AND WORDS

UNIVALENT FUNCTIONS MAXIMIZING Jel.f’(,) + .f()]

INTISAR^QUMSIYEHHIBSCHWEILER

Daemen

College 4380 Main

Street

AmherstNewYork 14226

USA

(Received

April 11, 1994 and in revised form

September 28, 1995)

ABSTRACT. We

studytheproblem

maxhes[h(z,) + h(z2)]

with zl,z2 in

A. We

showthat norotationof the Koebe functionisasolution for thisproblem except possiblyitsrealrotation, andonlywhenzl

e

^orz,z2 areboth real,andareinaneighborhoodof the x-axis.

We

prove that if theomitted set of the extremal function

f

^{ispart ofastraight line}that passesthrough

f(z)

^or

f(z)

^then

f

^isthe Koebe functionoritsrealrotation.

We

also show theexistenceof solutionsthatarenot unique and^aredifferent fromtheKoebefunctionoritsreal rotation. The situationwhere the extremal value isequaltozero canoccurandit isproved, inthis case, that the Koebe function isasolutionif andonlyif

z

^and

ze

^are both real numbers and

zz <

O.

KEY WORDS AND PHRASES.

Univalent Functions,

Support

Points, Quadratic ^Differ- ential.

1991

AMS SUBJECT CLASSIFICATION CODES:

Primary 30C55. Secondary 30C70.

1.

INTROD UCTION

Let H(A)

^denotethesetof all functions analyticinthe openunitdiskA

{Izl < },

endowed withthetopologyof uniform convergence^oncompact subsets.

Let S

denote the subsetof

H(A)

thatconsistsof functions thatareunivalentinA_andsatisfy

f(0)

^{0 and}

if(O)

^1.

^It

^isknown

[6]

^that

^S

^{is a} compact subset of

H(A). H(A)*

will denote the space of allcontinuous linear functionalon

H(A).

A

function

f

ⁱⁿ

^S

^{issaid to bea supportpointof}

^S

^{if thereis acontinuouslinear functional}

n

ⁱⁿ

H(A)*,

^{notconstant on}

S,

such that

L(f) ^>_ L(h)

for all h in

S.

Ifthisis thecasewe will simplywrite

f ^L.

An

expression

Q(w)dw ,

^where

Q(w)

^is meromorphic in a region

G,

^is called a quadratic differentialin

G. An

analyticarc

w(t)

^forwhich

Q(w)dw >

⁰

(i.e. Q(w(t))(w’(t)) > 0)

^iscalled a trajectory ^arc.

A

trajectory is a maximal analytic arc

w(t)

such that

Q(w(t))(w’(t)) >

O.

Thezerosandpolesfor

Q(w)

inGarecalledcriticalpoints.

A

critical pointiscalledaninfinite critical point ifithas order-2 or

less;

^{otherwiseit is} calledafinite criticalpoint.

Itisknown

([4], [9], [7])

^{that eachsupport}point of

S

maps the^diskontothecomplementofa

singleanalyticarc

P

with increasing modulus andanasymptotic direction at o.

It

wasshown

[11]

that the omitted set

F

ofasupport point is atrajectory arc for the quadratic differential

L(fe/(f- w))(dw/w) 2,

i.e.

F

is ananalytic ^arc

w(t)

^satisfying

L(f2/(f w(t))(w’(t)/w(t)) > O. ()

Thisis called the Schifferdifferentialequation. Furthermore

F

has the property that theangle between the radius andtangentvectorneverexceeds

n/4.

The problem offinding support points associated with ^a certain given functional had been studied.

For

examplein

[5], [2]

point evaluation functionalswerestudied, it wasshown that the Koebe functionisthe solutiontotheproblem

rnaxhesHh()

if andonlyif

(

in

((1- e)/(1 +e), 1)

and that no rotationofthe Koebe function is asolution. If is in

(-1, (1 e)/(1 + e))

^then

thereare two solutionsrelatedbyconjugation.

