further generalizations of the grunsky inequalities

FURTHER GENERALIZATIONS OF THE GRUNSKY INEQUALITIES*

By

R. N. PEDERSON AND M. SCHIFFER in Pittsburgh Pa., U.S.A. in Stanford, Calif., U.S.A.

Introduct ion

Grunsky's inequalities are based on the obvious fact that a function

(0.1) f ( z ) = z + ~ a,,z", ) ) = 2

analytic in the unit disc, is univalent if and only if the series generated by

(0.2) .log f ( z ) - f(~) _ ~ djkZl~ k z - - ~ j , k = O

converges for I zl < 1, Icl < 1 Grunsky i-11] showed that this is the case

if and only if for each complex vector (xl,x2,. . . ,x, ,) we have

/1 rl

(0.3) ~E dj xjxk [ X I x, I s l,k=i - - k=t k

In recent years, the necessity of the condition (0.3) has played an important

role in the solution of various coefficient problems. The most famous of these

problems is the Bieberbach conjecture which asserts that if a function nor-

malized as in (0.1) is analytic and univalent in the unit disc, then

(0.4) la.I _~.

with equality holding only for a Koebe function

* This work was supported in part by contract AF 49(638) 1345 at Stanford University and by NSF grant GP-7662 at Carnegie-Mellon University.

353

354 R. N. PEDERSON AND M. SCHIFFER

_ z ] ~ 1 = 1 . K,(z) (1 - ~z) 2 '

The conjecture was settled for n = 2 by Bieberbach [3] and for n = 3 by

Loewner [16]. The original proof for a 4 is due to Garabedian and Schiffer [9].

In 1960 a simplified proof that l a,] ___< 4 was obtained by Charzynski and Schiffer [5]. In this paper, as in Bieberbach's, it was the Grunsky inequality

for the odd function x/f(z z) which played a decisive role. The work of

Charzynski and Schiffer revealed two important notions. It showed that a

two by two truncation of the Grunsky inequality is sufficient to settle the

question of a4. It also indicated the feasibility of obtaining an estimate for

Re{a,) which depends only on Re{a2}. Garabedian, Ross and Schiffer [10]

obtained the local Bieberbach theorem for the even coefficients a2, by using

an n x n truncation of the inequality (0.3) corresponding to the function

x/.[(z2). Ozawa [19] extended the local result for a 6 to the global theorem.

Pederson [20] obtained an independent proof of the latter result by using

an equivalent formulation of (0.3), see [2], [13], [17], [21].

It appears that the classical form of the Grunsky inequalities sheds little

light on the total solution of the coefficient problem for the odd coefficients.

Garabedian and Schiffer [9] obtained a new set of inequalities which are

similar in form to (0.3) but where the coefficients depend on an omitted value

1/u of the function f(z). Whereas in the inequality used for the even coeffi-

cients, the univalent odd functions played an important role, the new inequality

stems from the Bieberbach-Eilenberg functions.

The beginning point of the investigation of Garabedian and Schiffer [9]

is to replace the generating function (0.2) by

(0.6) M ( z , ~ ; u ) = l o g - - x / l - uf (z ) - ~ ) I - u f ( ( )

[ , / i - . f ( z ) + 4 1 -

where 1/u is outside of the range o f f . Near the origin M has a power series

expansion

(0.7) M(z,~;u) = ~ Cjk(U)ZJ~ k. j,k=O

FURTHER GENERALIZATIONS OF THE GRUNSKY INEQUALITIES 355

The authors showed that if f(z) is univalent in the unit disc, then for each

set of complex numbers (2o,2D...,2n), 40 real, the bilinear form

(0.8) p(u) = ~ Cjk(u)2j2k ].k=O

satisfies the inequality

n 12 12 (0.9) Re{p(u)} =< Y~ j = l J

The proof of Garabedian and Schiffer used variational methods. An elementary

proof, related to the area theorem, was obtained by Hummel and Schiffer [12].

Nehari [18] gave a generalization of the method in [12] and obtained a large

number of important inequalities relating coefficients and omitted values of

univalent functions. It appears likely that the inequalities (0.9) would be

useful in attacking the odd coefficients provided that one had sufficiently

precise information about the omitted values o f f . Such information, however,

is difficult to obtain.

The problem of locating the omitted values of f was circumvented by

Garabedian and Schiffer by imposing the constraint

(o.lo) Op 0u = 0

and by requiring that the rational function

(0.11) Q(O = {: 2 ,z - ' , X, = 2 _ , , v ~ - - r l

have no roots off of the unit circumference. Under the conditions (0.10) and

(0.11) the inequality (0.9) was proved to be valid with 1/u not necessarily

an omitted value of f .

