c o n n e c t e d

of

152

For N(f,A)

A c G

and

y E Rn

= sup N(y,f,A)

local topological i(x,f).

Since

over all

inde~ of

f

let

f

is open,

N(y,f,A) y E Rn

= card f-l(y)

and

at a point

N(f)

x 6 G

n A,

: N(f,G).

The

is denoted by

discrete and sense-preserving,

i(x,f)

may be defined by (3.1.1) where

i(x,f) U

:

is any normal n e i g h b o r h o o d

N(f,U) of

the same for all normal neighborhoods [MS I].

With this d e f i n i t i o n

is open,

of

f,

In fact,

x.

N(f,U)

For more details

one can show [MS l] that if

discrete and sense-preserving

domain for

x.

and

D c G

is see

f: G +

Rn

is a normal

then

(3.1.2)

i(x,f)

=

N(f,D)

0

the

family

and

{tel:

4.8.

Now

condenser is a g a i n A

of p a t h s

let in

mapping.

Then

in

f: D + R n D

(meaning

a condenser,

Modulus

only

which

1 ~ t ~ ~}

is a n o r m a l

4.7.

depends

domain

and

on

n.

join

the

R n,

be a

In fact, line

[V2].

qm

mapping

A c D);

we

say

E = (A,C)

for

f.

capacity

inequalities.

is the m o d u l u s

segments

see

the

an

and

then

{tel:

a

= (f(A),f(C))

is a n o r m a l

Let

-i s t 5 0}

E = (A,C)

f(E)

of

condenser

f: G + R n

be a

if

qm

155

(4.7.1)

M(fF)

for all path families

~ KI(f)M(F)

F

in

G;

heme

fF = {foy : y E F}.

Further-

more cap f(E) ~ Ki(f) N(f~A)n-i M(f,C) n

(4.7.2)

for all condensers (4.7.3)

E : (A,C)

G

and

cap E ~ Ko(f) N(f,A) cap f(E)

for all normal condensers

E : (A,C)

Of these inequalitites, (4.7.2)

in

cap E

to Martio

[MI] and

(4.7.1) (4.7.3)

in

G.

is due to Poleckii

to Martio,

[P]~

Rickman and V~is~l~

[MRV i].

5.

5.1.

DISTORTION THEOREMS FOR

In these sections,

capacity

inequalities

tortion theorems are contained 5.2. N.

For

and

for quasiregular

Let

r ( (0,i) m(r)

MAPPINGS

we shall illustrate

(4.7.2)

in [M].

THEOREM.

qr

=

(4.7.3)

IV3].

f: B n ~ R n

be a

qr

denote

inf

If(x) - f(0) I

IxJ =r M(r)

:

sup

If(x) - f(0) I.

Ixl =r The~ there is an

r

o

> 0

such that

(5.2.1)

Alra ~ m(r)

~ A2r6

(5.2.2)

A3r~ ~ M(r)

~ A4r8

in proving

mappings.

See also

the use of the two several dis-

Some of these results

mapping with

i(0,f)

:

156

i/n-i

for all Ai,

r ((0,to)

~

and

Proof. that

8

= U(0,f,r),

neighborhood

constants

that

the

f(0)

and

which depend on

= 0.

0-component

with connected

f.

The

complement

Choose

of

R > 0

f-iBn(r),

whenever

such

is a n o r m a l

r ( (0,R]

(See

Let

n ~ = inf {IxI:

Fix

{ N ]i/n-i , 8 = k~ii ]

~ : (KoN)

are the best possible.

We m a y a s s u m e

U(r)

8.1).

where

are positive

i = 1,--.,4,

exponents

,

r ( (0,r o)

is a n o r m a l (3.i.i),

x ( ~U(R)}

and w r i t e

condenser

N(f,U(R))

in

and

r I = sup {IxI:

M = M(r) Bn

with

= M(f,U(m))

and f(E)

= N,

x E ~U(R)}.

m = m(r).

E = (U(R),U(m))

= (Bn(R),Bn(m)).

hence

from

(4.7.2),

KI KI = cap f(E) ~ -~- cap E ~ -N-- "

~n-i

By (4.5.2)

and 4.8 f o l l o w s mn- i n-i

This yields

the right

For the right er

inequality

E = (U(N),U(m)).

and 4.3,

inequality

Then

in

(4.7.3)

"

(5.2.1).

(5.2.2), f(E)

n-I

consider

the normal condens-

= (Bn(N),Bn(m)),

(4.5.3)

and

0 < on ~

cap E ~ N K 0 cap f(E)

N(f,U(M))

= N

give = NK 0

mn-1

(og

n-l"

consequently

(5.2.3)

where and

M ~ Cnm ,

cn N.

is a p o s i t i v e The r i g h t

and the right

f(E)

inequality

inequality

Next consider

constant

of

and

(5.2.2)

follows

o n l y on

n,

now f r o m

K0

(5.2.3)

(5.2.1).

the n o r m a l

= (Bn(R),Bn(M))

of

which depends

condenser

(4.7.3),

E = (U(R),U(M)).

(4.5.2)

and 4.3 y i e l d

Then

!57

~n-i

< cad E < NI 0

g = h 9 fN

where

and

h: R n § R n

is the r a d i a l stretching

h

is a

branch

qc set

direction, and

Ixl a-I

oNlxl n(~ and

mapping Bg,

g

and

Ig'(x)l

e = (N KO )I/n-I holds

Bg

BfN.

=

= q.

n-2

major

= olxl O-I

on the left

in

h(x) x

Consequently,

and

above

= Ixl~

in the radial direction J(x,g)

Ko(g) = m(r)

(5.2.2).

and

off the

Thus

M(r)

(5.2.2)

described

to the r a d i a l

directions.

(5.2.1)

and

o > N

olxl O-I

On the o t h e r hand

BfN =

(5.2.2).

by

At e a c h p o i n t

normal

To com-

set

Choose

defined

stretching

in a d i r e c t i o n

in all o t h e r

and

is the w i n d i n g m a p p i n g

has the m a j o r

Nlxl O-I

and

so e q u a l i t y

fN

p m 0

= N.

in (5.2.1)

be an integer.

let

sin %,x3,

I,N,I,--.,I. Thus ( N_N_~I/n-I and ~ : ~KI] =

in (5.2.1)

To s h o w th a t the left h a n d i n e q u a l i t i e s are best possible,

r

i(0,f)

(5.2.2)

the w i n d i n g

off the b r a n c h

Hence

and

.-., Xn),

has the m a j o r s t r e t c h i n g s

~(f~(x))

so e q u a l i t y

(p cos

p sin N%,x3,

= m(r)

(5.2.1)

N > O

sends e a c h p o i n t

M(r)

fN

of

for each i n t e g e r

n o t e that at each p o i n t

J ( x , f N) = N and

(5.2.3).

inequalities

(p cos Nr

Here

{x: x I = x 2 = 0},

1

which

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

=

_ on-iN = rO

and

This c o m p l e -

tes the proof.

5.3.

Next consider

[MS1]

that

n e x t two

f(x) + ~

sections

lim f(x) as

qr as

mappings x § =

f: R n § R n. if and o n l y

we s t u d y the r e l a t i o n

n + ~,

the d e g r e e

N(f)

It is shown in

if

between of

f

N(f)

< ~.

In the

the e x i s t e n c e

of

and the g r o w t h of

f

158

near

~.

For related results

m(r)

For

{lfCx)l:

inf

=

see [V3,w

r > 0,

let

= r}

Ixl=r M(r)

5.4.

THEOREM.

:

Let

sup { I f ( x ) l : Ixl:r f: R n § R n

Ix I : r } .

be a qr mapping with

N(f)

= N < ~.

Then

(5.4.1)

Alr8 ~ m(r) ~ A2ra

(5.4.2)

A3r8 ~ M(r)

for all sufficiently = (N/KI)I/n-I on

large value8 of

and

Ai,

r,

where

i : 1,''-,4,

~ = (KoN)

are constants

i/n-i

which depend

f. The ezponents Proof.

~

and

8

The fact that

qm extension, Rn

~ A4ra

N < ~

denoted again

is compact,

are best possible.

f,

f: ~ n § R n

implies to

Rn

[MS1] with

that f-l(~)

is a closed mapping;

f

has a

= {~}.

hence,

Since

by

(3.1.2) i(~,f)

Let with

iS(x)l

g = Solos -I _

1

=

/. xEf-l(~) where

for all

i(x,f)

= N.

S:R n § Rn

is a M~bius t r a n s f o r m a t i o n

x E R n \ {0}.

Then

g

is a

qm

map-

Ix; ping with the dilatations i(0,g)

= N.

sup {Ig(x)l: I/M'(I/r).

Letting Ixl = r}, Thus,

of

m'(r)

f,

N(g)

= inf {Ig(x)l:

we see that

applying

= N(f),

M(r)

g(0)

Ixl = r} = i/m'(i/r)

(5.2.1) and (5.2.2) to

g

= 0

and

and

M'(r)

and

m(r)

= =

we obtain

(5.4.1) and (5.4.2). In order to see that

(5.4.1) and

(5.4.2)

are sharp,

one can

159

take the m a p p i n g s (5.2.2)

fN

and

g

w h i c h give e q u a l i t y in (5.2.1) and

and form the m a p p i n g s

s-lofNoS

and

s-logoS

w h i c h will

give e q u a l i t y in (5.4.1) and (5.~.2).