For

any other the solutionisunique.

In

this paper^westudytheproblem:

maxhesHgl,2(h), (2)

where

J,2(h) h(() + h(2)

^{with 0}

< [(,[ <

1,0

< [2[ <

^1.

We

show thatnorotationoftheKoebe functionisasolutionfortheproblem

(2)

except possibly its real rotation, and only when

( 2

^{or both and}

⁽²

^{are real.}

^We

^{also show that if}

f J,

^andthe omitted set of

f, F,

^is part ofa straight line segment that passes through

f(,)

^or

f(.)

^then

f

^{isthe Koebefunctionorits}real rotation.

We

also studythecase where

rIaaxhesHJ,(h

⁰ and in thiscase the Koebefunctionis thesolution if andonly if and

(

are both realand

( <

0.

2.

THE OMITTED SET F

Usingthe Schifferdifferential equation

(1)

^{wecanconcludethat if}

f ^J1,

^then

^F

^satisfies

[a-w +b-w](-)w

a ^{>0 forwinF}

⁽³

wherea-

f()

^b-

Let

k, denote the function

z/(1 xz) 2,

^orforx 1 simply k.

Let L(h) h() + h()

^with0

^< I1 ^<

^1.

We

willneed the following

Lemma:

Lemma

1:

Let F(z)

^{bea functionthat is} analyticinaneighborhood ofthe origin.

Suppose

that there exist asequence ofreal numbers suchthatt, 0and

F(t)

^is

real,

then

F

is real ontherealaxis.

To

prove thelemmashow that all the coefficientsoftheTaylorseries expansionof

F

at the originarereal.

Theorem 1:

(a)

If

k

^{isasolutionfor}

(2),

^thenx =kl and

1 2

^{or areboth real.}

(b) Iff n,

^then

] L

^where

^{](z) f(2). In fact,}

^if

^f,-- L

^{uniquely then}

f

^{k or}

(c) Let (:

and

(

both be real. Ifneitherknork_ aresolutionsfor theproblem

(2),

then the

problem (2)

^{has at least}two distinct solutionsrelatedbyconjugation.

Proof: Parameterizetheomitted set

F

of

k

by w 5:t with t

_< -1/4.

Substitution in

(1)

gives the inequality

a 5

a

v. (a)

Define

a 5

F(t) + ()

a-

Weclaim that

F(t)

^{is realfor} all except possibly

a/"

^or

b/2.

The claim followsfrom the fact that the function

F(z)

^is meromorphicinaneighborhood of the realaxisand, from

(4)

maps the linesegment

< -1/4

^ontothe positive realaxis.

We

canapply

Lemrna

1,if necessary twice, ^toshowthat

F(t)

^is real inaneighborhood of

a/hc

^and

b/hc. ^For

small values oft,

F

can be rewrittenas

F(t) (a + b) +

^2t

+ ^2=t (1/4 + ) + (6)

From

thisfollowsthefactthat is

real,

orx 5=1. Thisisbecause

F’(0) :

isreal. This also showsthat a

+

^{b and}

1/a + _lib

^{are both real.}

Consequently,

either

z

^{or and areboth}

real. This provespart

(a).

To

prove part

(b)

note that the definition of

L

^{implies that}

^{YtL(f) L(f).}

^{If is a}

unique solutionfor theproblem

maxhesL(h)

^then

f f,

^sothat

f

^{k or}

f

^k_l.

Part (c)

^followssince

[f(,) -I-- f(2)] [f(,)

^-t-

f(2)]

This finishestheproofof Theorem 1.

The following theorem shows that the problem

(2)

has solutions other than k or its real rotation.

Theorem 2: Given r in

(-1, (1- e)/(1 + e))

^{we can finda} neighborhood

Ur

^ofrsuch

^that,

whenever

1, ff

^{are in}

^Ur,

^{k andk_l are} not solutionsfor theproblem

(2).