5.5.

COROLLARY.

Let

(5.5.1)

or

f: R n § R n

be a qr m a p p i n g . If(x) l

log

lim sup x§

log

If

-

Ix I

if lim inf log ]f(x)l _ 0 x§ log Ix I

(5.5.2)

then

f

has no

Proof. [MS I]);

l i m i t at

If

f

and Thus

~.

has a limit at

~,

then

N(f)

-- N < ~

(see

(5.4.1) would imply that 1

lim sup log

If(x) I / log

and (5.4.2) that lim inf log contradicting

6.

If(x)l / log

Ixl >_ 8 = (KoN)I/n-I > - 0,

(5.5.1) and (5.5.2).

THE CONFORMAL MEASURE

6.1.

A,

Ix] _ e = kKi/

Let

D c Rn

A c BD,

be a domain.

w i t h respect to

The c o n f o r m a l measure

D

at the point

x ( D

@(x,A,D)

will be

defined bY

r

where

E c ~

A = 0

we set ~(x,A,D)

~(x,A,D) § 0 ~(x~A,D)

: inf M(P(E,A;D)) E is a eontinu~n w i t h r

and

E n ~D = 0.

If

= 0.

is n o n - n e g a t i v e , as

x ( E

d i a m A § 0.

and for Also,

D # Rn

and fixed

x ( D

it is easy to see that

is a c o n f o r m a l invariant for every

n ~ 2

and

monotone

of

160

w i t h respect to ~(x,.,D)

A

for fixed

x

and

D.

However,

is not a d d i t i v e as a set f u n c t i o n on

a m e a s u r e on

8D

in the c o n v e n t i o n a l

BD

in general and thus is not

sense.

In certain proofs the conformal m e a s u r e can r e p l a c e the harmonic m e a s u r e w i t h the clear a d v a n t a g e that,

unlike h a r m o n i c measure,

c o n f o r m a l m e a s u r e is a c o n f o r m a l invariant in all d i m e n s i o n s

~ 2.

We shall i l l u s t r a t e the use of c o n f o r m a l m e a s u r e and the m o d u l u s inequality pings in r e m in

(4.7.1) in proving a two constant t h e o r e m for R n,

C

n ~ 2.

qr

map-

Recall that the c l a s s i c a l two c o n s t a n t theo-

is proved by the use of h a r m o n i c measure,

see [N,III2.1].

We shall need the following n o t a t i o n for our t w o - c o n s t a n t theorem.

6,2.

For

0 < m < r < 1

let

D = {x E Rn: m < lxl < i}. show that

~(r)

~(r)

1

~

= M(r(E,Sn-I,D))

and t h e r e f o r e

on

THEOREM.

subset

~-l(t)

extension

x ~ A,

Let

3D,

of

D

to

D U A

~(r)

E = {tel: m ~ t ~ r}.

It

is strictly increasing from

is strictly increasing

R n,

in

a nonconstant and

for all

from

m

0

to

A

qr

a non-empty mapping

with

proper a contin-

m ~ (0,1). x E D

If(x) l ~ m

and

for all

then

If~x~l ~

~-l(Ki(f)r

is d e f i n e d

Proof. x ~ D

where

be a domain

f: D § R n

If(x) I < 1

If

where

where

(0,|

6.3.

uous

: ~(rel,Sn-l,D),

By s y m m e t r i z a t i o n it is not hard to

is also not hard to show that to

: ~(r,m)

and

r a continuum

in S.1 and

\ A,D))

~

in 8.2.

By L i o u v i l l e ' s t h e o r e m for E > 0. < ~, E

We may assume that

qr

mappings,

if(x)J

> m

D ~ R n.

and that

since o t h e r w i s e there is n o t h i n g to prove. in

5

w h i c h meets

~

and

A

and

Let

such that

Choose

161

M(F(E,~DkA,D)

< r

+ e.

Denote

E' = f ( E ) \ B n ( m )

and let

r' = r ( E ' , ~ f ( D ) \ B n ( m ) , f ( D ) \ B n ( m ) ) . For each

y':

[a,h) + f ( D ) ~ B n ( m )

of

F',

with

y'(a)

E E'

and

lim y'(t) ~ ~ f ( D ) ~ B N ( m ) , choose a point z in E 0 f-l(y'(a)) t§ and let y: [a,c) § D be a maximal lift of y' from the initial point

y(a)

= z.

discrete mappings maximal and

[Rill.

Since the lift

f(A) c Bn(m),

and consequently subpath of

Such a lift exists by Rickman's

7'

it follows

~ ~ F(E,~D\A;D).

f(D) c B n,

7(t) + ~ D \ A

Note that

s M(fF) ~ KI(f)M(F) < KI(f)[r

F

that

was assumed to be

toy

In any case by virtue of 4.3 and

M(F')

where

y

is the family of lifts it follows,

Lemma for open

as

t § c;

may be a proper

(4.7.2)

~ KI(f)M(F(E,SD\A;D) + c],

y

of

by 4.3 and 6.2,

7'

for

7' E F'.

Since

that

~(If(x)l,m) ~ M ( F ( E ' , ~ B n , B n ~ B n ( m ) )

~ M(F').

Hence

~(If(x)l,m) and the result

~ KI(f)[r

+ e],

follows by 6.2 and letting

e + 0.

ACKNOWLEDGEMENTS

I with to thank the Mathematics Department at the University of Maryland for its hospitality during the special year in complex analysis.

REFERENCES

[G]

F.W. gehring,

[H]

J. Hesse, A p-extremal length and p-capacity equality, (to appear).

[M1]

0. Martio~ A capacity inequality for quasiregular mappings, Ann. Acad. Sci. Fenn. A.I. 474 (1970), 1-18.

[M2]

0. Martio~

[MR]

O. Martio and S. Rickman~

Extremal length definition8 for the conformal capacity of rings in space, Mich. Math. J. 9 (1962), 137-150.

On k-periodic mappings~ (to appear).

Measure properties of the branch set and its image of quasiregular mapping6, Ann. Acad. Sci. Fenn. A.I.

541 (1973),

1-16.

S. Rickman and J. V~is~l~, Definitions for quasiregular mappings~ Ibid. 448 (1969)~ 1-40.

[MRVI] O. Martio,

S. Rickman and J. V ~ i s ~ l ~ Topological and metric properties of quasiregular mappingsj Ibid. 488 (1971),

[MRV2] O. Martio~ 1-31. /MS1]

O. Martio and U. Srebro, Periodic quasimeromorphic mappings, J. d'Analyse Math. (to appear).

[MS2]

O. Martio and U. Srebro, Automorphic quasimeromorphic mappingsj Acta Math. (to appear).

[N]

R. Nevanlinna,

[P]

E.A. Poleckii,

Analy%ic

Funciions,

Springer Verlag,

1970.

The modulus method for non-homeomorphic quasiconformal mappingsj Mat. Sb. 83 (1970), 261-272 (in Russian).

[Rel]

J.G. Reshetnyak~ Space mappings with bounded distortion, Sibirsk. Mat. Z. 8 (1967), 629-658 (in Russian).

[Re2]

J.G. Reshetnyak,

On the condition of the boundedness of index for mappings with bounded distortion, Ibid. 9 (1968)~

368-374

(in Russian).

[Ril]

S. Rickman~ Path lifting for discrete open mappings, (to appear).

[Ri2]

S. Rickman,

[Sa]

J. Sarvas, Multiplicity and local index of quasiregular mappings (to appear).

[St]

U. Srebro, Conformal capacity and quasiregular mappings, Ann. Acad. Sci. Fenn. A.I. 529 ( 1 9 7 3 ) ~ 1-13.

(to appear).

163

[VI]

J, V~is~l~, Discrete open mappings on manifolds~ Ibid., A.I. 392 (1988), i-i0.

[V2]

J. V~is~l~, Lectures on n-dimensional quasiconformal mappings, Lecture notes in Math. 229 Springer Verlag, 1971.

[V3]

J. V~is~l~, Modulus and ~apacity inequalities for quasiregular mappings, Ann. Aead. Sei. Fenn. A.I. 509 (1972), 1-14.

[V4]

J. V~is~l~, XVI Seand. Cong. of Math.

(1972).

TECHNION HAIFA, ISRAEL

STARLIKE FUNCTIONS AS LIMITS OF POLYNOMIALS T,J,

SUFFRIDGE*

INTRODUCTION, This paper is a study of functions w h i c h are starlike of order (functions

satisfy the c o n d i t i o n of functions

a n a l y t i c in the unit disk w h i c h

f(z) = z + a2 z2 + .-. Re[zf'(z)/f(z)]

starlike of order

~

> e),

where

e ~ 1.

The class

is first c h a r a c t e r i z e d as the class

of limit functions of sequences of p o l y n o m i a l s having a simple r e s t r i c t i o n on the location of their zeros. from a study of these polynomials. te proof of the result of Brickman, functioDs

starlike of order

~

Our results then f o l l o w

These techniques yield an alternaet.al,

that the class of

lies in the convex hull of the

c o l l e c t i o n of r o t a t i o n s of the f u n c t i o n

z/(1 - z) 2(l-e)

Additional

information is o b t a i n e d c o n c e r n i n g the p r o b a b i l i t y m e a s u r e s [0,2~] ~.

for which

For each

IO~ z/(l - zeit) 2(I-~) d~(t)

e ~ i,

the p a r t i c u l a r case

~ = 0,

on

is starlike of order

a c o n v o l u t i o n - t y p e t h e o r e m is obtained.

Schoenberg conjecture Small).

U

For

this yields a proof of the Pdlya-

(recently proved by R u s c h e w e y h and Sheil-

Further results are obtained,

some of w h i c h bear on the

geometric effects of c o n v o l v i n g certain convex functions.

i.

A C H A R A C T E R I Z A T I O N OF FUNCTIONS STARLIKE OF ORDER C o n s i d e r the class

P

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

P(z)

~, n ic~ = H (l+ze J), j=l

where

(i)

*

2~/(n + 2) ~ ej+l - ~j'

i ~ j ~ n,

~n+l : el + 2~.

This work was supported in part by the National Science Foundation under grant number

GP-39053.

165

For such a polynomial

P,

zP'(z)/P(z)

n " [ zeleJ/(l j=l

=

Using the m a p p i n g properties

of

i@j + ze

). we see that for fixed

w/(l + w),

r < 1 min[

min Re[zP'(z)/P(z)]

]

n

is attained when equality holds m i n i m u m occurs for Let

(i.e.,

Then for

P E Pn'

p(zP(z)) ~ p(zQn(Z)). subsets of

Thus we conclude

u n i f o r m l y on compact

Thus,

the

= (i + z n+2) / (i - 2z c o s w ( n + 2) + z2).

P(f) = sup { r : Re [zf'(z)/f(z)]

uniformly on compact n § ~.

1 ~ j ~ n-l.

denote the radius of stamlikeness of functions

p(f)

Izl < 1

Qn(Z)

in (i) for

Izl < i,

that if

subsets of

Similarly, define Pn(8) n iej P(z) = ~ (1 + ze ) where

But so

> 0,

f

analytic in

Izl < I}).

lim ZQn(Z) = z/(l - z)2 n§ p(zQn(Z)) § 1 as

P

E P and zP (z) § f(z) nk nk nk Izl < 1 then f is starlike. to be the class of polynomials

j=l (2)

2e ~ aj+ 1 - mj,

Here we require

1 ~ j s n,

0 ~ e ~ w/n. min P~P (8)

Again

en+l = ~i + 2~.

it is clear that for

[ min Re[zP'(z)/P(z)] Izl-r

r < i,

]

n

is attained when equality m i n i m u m oecurs for

1

in (2) for 1 ~ j ~ n-l.

nH (i + ze i(2j-n-l)8) j=l n Qn(Z;8) = [ C(n,k,8) z k k=0

Qn(z;e)

prove by induction that

C(n,k,e)

holds

It is easy to

=

if

where

k = 0,n

= k sin(n - 4 + i)8 sin ~

j=1

We wish to show that for

1 < k < n-i

e = ~/(n+2-2~),

Thus the

166

(3)

lim n+~

ZQn(z;e)

For this value of

8,

n § ~.

while

we have

Further~

0 ~ C(n~k~8)

~ 1

8.

we see as before

that

for

Let

uniformly

~

such that starlike

be fixed.

for Sk

g

has degree

s

Define

g

as

> ~

for the

Izl < r}~

n + ~.

: z + a2z

The

2

+ "-"

is

> ~

be the

nk-i

: z + a2z2

+ "'"

and

= f(rs163

partial

ZSk(Z)

ZPnk

is starlike

be an increasing Then

and it is sufficient

nkth

and

Izl < i.

f(z)

g(z)

nk § ~

is the limit of polynomials

nk > k

g

is

to show that for of the required

sum of the power

is starlike

sequence

of order

series e.

Then

so that

< Re[z(ZSk(Z))'/ZSk(Z))]

=

1 + Re[zS~(z)/Sk(Z)]

which can be made arbitrarily

small

Izjl ~ 1

{zj}

for some

follows

been proved.

such that

0 < rs < i,

e + ~

ZSk(Z)

where

Assume

{rs

lim rs = i.

Let

f(z)

e =< 89

if and only if there exists a sequence

and let

arbitrar~ but fixed form.

Then

on compact subsets of

of order

§ 1

theorem has therefore

Proof of "only if". of order

(3)

p (f) ={r:Re[zf'(z)/f(z)]

Pnk E P nk(~/(nk + 2-2a)),

{Pnk}k~ I, f(z)

~

for

Hence

0~(ZQn(Z ; z/(n+2-2e))

~ ~ 1

starlike of order

89 ~ e ~ i.

Letting

"if"part of the following i.

F(2+k-2~) P(2-2~)F(k+l)

0 ~ C(n~k~8) ~

given choice of

THEOREM

-- ~ r(2-2~)F(k+l)F(2+k-2~)zk+l k=0

s i n ( n - j + i)8 : sin(n+2-2a-(l+j-2~))@ F(2+k-2~) = sin(l+j-2e)@ so that C(n,k,@) § F(2-2e)F(k+l)

= sin(~-(l+j-2~)@) as

: z/(l - z)2(l-~)

j.

Here

(near

=

-~)

~ - I -z/z. l 1 + Re[j_~l ] in

are the zeros of

Izl < 1 S k.

if

Thus we

conclude Izjl > i, 1 ~ j ~ nk-1. Now set ZPk(Z) : ZSk(Z) + 2n k z Sk(I/z). Then Pk(Z) has degree 2nk-l, and we wish to show ZPk(Z)

+ g

and

Pk E P2nk_l(~/(2nk+l-2a)).

Since

167

iznk r llon zJ l Sk(Z)

we conclude

Iz

Izl ~ i,

uniformly bounded.

z = e ir =

uniformly there so

Since {S k}

g(z) is

Izl < i.

Pk(Z) = 0

e -i(nk+ 89 ~ [eir

-i(nk- 89162

)

Pk(ei$ ) = 0

That is,

zS k + g

Izl ~ i.

ZPk(Z) = lim ZSk(Z) = g(z) k§

on compact subsets of

= e

when z = 0

It follows that

lim k~

0

0

2n k_ nk+l Sk(i/z) I ~ Izl ISk(Z)l,

is analytic on

For

=

+ e

l(n k

if and only if (ei~) + e i(2nk~) Sk(ei~)]

8 9

= 2 Re[e

if and only if

-i(nk- 89162 Sk(ei$)].

e -i(nk- 89162

)

is pure

imaginary. Since _i(nk_89 ~ d (arg[e d-~ and

]zj] > i,

(4)

Sk(eir

= -(nk- 89 + Re

nk-i _eir [ j=l l-eir

we conclude

-n k + ~ - 8 9~

(arg[e _i(nk_ 89162

]) < -nk/2,

where the left inequality comes from the fact that of order

~

equation.

elude that

Pk

Pk

Izl = i).

p(eir

zS k

is starlike

and the right inequality comes from the preceding -i(nk- 89 Since the argument of e Sk(e i~) is decreasing

and the total decrease in

lie on

,

has

~2 > 91'

2nk-i If

[0~2~]

is

zeros on 92

and

2~(nk-89 = (2nk-l)~ Izl = 1

~i

we con-

(i.e. all zeros of

are consecutive zeros of

then from the left inequality in (4) we have

168

(~2 - ~ l ) ( n k completes

that

is t r i v i a l function

2.

the above

for

~2

proof

a

= i, for

starlike

of o r d e r

= {i

+ yz n

[ii]

from

Pn(8)

a zn n

such

the

: IYI

This

2n k + 1 - 2c~

applies

in this 1

even

case

while

for

f(z)

~/(n

~
-z.

i n e q u a l i t y t h e r e f o r e also holds on the circle of radius i,

and hence

P

is c l o s e - t o - c o n v e x .

the zeros are separated by exactly

2~/(n+l),

family of c l o s e - t o - c o n v e x functions

implies that

convex.

(z = e i8)

if y < 2~/(n+l).

varies f r o m

s u f f i c i e n t l y near

(in

It was shown in [i0] that for u n i v a l e n t

p o l y n o m i a l s of the given form,

,

so the univalence

Izl = I.

Izl = i,

n-i

2w/(n+l), This r,

r

If some of

c o m p a c t n e s s of the P

is c l o s e - t o -

170

T H E O R E M 3.

class

K

Let

g(z)

: z + A2 z2 + .--,

Then

Izl < i.

g

is in the

of functions which map the unit disk conformally onto

convem domains if and only if to-convex polynomials

g

is the limit of a sequence of close-

Inl

P(z) = z + a2z2 + ... + ~1 q zn,

: l,

having strictly increasing degrees. Proof.

It is w e l l - k n o w n that

only if

zg'(z)

only if

g'

is starlike

g

is in the class

(of order

0).

{Pnk }

is the limit of some sequence

Pnk_l(w/(nk+l)), if and o n l y if

nk § ~ P

as

k § ~.

Hence

By Theorem2,

is c l o s e - t o - c o n v e x .