Proof:

Let f

ⁱⁿ

^S

^{be such that}

f(r) > h(r)

^{for all h in}

^{S. It}

^{is known}

([5]

^and

[2])

that

f

^is not unique,

f

^{k and}

fr

^k_.

^A

^continuityargument shows that thereexistsa neighborhood

U

ofr such that

Ytf,() > k()

^{for all in}

U,. Consequently [f() + fr(ff)] >

[k() + k(2)],

^{whenever and arein}

U. A

similarargument appliesfork_

We

note that if

(1 e)/(1 +e) < , <

^{then k is}the unique solution for the problem

(2).

^{This follows because if}

Yt[f(l)+ f()] > Yt[k()+ k(2)],

^{for some}

f

ⁱⁿ

^S,

then either

f() > k()

^or

f(z) > k(’). ^But

^kmaximizes

{h(r):

h E

S}

^uniquelyfor anyrwith

[(1- e)/(1 + e)] <

r

<

¹

(see [5]).

Corollary:

Let

r E

(-1, (1 e)/(1 + e)).

^{Then there exists}

U*,

a neighborhood of r, such thatwhenever

, 2

^{are realin U*or}

1 2

ⁱⁿ

^U*,

^{the problem}

(2)

has at least two distinct solutions

f

^{andg related by}conjugation.

Thecorollary followsasaconsequenceofthe previous Theorems.

Theorem 3: If k

L

^then

[ (i 4)’ >- ^O, (7)

and

[(1- 4)] ^{> o, (8)}

and

[(1 4)] ^{>-0’ (9)}

Proof: To prove

(7)

parameterize theomittedarcby

w(t) -t, >_ 1/4,

and substitutein

(1),

weobtain

a 5

>

0 for

> 1/4. (10)

a+t +o+t

Multiply

(10)

bytandtake the limitas ttends toinfinity to obtain a

+

^b

^>

^{0. Sincea}

,

from Theorem it followsthat

a ^>

^0andthisisexactly

(7).

To

proveinequality

(8),

weusethevariation

NL(k) ^<_ NL(zk’P), (11)

where

P

hasapositive real partand

P(0) (see[12] p.82). From (11)

^weobtain

2(<) ^< [( ₍₁

i_^/

() ^{<) (P() +} P())]"

Substitute

P--

1,

(12)

^becomes

orequivalently

From

thisit followsthat

( + )

4 4(I/

[( _0 f-] ^<

^0.

Derine

a2 b2

_lr

-- ⁺

^b

^{Zzo + Zz}

Notice that

F(z)

^is meromorphic in a neighborhood of the x-axis.

Apply Lemma

1 and use similar

argument

^asinTheorem part

(a)

^{toconcludethat}

F(t)

^is real for all exceptpossibly

>0.

(1 -()3

To

obtain

(9),

note that for any^cwith

Icl

the function

g(z)

^z-

:(c/ 1)z

( z)

is in

S (see[3]).

Consequentlywehavetheinequality

(with c--i)

)] ^{> [- ( + )} - ^{( + )}

[(i C)

^/

(1 (1 ’)’

^/

(1 )2 ]"

From

thisinequality

(9)

^follows. Thisends the

proof

of Theorem3.

A

similar statement holds whenk_

L.

Corollary

Theorem 3 and the results in

[5]

^showthatanecessary conditionfor k

L

^isthat

(

^isina neighborhoodof the linesegments

((1 e)/(1 + e), 0)

^and

(0, 1).

Thisisbecause

z/(1- z)

maps the circle determinedbythe points1, 0,^{-iontothe lineu}

v, (where z/(1 z)

^u

+ ^iv).

^Inequality

(9)

^{thenimpliesthat}

-r r 3n

(

5r

-- ^{< " ( ’0 <} -

^o

^{< , (i-) <}

That is,

(

must be in aneighborhood ofthex-axis.

A

similarargument givesa regionfor in orderfor k_

L.