K

if and

g E K where

if and P' E nk

P'nk E % _ l ( W / ~ + l ) ) ,

Hence the t h e o r e m follows.

nk

3. C O E F F I C I E N T REGIONS FOR THE CLASS If

P(z) = i + "'. + a n Z n E Pn(e),

since all zeros of P(zei8))

P

lie on

[zl = i.

a

n

= I.

then

We may r o t a t e

P

It is clear that

so that the c o e f f i c i e n t s satisfy the c o e f f i c i e n t s = C(n,k)

1 ~ k ~ n-i

and

C(n,k,8)

an_ k = a k.

(i.e. form

are p a r t i c u l a r l y

C(n,k,n/(n+l))

and

Pn(0)

9 -. + z n

w i t h all zeros on

~ i.

n = 3:

Let

C(n,k,n/n)

For example, = 0,

Pn(w/n)

= 1 + a2z 2+

Iz[ = I.

P (e), n

1 + az + z 2 E P 2 ( e ) ,

-2 cos e ~ a ~ 2 cos 8.

{Q2(z;8),Q2(-z;8)}

simple.

is the c o l l e c t i o n of all p o l y n o m i a l s

detail the c o e f f i c i e n t regions for We have

P(z) = znp(I/~)

Thus we have

In order to m o t i v a t e the next section,

n = 2:

P (e) n

For c e r t a i n values of

(the usual combinatorial),

{i + z n}

if

[anl = i

We m a k e this a d d i t i o n a l r e s t r i c t i o n on

in the r e m a i n d e r of this paper.

C(n,k,0)

0 ~ e ~ n/n,

w i t h o u t c h a n g i n g the s e p a r a t i o n of the zeros and hence we

may assume

8,

Pn(e),

Hence

(recall that

P E P3(8),

we now d e s c r i b e in some n = 2,3~4.

0 ~ 8 ~ g/2,

P2(8)

if

and only

is the c o n v e x hull of n Qn(Z~8) = [ C(n~k,e)zk). k=0

0 ~ 8 < n/3.

Then

w e have

171

P(z) = i + az + az 2 + z 3,

P(z)

Thus

A : i,

values for

= i + ~

e 2i~/3,

Q3(ze-2i~/3;8)

which we prefer to write

Az + ~

e -2in/3

respectively.

A

A z2 + z 3

yields

P(z)

of this polynomial

by

the closed triangle 8

8 § ~/3.

P3(9)

for the region of

8 ~ a ~ (2~/3)-8

-2~/3.

with vertices

increases with Thus

and

I,

The region e 2i~/3

,

e

and rotations

is contained

-2i~/3

and fills out the entire triangular is contained

Q3(zei2~/3;e),

the polynomial

+ ze-2ia),

2~/3

: Q3(z;%),

The b o u n d a r y curves

are found by considering

(i + 2ze ia cos e + z2e2ia)(l

in the form

in

it

;

region as

in the convex hull of

{Q3(z;e),Q3(zei2~/3;e),Q3(ze-i2~/3;e)}.

n : 4:

0 ~ 8 ~ ~/4,

B

is real.

The coefficient

contained

in the tetrahedron

(-i,0,i)

and

8

increases

(0,-i,-I) to

7/4.

special case of Theorem

4.

FURTHs Since

so that

has the form

sin 40 sin 48 sin 38 Bz 2 + sin 4e ~ z 3 4 = i + ~ Az + sin 8 sin 28 ~ + z ,

P(z)

where

P 6 P4(8),

PROPs P(z)

P(e ir

region

having vertices

(ReA, I m A , B ) (i,0,i),

is (0,i,-I),

and increases

to fill the whole region as

These assertions

will be verified below as a

6.

OF THs CLASS = zn P(i/z),

= einr162

pn(e),

we conclude where

R

e-in~/2p(eir

is real.

Consider

is real the

expression

(5)

P1 (z'~)

=

p(zeir _ p(ze-ir z(2i sin n~)

from which we see that

-

n [ C(n,k,e) s~--~---~zk-i k:l sln n~ ~ '

0 < ~ < e, --

172

Pl(Z,@)

:

n 7 C(n-l,k-l,8)Akzk-i k= 1

We have the following theorem. n [ C(n,k,e)Akzk where k=0 Ak : An-k (0 < k < n), and A 0 : i. Then P n k-i all zeros of Pl(Z,e) : [ C(n-l,k-l,8)AkZ k=l THEOREM 4.

P(z)

Let

Proof.

Assume

=

P E P (8). n

all zeros of

Pl(Z,~)

on

it follows

Izl = i,

pl(z,0 ) = in P'(z) 8 = 0.

For

z = e ir

IzI < i.

from Lucas'

if and only if

IzI < i.

lie in

Izl =< 1

0 < ~ < 8 < _w ~ = n

Since all zeros of

P

lie

theorem that all zeros of with equality possible only when

0 < ~ < 8 < w/n,

0 = P(ze i~) - P(ze -i~)

E Pn(e)

We will show that for

lie in

lie in

0 < e < w/n,

Pl(Z,@)

= einr162

= 0

if and only if

- e-in~/2R(r

But this is true if and only if (cos n~/2)[R(r which

implies

only for

R(r

~ ~ e

P~z,~)

- R(r : R(r

since

P ~

= 0.

Now assume Izl > i.

~,

It follows that for

Izl = I.

Since the zeros of

We wish to show that

We a c c o m p l i s h

vary the coefficients

of

of zeros of

Pl(Z,8)

to obtain

IzI > i

= A 1 + ... + z n-l,

Izl > i.

Recall that

if

has two zeros on

P(z)

exactly

28.

P

without

this implies

Pl(Z,8)

Izl = 1

Pl(Z,e)

Pl(Z,e)

in

changing

IAII > i.

Pl(Z,e)

has a zero on

Since the coefficients

the number of zeros of

Pl(Z,~)

has a

this by showing that we may

continuously

Pl(Z,8)

0 ~ ~ < e.
1

vary continuously, increases

if

there must at

173

some point

be a zero on

CASE i.

Assume

also has the factor P(z)

Izl

P

= i.

has a factor

(i + ! eiez) P

= (i + BlZ + ...

(I + pe!az),

since

P(z)

+ e-2iezn-2)(l

p > I.

= z n P(I/~)

+ (p+~)eiaz

Then

P

so

+ z2e 2ia)

a [~d sin ne

which

= IB1 + (p + ~) ei~l, P

can be made a r b i t r a r i l y

tending in

IAll

to

0

Izl > i,

large

does not a f f e c t the d e s i r e d

by letting

the n u m b e r

result

p + O.

of zeros

follows

for this

of

Since lying

Pl(Z,e)

case.

CASE

2. A s s u m e all zeros of P lie on Iz] = i, P(z) = iaj (I + ze ), a I < a 2 < ... < a < al+2~ : an+l, w h e r e for some j=l = = = n 9 < 28. A s s u m e the n o t a t i o n is c h o s e n so that j' aj+ 1 - a 3 n

an+ 1 - a n > 2e.

Choose

al~

a]2. - ~31. ~ 28,

a.3s

- a.3Z_ I < 2B,

aZ 0

by to

~s

either

~J2_l ...

" 'eJ2 ,''',a.]k a31

- ajl < 28,

, e.3k = ~n+l .

..-,~js For

~ 28~

and let

t

replace vary from

menormalize

that

'AI'

= slnSine9i S JIn e~

> sin Sinns8

I ~ ei(2j-n-l)el

j =l and the proof

P~(8) = {Pl(Z,e) n = {Q(z) = k=l[ Ak zk-I

all zeros

of

= i,

j =i

is complete.

We define P'(w/n)

- ~js

a.31 =

by r o t a t i n g 9 to o b t a i n A n = i. W h e n n P(z) now has the form ~ (i + ze 18j) where 8j+ 1 - 8j is j=l 0 or 28 and for at least one Jr 8j+ 1 - 8j = 0. Hence it

is clear

Finally~

SO that

jp =< s < Jp+l ~

= as + t(~ 1 + 2(p - 1)8 - es

i.

t = i,

successively

Q

lie in

: P

~ Pn(e)}

: An = i,

Izl ~ I}.

for

0 ~ e < W/n

An_ k = Ak,

Then for

and

1 =< k =< n-l,

0 ~ e ~ w/n

we have

and

174

the relation n

n

Pn(e) = { [ C(n,k,e)Akzk k=0

: k= I

C(n-l,k-l,e)Ak zk-I E P'(8), n

To simplify the notation,

or

{Ak}~= 0 ( P n ( e )

6 P'(8), n

if

n we will say {A k}k:0 ( P~(e) n A 0 = 1 and [ C(n-l,k-l,8)Akzk-i k= 1

A 0 : i}.

0 < 8 < ~/n. We wish to show that if then

{Ak}~= 0 ( P~(8 2)

0 < e I =< e 2 < ~/n

and that

P~(~/n)

and

n 0 ( p~(81) {Ak}k=

is the convex hull of

{ n[ ei2kj~/n zk_l : 0 ~ j ~ n-l}. This will verify the assertions k=l made in the previous section concerning coefficient regions for n = 3,~ where

and will imply that co(G)

pn(8)

co{Qn(zei2j~/n;e)

c

is the convex hull of

: 0 ~ j ~ n-l},

G.

0 =< 8 < ~/n. LEMMA I. Let qn~(k)(z;8) = Qn (zei2(k-l)W/n;e)' i =< k =< n, .^(k) n Then tqn (z;8)}k: I is a basis for the vector space (over the real n

~ Ajz 3 such that ~j : An_~. 0 < j 0

for some

j.

n

: 1 < k < n} = = "

as < 0

for some

Replace

Q

by

given by (6) with

as

and all other

remain the same.

ak

replaced by

Al(t)

Qt

t.