It

is notknownwhetherkistheonly rationalfunction that maximizestheproblem

(2). ^We

prove thefollowing:

Theorem 4:

Assume f

^isin

^S

^with

f- J,,,

^andsuppose that

f(z)

^{is a}rational function in z.

Assume

further that the analyticcontinuationof

F

passes throughone ofthe simple poles a

f(),

^b

f(()

^ofthequadraticdifferentialin

(3).

^Then

^F

^{is a}horizontal linesegment.

Proof:

It

isknown

[13]

^{in thiscase,that}

^F

^isastraight linesegment. Withoutlossofgenerality parameterize

F

by w a

+ fit

^anduse

(3)

^toobtainthe inequality

a b

[_-- ⁺

^b_a_

/t] ^[a / t]

^>_0

^for

^a

^{+ t e r.}

when

(b- a)/, -a//

^{or 0.}

^It

follows that

tF(t)

isreal^[orall

(b a)/[, -a/l,

i.e.

a

b’t

[-+b-a-fit ][a+t ⁽¹³⁾

is real for closeto 0. Take the limitin

(13)

^{as tendsto zero}to conclude that

[-a/][/a]

is realor

-

^isreal.

^{Hence F}

^ahorizontal linesegment.

Note

that by takingthe limit in

(13)

^{tendstoinfinity,we}obtain

(a + b)/

^{is real,i.e.}

(a +

^b

)

^is

real,

whenever the assumption in Theorem 4 holds.

One

wouldlike toshow that the solution to problem

(2)

^is unique for any

,

^with

,

not real. The previous Theorems andargument support this conjecture.

However,

it renains anopen problem.

3.

THE EXTREMAL VALUE

In

thissectionwewillstudythesituation:

f ^J,

^and

^J,(f)

^O.

This situationoccurs, forexample if

1

^rl

>

0 and

2 r: <

0 with

k(rl) + k(r2)

^{0 and}

r: > (1 -e)/(1 + e).

^{Thiscase is ofspecial interest for the} following reason: If

g,:(f)

0, the quadratic differentialin

(3)

becomes

--2a dw

(14)

w(w-a)(w+a)

where a

f(l) -f(e). ^Let

^{a re}

’

^{andw ve}

’

and substitute in

(14)

^{to obtain}

_2re,

Odv

( )( + ) ⁾

The trajectories in

(14)

^can be obtained from the trajectories in

(15)

^{bya rotation.}

^It

^is

known

([7],[8])

that if 8 is an irrational multiple of2 then every trajectory of

(15)

^{is dense}

in the whole complex plane, i.e. it comes arbitrary close to any complex number.

It

follows that the same is true for

(14).

Therefore if thissituation occursand

argf()

^{is an irrational} multipleof

2,

thenwe canconclude that thereexistsasupportpoint

f

ⁱⁿ

^S

^{withtheproperty}

that itsomittedset

F

has an analyticcontinuation that is dense in the wholecomplex plane,

(this

^seemsunlikely, butremainsas a

conjecture).

Weprove thefollowing

Theorem 5:

Suppose f J,,

^and

Jl,(f)

^0.

f

^{k if and onlyif and arereal and}

6

^<0.

Proof:

Assume

firstthatkisasolutionfor

(2)

^and

gl,(k)

0. Parameterize

F

by

w(t) -t,

>_ 1/4,

and substitute in

(3)

to obtain that

[(a/(a + t)) (a/(a t))]

^is real and positive for

>_ 1/4.

^Define

F(z) a/(a + z) a:/(a z),

and note that

F

is meromorphi in a

neighborhood ofthe realaxis.

Therefore,

we canconcludethat

F(t)

^isrealfor all exceptwhen a isreal.

However,

ifais real then and arereal and

12 ^<

^0. Considering

F(t)

^forsmall

t,

wehave

-2t 2t 2t

Y(t) a[

a a a

-’" ^"]

^{for near}

^0,

so thata is real. Multiply

F(t)

byt andtakethe limit as oc to conclude that a is real and positive. Therefore either^{a is} real and positive ^{ora is} real and negative.

In

any case we conclude that

1

^and

2

^{arereal and}

^,2 <

0.

To

provetheonlyif part, weneed thefollowingobservationdue to

Leung.

Lemma A: Let f

ⁱⁿ

^S

^{maps A ontothe} complementofan analyticarc extendingtoinfinity.

Suppose f(1)

^o,and

f(e ’) f(e-’)

^forall0in

[0, r].

^Then

f

^k.

Proof of

Lemma A: Let g(z) (f(z)(1 z)2)/z.

^Then

g(z)

isanalyticonz

_<

and

g(z)

0 in

A.

Also

g(1/z)

^isanalyticin z

_>

^1. Thegiven conditionsyield

g(1/z) g(z)

^on

Izl

^{1. Thus}

g(1/z) g(z)

^{for allz.} Thisimpliesthat

g(z)

isboundedinthecomplex planeandthereforeis aconstant.

We

also need thefollowingwell known factaboutquadratic differentials.

Lemma B: Suppose B(z)dz

and

A(w)dw

^are quadratic differentials in the z-plane and w-plane respectively.

Suppose

under a slit mapping w

f(z)

^where

f

^{is in}

^S,

^{we have}

A(w)dw

²

B(z)dz :. Assume

further that

f(-1)

^{w0, where}w0 isthe finite tipof the slit. If

f-(w) {e’l,

^e’2

}

forw ontheslit,then

We

would like to prove that if

(,

and

(=

^are

real, (,(2 < 0,

and

f J,,2

^with

J,,(f) ^O, th,en f

^k.

Substitute w

f(e ’)

in

(14)

^toconclude that

-(z) z(z))

(z) ((z) )((z) + a) ((z) ⁽¹⁶⁾

is real and nonpositivefor

Izl

^{1. Recallthat}

^f(4,)

^aand

^f(6)

^{-a andnotethat bythe}

Schwarz reflection principle,

Az(z- e’)

O(z) (z 4,)(1 41z)(z ’2)(1 4=z) (17)

where e isthe point on

Izl

^{1 that} correspondsto the finite tip of the omitted setof

f, F

fora similarargument see

[51). ^Because O(e ’) _< 0,

^wehave

(,) ^Ae

, (e

^,o

e,,) (e ’

^e,,

(e ’ ^{’,)(1 <,e’)(e} ’ ^{<2)(1 (e ’)}

l1 4,e’oll 4e’ol

Therefore

Ae >

^0.

Equate

the two expressions for

O(z)

ⁱⁿ

(16)

^and

(17)

^{and divide the}

resultingequationby z, then take the limitasz 0weobtain

Ae

^2’

2( 2.

^If

¹⁽ < 0,

then

Ae ’’

^isnegative. Thereforee is negative. Thisimplies that

f(-1)

^is the finite tipof

F

and also that

O(e ’) O(e-’o). ^We

^{can apply}

^{Lemma B}

^{with with}

B(e ’) O(e ’)

to show that if

f-’(w) {e’,e’}

for w in

F, then/ -0

^{and hence}

f(1) .

^All the conditions in

Lemma A

arefulfilledso wemay conclude that

f

^k.

Remark:

The problem

(2)

^remainsundeterminedformany valuesof

4

^and

. ^We

^conjecture

that if

[(1 e)/(1 + e)] < <

^and

[(1 e)/(1 + e)] < <

1, then the omitted set

F

hasan

analytic continuation that is the real axis.

Otherwise,

^for the allother values of and

, ^F

hasananalyticcontinuationthatspiralstoward the origin.

Acknowledgments. This paper formspartof the author’s doctoral dissertationwritten at the

State

University of

New

York at Albany. The author wishes to thank Professor Donald Wilken and Professor

Y.J. Leung

for their helpwhile this workwas on progress.

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