Hence

Q

and

Then

aj

Qt

replaced by

and

Izl < i,

Q

aj + t

have the same

Izl = 1

and

Izl > i.

satisfies

IAl(t) I = IA1 + t(e 2i(j-l)~/n - e2i(s

for large

n [ a k = i, k=l where Qt is

Since (t > 0),

az-t

number of zeros in each of the sets But the new constant term

s

I > 2t Isin(j-s

has a zero in

Izl > 1

- I~I > 1

and

Q I P~(~/n)-

This completes the proof of the lemma. The following result,

which we state as a lemma,

of Szeg~ [13] and [5,p.47].

is a result

It is a consequence of Grate's theorem,

which has a beautiful generalization

to higher dimensions

[3].

It

will serve to motivate the theorems which follow. n

LEMMA 3. R(z) =

P(z) =

region

A.

n ~ k=O

c~rcular where

Let

~

n

[ C(n,k)Akzk , Q(z) = [ C(n,k)Bkzk, and k=0 k=0 C(n,k)AkBkZk a n d s u p p o s e all zeros o f P lie in a Then

i8 a s u i t a b l y

every

chosen

zero

y

point

in

of

R

has

A

and

8

Using this lemma we readily see that if R (Pn(0).

the

is a zero

P,Q ( P n ( 0 )

y = -aS, of

Q.

then

This suggests the following theorems which will be proved

in section 8. THEOREM 5.

form

If

{Ak},{B k} ~ Pn(e)

then

{AkB k} ~ Pn(e).

An obvious corollary to Theorem 5 is the following.

177

i. If

COROLLARY

{Ak},{B k} (P~(8)

then

{AkB k} E P~(e),

Corollary 1 is clearly equivalent to Theorem 5 when However Theorem 5 is trivial for for

1 ~ k ~ n-l)

8 = ~/n

(since

0 ~ 8 < ~/n.

C(n~k,~/n)

while Corollary i is not so obvious.

it is easy to see that Corollary 1 is true for

0 ~ e

= 0

Nonetheless

e = ~/n

by means of

Lemma 2. Observe that if where

B k = i,

n 0 {Ak}k=

0 ~ k ~ n~

P~(8).

THEOREM 6.

If

E P~(0),

{Bk} ~=0

E

P'(e)n

Szeg~'s theorem implies

This is a special case of the following theorem.

Further,

If

{Ak}~ P~(82).

then since

0 ~ 8 ~ ~/n~

0 ~ 81 ~ e 2 ~ ~/n

{A k} E Pn(82).

COROLLARY 2.

{Ak}k= 0n

and

{A k} E Pn(81),

then

Pn(8) r eo{Q~k) (z;8)} k=l n 9

0 ~ e I ~ 8 2 ~ ~/n

P~(e)

Eurther,

and A k (

P~(el),

then

c co{ei2(k-l)~/n ^(k) qn-1 (z'8)}~ ' : I"

Note that the last statement follows readily from the first part of the Corollary and Lemma 2.

5.

CONSEQUENCES OF THEOREMS 5 AND 6,

Let

Ck(~)

be defined by z/(l-z) 2(I-~)

=

~ Ck(e) zk+l k=0

i.e. F(2+k-2e) Ck(a) : F(2-2e)F(k+l) and let

S (e)

denote the class of normalized functions

origin with derivative order

~ < i.

Let

1

at the origin)

(0

at the

which are starlike of

178

co

(7)

f(z) =

and set

~ Ck(e)akzk+l , k:0

g(z) =

[

Ck(a)bkzk+l

(

S~(e)

k;O

f~g(z) =

[ Ck(m)akbkzk+l The operation e depends on k=0 but its meaning will be clear from the context. Then by Theorem i, there are sequences {Pnk},{Qnk } E Pnk(~/(nk+2-2~)) ZPnk § f Hence

and

ZQn k § g

uniformly on compact subsets of

Z(Pnk * Qnk) + f*g

Theorem 5)

such that

(Pnk e Qn k

IzI < I.

is the operation implied in

uniformly on compact subsets of

Izl < i, and Theorem 5

implies that

Pnk ~ Qn k E Pnk(~/(nk+2-2e)).

f*g E S (a).

We have thus obtained the following result.

THEOREM 7.

If

f

f~g(z) =

then

and

g,

given by(?),

[ Ck(~)akbkzk+l k=0

The special cases

~ = O

Therefore we obtain

are starlike

is starlike

and

~ = 1/2

nbnzn

are starlike then

of Theorem 7 were

if

~ anzn n=l

With

and

~ = i/2,

of order

1/2

~ bnzn n=l

then

Brickman,

[ anbnzn

Hallenbeck,

If

~ anbnzn n=l and

the

is convex.

[ bnzn

is starlike of order

showed that the extreme points of z/(l_zeiY) 2 (i-~)

[ a n zn

MacGregor,

e = 0

~ nanzn and n=l is starlike; equivalently,

are convex then

the result is:

With

If

~ nanbnzn n=l

n=l

~

of order

recently proved by Ruscheweyh and Shell-Small [7]. result is the Polya-Schoenberg conjecture:

of order

are starlike

1/2.

and Wilken [i] recently

S*(e)

are the functions

0 < y < 2~, and that each

f E S * (~)

has the

repre sentat ion (8)

where

f(z) =

~

I~

" 2(i-~) dw(t) z/(l-ze It)

is a probability measure on

[0,2~].

Using the fact that

179

each of the functions point of

z/(l-zeit) 2(I-~),

~(k). {{qn ~z ;~/(n+2-2e)) }n k=l }~ n=l

0 ~ t < 2~,

is a limit

together with Theorem 6,

we

obtain the following theorem. THEOREM 8.

If

[0,2~]

on

f ( S (~)

then there exists a probability measure

such that (8) holds.

set of probability measures on by (8) are starlike of order

Further,

[0,2~] ~.

If

let

U(~)

denote

the

for which the functions given ~ ~ 8 ~ ij

then

U(e) c U(8).

Theorem 7 can be expressed in terms of convolutions of measures in the class

U(e)

defined above.

If

f(z) : I ~

z/( i - zeit) 2(I-~) d~(t)

I~

z/(l zeit) 2(I-~) d~(t)

g(z)

:

and

then f,g(z)

=

12~ 12~ z/(l-zel(S+t)) 2(I-~) dw(s) dg(t) 0 0

=

IO ~ ~t+2~z/(1 zeiT) 2(I-~) d~(T-t) d~(t)

=

121T[ ; 21~z/(1-ze IT) "2 0

=

Here

~

= ~*v.

(l-a)dg(T-t)]dv(t)

0

I~

" 2(i-~) d~(T) . z/(l-ze IT)

has been extended to be periodic with period

2~.

We write

It is clear that Theorem 7 is equivalent to the following

theorem. THEOREM 9. U(e)

i8 closed under the operation

It is well-known it is starlike of order

([6],[9]) 1/2.

that if

*.

T[ akzk k=l

Equivalently,

is convex then

[ kakzk k=l

starlike

180

implies

[ akzk T is starlike of order i/2. This result is a k=l special case (a = O, 8 = 1/2) of the following easy consequence

of

Theorem 6 .

If

THEOREM 10.

Ck(e)akzk+ 1 E S * (~)

and

~ ~ 8 ~ 1

then

k=O Ck(8)akzk+l

E S~(8).

k=0

6.

SOMEGEOMETRIC PROPERTIES OF THE CONVOLUTION,

EXAMPLE.

Let

f(z)

=

[ kakzk ~ k=l

g(z)

=

be starlike

(of order

0)

and let

akzk = [O f(w)/w dw. k=l

Then

g

zg'(z) g

maps the unit disk onto a convex domain. : f(z),

[z I : r,

is normal to The curve

Also, g({Izl

the vector : r}),

maps an arc of the unit circle onto a straight line segment

only if

f

maps that arc onto a radial

normalized minus

=

=

For

~ = 0

(9)

slits along the real axis,

ze i8 1 [.----I~ 2i sin 8 l-ze ~

be The

i.e.

= l_ze_-~

sin k8 z k k=l ~ ,

0 < 8 < w.

we obtain

f~h(z)

ak ~._--;T:-~-zsin ~ • k8

=

From (9) we see that if

then

h(z)

z (l-zeiB)(l-ze -i8)

k:l

length

Let

if and

starlike function which maps the unit disk onto the plane

two radial h(z)

segment.

so

y

28

is constant when

7-28).

If,

_ g(ze-iB)].

is an amc of the unit circle whose

is greater than

arg f*h(z)

arc of length

F

k = 1 [g(zeiS) 2i sin 8

and ze i8

for example,

g(F) and g

is a line segment~ ze -i8

lie on

F

(an

maps the unit disk onto

181

the interior of

polygon and

a

B

is small,

then the Hadamard

product of

12 with

g

[

h(z)/z dz =

i sin k8 k [ sl~---~--~u z

k=l

is a m a p p i n g w h i c h smooths out the corners of the polygon

w i t h a slight amount of shrinking and stretching. The above example is a special case of a more general result which we obtain below lemmas concerning LEMMA 4.

Let

(Theorem 13).

be a given real number and let

{Qn(zei(0+2kS);8 )}k=0 n

of degree

less than or equal to

Since the d i m e n s i o n of the vector space is

sufficient

to prove that the given polynomials n independent. Assume 7. akQn(zel(~+2kS);8) k=0

Qn(zei(~+2k8);8) setting

has the

an

LEMMA 5. If

=

0,

Then

an_ 1

0 < 8 < ~/n

with :

0

, ...

and

,

:

P (Fn(8),

0

Since

0.

n

)}j:l'

we obtain in

9 then

for all real

has the r e p r e s e n t a t i o n 9

P(z)

where

ak

is real,

z = -e -(~+(n-2s

0 < k < n. ,

§

nke ) Qn(zei(~+2kS) ;8)

Further,

s = 1,2, .. .,p

an-p+ I 9 Proof.

n

n -l(~ 7 ak e = k=0

By Lemma 4,

then

it is

are linearly

s = 1,2,.-.,n+l, a0

n.

n+l,

{ (l+zei[~+ (2j-n-l+2k) 8 ]

factQrs

z = -e -i(~+(n-2s

succession

P

0 < e <w/n.

is a basis for the vector space of polynomials

Cover the complex numbers) Proof.

some preliminary

Pn(e).

the polynomials

~

We require

if

P(z)

= 0

for

0 = a n = an_ 1 : ...

~,

182

P(z) =

n[ ak e -i(~r k=0 n

+ i ~ bk e

nke)Qn(zei(~+2ke);e)

-i(~ @+ nke)

k=O

where

{a k}

R(r

where

R

=

and {b k} e- i n ~r

and

ir

Rn

n -~r [ bk e k=0 lemma is p r o v e d .

are real.

=

Setting

Hence

(zei(~+2ke)

The r e s t

(zei(~+2k8)

z : e ir

n [ a kRn(r k=0

are real. Qn

Qn

+ i

of the

we have

n [ b kRn(r k=0

n [ b kRn(r k=0

;8) ~ 0.

;8)

~ 0

and

Thus the first part of the

lemma i s

proved

by t h e

same d e v i c e

used in proving Lemma 4. THEOREM ii. A s s u m e {Ak},{B k} ~ Pn(e), 0 < 8 < ~/n. Further,. n assume P(z) = ~ C(n,k,e) A k z k has p zeros, z = -e lSj, w h e r e k=0 n ~j = r + 2(j-l)e, 1 < j < p, a n d Q(z) = [ C(n,k,8) Bkzk has k=0 q zeros, z = -e i0j, w h e r e ~2j = ~i + 2(j-l)e, i =< j =< q. I f n p + q > n,

then

P~Q(z) =

~ C(n,k,8)AkBkZk k=0

has

p+q-n

zerosj

i(r z = -e

,

Proof.

P(z) =

P

and

Q

1 < j < p+q-n.

have the representations

n-p i(~(r [ ake k=0

)

i(-r Qn(ze

i(~(~l+(n-l)e)-nMS) Q(z) = n~q bke k=0 Hence P,Q(z) : 2n-~q k[ ak_Rb s ei(~r162162 k=0

s

and the desired result follows.

;B)

i(-~l-(n-l)8+2kS) Qn (ze

;e).

183

Remark. for

Theorem ll holds trivially

8 = 0.

To see this,

suppose

zeros of multiplicity

p

{Pk }

{Pn(Sk)}

Pk

and and

{Qk } Qk

3.

k § ~,

q

and

Q have

We choose sequences

e k § 0,

Pk § P'

of Theorem ii.

we obtain the result for

and Ql(Z;e)=

0 < e < w/n, and that

Cj+l - Cj = 2e,

Qk § Q

Applying

and

Theorem

8 = 0.

P1

1 < j < p-l, and

n ; C(n-l,k-l,e)~z k-I k=l has

p

Q1

Zj = e iCj

seros

has

q

seros

~j+l - ~j = 2e, 1 < j < q-l, where p+q >. n [ C(n-l,k-l,8)AkBkZk-i has p+q-n+l zeros z. = e 18j k=l ]

zj = e n-l.

P

Assume

Pn(8),

satisfy, ing

and is also true

and that

respectively. where

n Pl(Z;8) = [ C(n-l,k-l,e)~z k-I k=l belong to

e = w/n

P,Q E Pn(0)

satisfy the hypotheses

ii and letting COROLLARY

from

and

for

satisfying Then

8j+ 1 - 8j : 28,

satisfying

Proof.

For

and the remark.

1 0

occur above, we must

j+q : n-l.

the derivative

Thus

of

sin s

is

> 0

j+q-i R sin ~e) ~=I

is to be replaced for the values

the two denominators

show that

if

sin s

is

n-j-q R sin s A=l

the last factor and

for

1

n-j-q-i E s

j+q n-j-q-i ~ sin~e)( ~ sin s 1 Z:I

Similarly, the second denominator

(_l)J -I

by

(

by of

are of opposite

1 q

when and

sign.

j+q = i. s

which

Therefore,

lgl

m

m

n

~ aqap(-l)P+q(q-p)[ q=0 p=0

n

~ sin(~-j-q)e 4=1 s

~ sin(~-j-p-l)e] s=l s#j+p+l

>

0,

that is, m

q~l

n

g aqap(-l)P+q(q-p) q=0 p=0

n

[ H sin(~-j-q)e ~=i s

H sin(~-j-p-l)e s=l s#j+p+l

n

sin(s

H sin(~-j-p)8] s=l s#j+p

s s m

qzl j'+q-i [ aqap(q-p)( ~ sin s ~=0 p=O s

n-j-q-i ~ sin s s

n-j -p-i 9( H sin sS)[-sin(n-j-q)8 s=l

The quantity sin n8 sin(q-p)8

in brackets > 0,

sin(j+p)8

is

P(z)

first observe P

and

values a

q

and

Q

ii

k-m

and

Q.

course,

=

n [ C(n,k,%)AkBkZk, k=0

different

m

We will use the coefficients

b

(10) for

that

zeros respectively zeros

independent

Q

and replace

P,Q(z) each

k = m

or none at all.

and

t

k = m = n/2, These

zeros and the other

by

k

m ~ k ~ n/2. of

separation

in argument

t

instead of

q

We assume

having minimum

or

m

and

has two collections

(i.e., they are separated

we could have

one such eollection called

P*Q(z)

in this representation.

then implies

consecutive

=

n n [ C(n,k,8)Akzk and Q(z) = [ C(n,k,8)Bk zk, we k=0 k=0 some ways of forming the convolution. We will assume

respectively

Theorem and

P

- cos(n+q-p)8]

> 0.

and the lemma now follows.

i n the representation s

+ sin(n-j-p)esin(j+q)8]

are given by (i0) with possibly

for

j+p-i ~ sin sS) s=l

1 ~[cos(n+p-q)8

In order to study the convolution where

=

by

n-m-k

between 28).

Of

in which case we have

zeros of

P*Q

will be

zeros will be called

dependent

192

zeros.

The independent

zeros of

P*Q

arise from the fact that

m

[ aq(-l) q e-i(q ne+mt(~-ne)) q=O

P*Q(z) =

so that if all

terms are

m+l

O~

is such that all the terms are for

z = e i(~+28)

Then

0

Q(zei(2mt(~/n-e)+2qe))

P,Q(z)

is zero.

Assume

z = e i~

but that the last term is not zero

e-i(n/2)~P,Q(eir

is real and has the

m

value

[ aq(-l) qs (r q:0

< ~ < ~+2e, P,Q(e i~) # 0.

(~/n-8)+2q8 )

where

Similarly,

P*Q(e ir

e i(@-28).

~ 0

if

If

consider the dependent

k

zeros are

Thus,

in order

we need only

with

n

C(n,k,e)B k = is the elementary

variables.

Then

P*Q(z)

=

and we may form this convolution by forming

n ~ C(n,k,e)Akzk , k=0

n-i [ C(n-l,k,e)(Ak+~iZAk+l)zk, k=0

n-s ~~ C ( n - ~ , , k , 8 )S(s [ p=0 k=0 P )( { l Z , . . . , ~ z ) A k + p Z k sequence of polynomials

by

9 .-, Pz(z;~iz,...,~s

....

-i~Sps

28.

then

~k-(n) = [~jl~j 2...~jk

symmetric function of degree

Ps163

but the first term is

of this type,

n Q(z) = H (l+{-z),j j=l

where

k=0~k ~ -(n)(~l , 9 .,~n)Akzk ,

e i@

where

zeros.

Observe also that if s(n), k ~ l , ~ 2 , ' . . , ~ n ),

4-28 < r < ~,

zeros by more than

to prove Theorem 5 for polynomials

(13)

real.

This implies that the independent

separated from the dependent

successively

is

then all terms in the sum have the same sign and

is chosen so that all terms are zero at not zero at

S

...

... ,

Denote this

P(z) = P0(z), Pl(Z;~iz),

P2(z;~iz,~2z),

Then

z) :

,

,~s

+ ei~SpE(ze-iS;~l

z,

~s +

= 2 cos~

n-s

8

193

Ps163

-

+ (~Z+I -i)

It is clear that function of

2i sin(n-s Ps163

~iz,-..,~s

is linear and symmetric as a hence the convolution

is independent of the order of 9

p~(ze,-ie;$1z,...,~zz)

61'''''~n"

P*Q = Pn(~iz,...,~nz)

Further,

if

k

Q(z) = Qn_k(zeiY) ~ (l+~jz),

j:l

then

P*Q(z)

is a rotation of

Pk(Z; ~i z, 9 "", ~k z) . We are now ready for the following lemma. LEMMA 8.

Theorem 5 holds for polynomials of the form

Qm(zeiS;e) Qn_m(ZeiY;e) ~ Pn(8).

Proof.

As in the above discussion,

are given by (10) with representation of where

Q

aq, m, t by formula

I~ll = I~k+iI = 1

and

we will assume

replaced by (i0).

Assume

~j = ~j_l e-2is,

P

and

bq, k,s

in the n Q(z) : ~ (i +~jz),

j:l

for

2 ~ j ~ k

k+2 ~ j ~ n. Consider Pn_2(z;~iz,''',~k_iZ,~k+iZ,''',$n_l z) n-2 =

k:0

s~n-2)(~l,--',~k_l,~k+l,''',~n_l)Akzk

+ 2(cos O)z

Q

n-2 ~(n_2)(~l, ", k [ bk "" ~k-l'~k+l'''''~n-l)Ak+l z k=0

+ z2 n~2 ~(n-2) k A(z) + 2(cose) zB(z) + z2C(z) k=0 bk Ak+2Z =

where C(z) : zn-2 $1~2...~k_l~k+l'''~n_ I A(I/~).

and

194

Then (14)

P*Q(z)

Assume

P,Q(z)

: A(z) + (~k+~n)ZB(z)

= 0 = P~Q(ze 2ie)

for some

+ ~k~nZ2C(z).

z.

Then we have

A(z) + (~k + ~ n ')zB(z) + ~k~n z2C(z) = O, (15) A(z) + (~i e2ie +~k+le2ie)zB(z)

Assume for the moment that the equation

+ $1~k+l e4i8 z2C(z)

IB(z)I < IA(z)I

A + (~+~)zB + {nz2C = 0. ~Z

=

-

IBI < IAI,

assumption

arg ~

~ = ~k

I+(B/A)~z .

ering the location of impossible to have The results imply that for

decreases as

when

~ = ~n

~j,

in Marden ~ = 0,

and

when

C"

arg n ~ = ~n

i ~ j ~ n,

~ = ~ie2i8

and consider

We must have

B/C + ~z Since

( : IC(z) I)

= 0.

increases and by when

n = ~k"

Consid-

we see that it is clearly n = ~k+l e 2i8

[5, pp.38-43]

IA(z) I > IB(z)I

in conjunction with (13) when

z

is a dependent zero.

Taking P(z) = Qn m (zei(me+2mt(~/n-e));e)

Qm (ze-i(~-m)o+2(n-m)t(=/n-e));8)

and ^

.

Q(z) = qn_k~ze with

m, k, s, t

false then

i(kS+2ks(~/n-%))

fixed and

8

8 ) Qk(ze-i((n-k)8+2(n-k)s(~/n-e));8 )

variable, we see that if Lemma 8 is

IA(z) I = IB(z) I = Ic(z)I

for some

z

and

P*Q(z) = 0, z a dependent zero of P*Q. Since ~(n-2)/2 ($1$2"--~k_l~k+l..-~n_l) 89 B(z) is real when becomes

A(z)(l +z~keir

+ Z~neir

for some

e

such that

Izl : i,

(14)

~, and one of these

195

factors is

0.

Thus the value of

z

is independent of either

or ~n' and we may assume it is independent of n [ s(n-l)(~l'~ '''''$n l)Ak zk-I 0, Izl = i. 2 = k=l k-i n [ C(n-l,k-l,e) Akzk-I k=l we see that

Pl(Z,e)

tion of n-m-i

This means

Since

P(zel8) - P(ze-le) 2i z sin ne

:

has one zero in

m-i zeros with consecutive

~n"

Izl < i,

zeros separated by

zeros with consecutive

{k

: Pl(Z,e)

one collection of 28,

and one collec-

zeros separated by

28.

Let-

ting Q(z) =

n-i ]I (l+[jz) : Q ( z ) / ( l + ~ n Z ) : 9:1

n-i [ C~(n-l,j,~)B.z j j :0 ]

and 9

m

Pl(Z,e) = Qn_m_l(zel@;8) where

Iqjl = i,

1 ~ j ~ m-i

the remark preceding Lemma 8)

E (l+njz), j=l

and Iqml > i, we have (in view of n (n-l) [ Sk_ 1 (~l,~2,---,~n_l)Akzk-i is a k=l

rotation of ^

Qm(Z;qlZ,q2 z,...,qm z) =

n-l-m m ^ 9 [ [ C(n-m-l,j 8) s(m)(qlZ,''',qmZ)Bj+pZ3 j:0 p:0 ' P

n-l-m m-i ~(m-l), zj : j[0: p=0[C(n-m-l,j,e) ~p ~nlZ,..',~m_lZ)Bj+ p

+ nmZ

= Since

l~jl = 1

ID(z) I = IE(z) I when

n-l-m m ^ 9 j!0 p!l C(n-m-l'J'8)S~ ll)(nlz'''''Nm-lz)Bj+pz3

D(z) + qmZE(z) .

(i < j < n-l)

and

I z I = i. Therefore~

lqjl : 1 (i < j < m-l), n ~(n-l),~ .. r )Az k-I A ~ 1 ~%1'" '~n-i "I< = 0 k=l

196

for some

z,

Izl : i,

implies

D(z) : E(z) : 0.

location of the zero is independent of

~(z)

form

Qn_m_l(zei~)Qm(zeiY)

Nm"

Thus we may change

for some

y

Q

are of degree

conclude that the zero an independent zero.

8.

PROOF OF THs

n-l, z

nm

nm

of

to has

SO that

a zero which does not depend on one of the zeros and

This means the

P.

Since

we may use an induction argument to

of

P*Q

for which

IA(z)I

= IB(z)l

is

This concludes the proof of Lemma 8.

5 AND 6,

We require the following lemma, extreme points in the class ( n-i {Qnm)(z;e)}m=0 9

P (8) n

which implies that the only are the polynomials Pn(e) c

This gives an alternate proof that

n-i co{Qn(m) (z;e) }m=0" L~MMA 9.

P E Pn(8),

Assume

where

0 < e < n/n,

and that

i~. P(z) :

K

Qkj(ze

J;8),

8j + (kj-1)e < 28+Sj+l-(kj+l-l)8

where

j=l (i < j < 9,-1) and P

into factors

1 < p r

We will show that < 0

for

the zeros of

1 ~ q ~ kl-l. Rp(r

Differentiating 0 : dRe(r de

~l,p

•

> "'" > Ck I' and

{Ak(S)}

A0(g) = i,

r162162162

-i ~ r 2 pe(elr

Re(C) = e

RI(r

r

[ H Rj(r j :I

satisfy the required coefficient relation

An_k(g),

r

kl + ie(kp(--~ RI(r162

s

2

k - kl(-~2 Rp(r162

9Rp(r

=

e

Ak-~-~=

be continuous Re(r

: 0,

r162

1 ~ q ~ kl,

are the zeros of

! Cq+l(0) - r

= 0

and

~q+l~"(0) -

The same result will clearly hold for

and the proof will therefore be complete. Re(r

and setting

: • [ [ ( H Rj(r162 s j~s

e = 0,

we find

s)

J

j~l,p,s

(kpR{(r162162162

H Rj(r j#l,p

8j)

so that

H Rj(r162162162

j~l

I) H R.(r j~l

]

I)

198

and r '( 0) = -kp,

i =< q ~ k I.

Differentiating once more and setting :

[R~(r

+ 2 ~l

l)

H

j~l,s

H Rj(r162 j~l

e = 0

+ [R[(r

R~(r162162

I)

yields H Rj(r j~l

+ 2(kpR~(*q+Bl)Rp(r

+ kpRI(~q+81) Rp(~q+BP) - klRi($q+Bl)Rp(~q+SP))(j~l,pll R.(r162

+ 2 kpRi(r162

sIl,p j~l,p,sH Rj(r162162

Hence

R"(r ~"(0) = k 2 I ~ -q p R{(r

I)

-

R~(r 2 kpk I Rp(r

Using the fact that ~+Sj+(2s-kj-l)8 k S ks R-(~+6~), = 2 H cos ( ) s=l and Cq = ~/2 - 81 + (-2q+kl+l)e, we have k 8n-81- (kn-2j- k]+2q) 8 = 2 9 cot(s-q)8 - kpk I ~ c o t = P s=l j=l s#q k1

r

=

q

so that

k 2

p)

199

0

~p-81- (kp-~+2q) T~ Cq+l(0)

- -q~"(O) : k 2p (cot (-q@) -cot(kl-q)8)-k p k I (cot ~p-~l+(k~§ 2

- cot

k sink 8 = -kp [sin qP8 sin(kPl-q)0

Since both quantities Proof of Theorem P*Q(z) true for

k I sin kp8 ~9-~y(kn-kl+2q)8 sin 92 in brackets 5.

Assume that

near

z/n

{8 E [0,w/n]

is therefore

is such that all zeros of

Izl = 1

when

P,Q (Pn(8)

: P,Q ~ Pn(@)

both open and closed,

=

C(n,k,8),

under the conditions

and the theorem follows.

> 0.

and

P

Rp(Z) = ze iBp Q~ (zeiBp;e) p

Let

P(z) + is[kpRq(Z)

~(e)

be a continuous

(P + ie[kpRq- kpRp]) property of

P,

, Q(e i~(s))

Q, and

~

8

fixed.

e-i(n/2)(~+2@)P*Q(e i(~+2@))

P,Q E Pn(8), If

@

and

Q

P*Q(e i@) = 0,

iBj E Qkj(ze j=l j~P

- kqRp(Z)]

E Pn(e)"

function

such that

= 0.

We

is

and

are both as in Lemma 8,

there is nothing to prove. Therefore, assume that s E (zelSJ;8) where s > 3 and that P,Q, and j=l Qkj = the above minimum is achieved. Let

By Lemma 9,

1 ~ k ~ n-i

P,Q ( P (8). n v ~ 6 [8,w/n]}

P*Q ( P n ( @ ) ,

0 < @ < 8 < w/n,

wish to show that the minimum of

d/d@(e -i(n/2)@P*Q(ei@))

(this is

We will show that this implies

To begin the proof, let

positive

the lemma follows.

since the coefficients

are small in this case). The set

0

~p-~l+(kp+kl-2q) '3]" 2

sin

are positive,

are distinct and lie on @

)

P(z) = ~

are such that

;0).

~(0) = ~

and

Then due to the minimizing

we have

ie i~ (PeQ)'(e i~) ~'(0) + i[kpRq - kqRp] , Q(e i~) = 0

200

and [-i~P*Q(e i(~+2r Fixing

p

and letting

Rq*Q(e i(~+2r gin.

+ iei(~+2r

Since

q # p

we may fix

Rq*Q(e i(~+2r

= 0.

we see that the vector

1 < q < 4.

q,

is contant.

~(e)

of the above type,

q

If

~'(0) = 0

this is impossible.

p

vary to obtain the

P

for every choice of

p

and

Taking a sufficient number of variations

we see that

(i.e., any two zeros of

and let

lies on the same straight line through

the origin for then

vary,

'(0) + i[%Rq-kqRp],Q(ei(@+2r

always lies on the same straight line through the oris > 3,

result that

'(ei(~+2r

P

is an interior point of

are separated by more than

Hence for some choice of

p

and

Pn(e)

2e); q,

but

~'(0) # 0.

Since s

s

Rq(Z) : zP'(z),

~ Rq~Q(z) = z(e~Q)'(z), q=l

q=l it follows that

Rq~Q(e i(@+2r

on the same line, or

i ~ q ~ 4.

ei(~+2r162

IP,Q(z)I

covered

as

lie

This implies either

P,Q(e i(~+2r

is real.

= 0

Since

maps the unit disk onto the half plane n

times, a point

z

on

is real (and therefore= n/2)

z(P~Q)'(z) /P*Q(z) of

ei(~+2r162

/ P~Q(e i(~+2r

z(P*Q)'(z) /P~Q(z) {Re w < n/2}

and

z

varies on

Izl = i.

Izl = 1

for which

is a local maximum

The set of

r ~ (0,8)

such

that the extremal problem under consideration has a positive solution is clearly open and non-empty. set of local maxima of bounded away from Similarly, 9P*Q(e i(~+2r

However,

Ip*Q(z)I,

it is also closed since the

Izl = i,

for

P,Q ~ Pn(8)

is

0. we may show that the maximum of

is negative when

d/d~( e-i(n/2)~ P*Q(ei~)) < 0. This completes the proof.

P,Q E Pn(8),

e -i(n/2)(~+2r P*Q(e i~) = 0,

and

201

Proof of Theorem proof of

Theorem

that Lemma

9.

5

6.

The proof of Theorem

except that

7 applies,

6 is identical

r

is allowed to vary in n Q(z) : ~ C(n,k,8+s)z k. k=0

and that

to the

(0,e+r

DISCUSSION AND OPEN QUESTIONS The proofs of Theorems

Possibly

there is a generalization

would lead to a simpler applies

5 and 6 are admittedly

proof.

of Grate's

Also,

for

theorem

e = 0,

to a much larger class of polynomials

Theorems

5 and 6 can be generalized

For instance,

rather

than

tedious.

[5,p.45]

SzegS's

result

Pn(0).

Perhaps

to a larger class than

for which polynomials

P

which

Pn(e).

is it true that if

W

and

9

are probability

measures

such that

P(ze It) d~(t)

and

9

P(ze It) d~(t)

have all their zeros on

P(ze ~(t+s))

d~(s)

d~(t)

It follows from Lucas'

has its zeros

theorem,

then for

1 < s < n-I

Izl < I.

We have shown that if

Izl > i,

Izl __> l? n ~ C(n,k)Akzk k=O

n ~ C(n-s163 k=~

zk-s

n ~ C(n,k,e)Akzk k=0

6 Pn(8)

all zeros of

In view of the representation likely that functions

in

that if

n [ C(n-s163163 has all its zeros in k=s this result true also for Z = 2,--., n-l?

it seems

then

Izl > 1

given by L e m m a

starlike

of order

a

lie in then

for

5 for

E Pn(0~

s = i.

Is

P ~ Pn(8)

have a represen-

tation

I~ f(z)

where

A

is a complex

not a probability length

y

such that

= A

constant

measure).

on the unit circle d~

ze i(l-e)t (l_zeit)2(l_a)

and

W

Further, there

d~(t),

is a positive if

Re ~

measure = e

on an arc of

should be representation

has no mass on an arc of length

y.

Such a

(but

as above

202

representation is b o u n d e d . bounded

does

This

convex

exist

follows

functions

for

e = 0,

readily given

at

from an

in [12].

least

when

integral

f

z

f ( w ) / w dw 0 r e p r e s e n t a t i o n for

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L. Brickman, D.J. Hallenbeck, T.H. MacGregor and D.R. Wilken, Convex hulls and extreme points of families of starlike and convex mappings, Trans. Amer. Math. Soc. 185(1973), 413-428.

2.

W. Kaplan, Close-to-convex 1(1952), 189-185.

3.

L. HSrmander, 55-64.

4.

S. Mansour, On extreme points in two classes of functions univalent sequential limits, Thesis, University of Kentucky, 1972.

5.

M. Marden, The Geometry of the Zeros of a Polynomial in a Complex Variable, Amer. Math. Soc., Math Surveys no. 3, 1949.

6.

A. Marx, Untersuchungen ~ber 8chlichte 107(1932/33), 40-67.

7.

S. Ruscheweyh and T. Sheil-Small, Hadamard products functions and the P61ya-Schoenberg conjecture, Helvet. 48(1973), 119-135.

8.

G. P61ya and I.J. Schoenberg, Remarks on de la Vall~e Poussin means and convex conformal maps of the circle, Pacific a. Math. 8(1958), 295-334.

9.

E. Strohh~cker, Beitr~ge zur Theorie Math. Z. 37(1933), 356-380.

8chlicht functions,

On a theorem of Grace,

Math.

Mich.

Scand.

Math.

J.

2(1954),

Abbildungen,

der schlichten

with

Math. Ann. of 8chlicht Comment. Math.

Funktionen,

10.

T. Suffridge, On univalent 44(1969), 496-504.

Ii.

T. Suffridge, Extreme points in a class of polynomials having univalent sequential limits, Trans. Amer. Math. Soc. 163 (1972), 225-237.

12.

T. Suffridg e, Convolutions 15(1966), 795-804.

13.

G. Szeg~, Bemerkungen zu einem Satz yon J.H. Grace ~ber die Wurzeln algebraischer Gleichungen, Math. Z., 13(1922), 28-55.

polynomials,

J. London Math.

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J. Math.

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Advances in Complex Function Theory

Complex Function Theory

Complex Function Theory

Complex function theory

Notes on Complex Function Theory

Geometric Function Theory: Explorations in Complex Analysis

Geometric Function Theory: Explorations in Complex Analysis

Geometric Function Theory, Explorations in Complex Analysis

Function theory in several complex variables

Classical topics in complex function theory

Function Theory of One Complex Variable

Handbook of Complex Analysis: Geometric Function Theory

Function theory of one complex variable

Function Theory of Several Complex Variables

Function theory of several complex variables

Handbook of Complex Analysis: Geometric Function Theory

Advances in Immunology Volume 74

Advances in Agronomy, Volume 74

Seminaire Pierre Lelong (Analyse) Annee 1973-74

Function Theory in Several Complex Variables (Translations of Mathematical Monographs)

A Scrapbook of Complex Curve Theory (University Series in Mathematics)

A Scrapbook of Complex Curve Theory (University Series in Mathematics)

An Introduction to Complex Function Theory

Function theory in polydiscs

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Arithmetical Algebraic Geometry: Proceedings of a Conference Held at Purdue University December 5-7, 1963

Advances in Virus Research, Volume 74

Geometric Function Theory: Explorations in Complex Analysis (Cornerstones)

Complex Analysis A Modern First Course in Function Theory

Advances in LES of complex flows: proceedings of the Euromech Colloquium 412, held in Munich, Germany 4-6 October 2000

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