Garabedian and Schiffer proved the local theorem for the odd coefficients

as a consequence of (0.9), (0.10) and (0.11). When n = 1, the result implies

an inequality of Jenkins [13] from which the global theorem for a3 can be


deduced. In the case n = 2, the inequality is equivalent to an inequality of

Garabedian [6], see also [7]. An independent proof of the local Bieberbach

theorem has been obtained by Bombieri [4], who used the Loewner repre-

sentation and shows that the Loewner parameter functions near x = - 1

leads always to lesser values for Re{an}.

It is the purpose of this paper to obtain a sharp inequality for Re{p(u)}

defined by (0.8), in the case where 1/u is a value assumed byf(z) . The inequality,

as one would expect, depends on the point at which the value 1/u is assumed.

We shall show that if we impose the conditions (0.10) and (0.11), the new

inequality reduces to the inequality of Garabedian and Schiffer. One novelty

of our approach is that we circumvent the lemma on solutions of complex

partial differential equations due to Berg [1] and Lewy [15] which was used

in [9].

1. Formulat ion of the Variat ional Problem.

Let ~ denote the class of functions, normalized by (0,1). which are analytic

and univalent in the unit disc. In this section we shall derive the differential

equation for the function which maximizes the left side of (0.9) with

1 (1.1) u = f ( r ) ' 0 < r < 1.

After this problem has been solved, the problem corresponding to u = 1/f(a) ,

t a l < 1, can be obtained by a rotation.

For fixed r and a given set of complex numbers (20, 21, " ' , 2n), we consider

the functional

(1.2) O[/ ] = Re Cjk(U ) t j , k = 0

within the class ~ and ask for its maximum in ~ . The coefficients cjk(u) are

defined by (0.7). By differentiating (0.6), we obtain the identities

(1.3) f ' ( z ) ~(1 -- uf(~)) 1

M ( z , ( ; u ) = f ( z ) - f ( ~ ) ~- uf ( z ) ) z -


and

(1.4) M(z, ~ ; . ) = 1 . . . .

Udu ~-~(1 - uf(z)) ( 1 - uf(O)

In order to characterize the extremum function J(z) , we may by the general

theory of [22] subject it to a boundary variation of the form

e i~p 2

+ f ( z ) {- O(p3) P > 0 (1.5) f * ( z ) = f ( z ) - - w ' '

where w is a boundary point of the extremal image domain. By definition

(0.6) and the identities (0.7), (1.3) and (1.4), we find that

(1.6) ~, 5Cik(U)ZJ( k = ],k=O

ei~ 2 / . p x/(,1 - u f ( z ) ) ( 1 - u f ( O ) + oC).

The variational result (1.6) leads us to define the generating function

(1.7) 1 #1 - u/(z)

- ~, mj(u ; w)z j , x/1 - uw f ( z ) - w) j=o

the series development being valid near the origin. As a consequence of (1.7),

we have the identity

(1.8) i~t 2 ~ W ? 5Ctk(U ) = -- e p mi~u; ) ng(u; w) + O(pa).

Thus, under the variation (1.5) of the extremum function, the functional

(1.2) varies according to

(1.9) 6 ~ [ f ] = - Re e'~p2 2jrrtj(u ; w -t- O(p3). ,,. L j = O

In view of the maximum property of the functional ~ [ f ] , the right-hand

side in (1.9) must be nonpositive. By invoking the general theory of boundary

variations [22], we deduce that the boundary of the image domain consists

of analytic arcs which satisfy the differential inequality

358 R. N. PEDERSON AND M. SCIMFFER

(1.1o) (~dw\ 2 r . 2 ) < o .

I f we parametrize the boundary by

w = f (e" ) , z = e",

the inequality (1.10) reduces to

(1.11) z f ' ( z ) ~ 27nj(u;f(z)) = real, lzl = 1. j=O

In order to study the analytic dependence of mi(u;w) upon w, let us write

(1.7) in the form

(1.12) ~ / 1 - u f ( O 1 = _ ~ mj(u;f(z))~l ' x/1 - uf(z) " f (z) - f ( O j=o

a series development which is valid for ~ near the origin and z close to the

unit periphery. Hence by (1.3)

(1.13) f ' ( z ) x / l -- uf(O _ = -- ~ mj(u;f(z))f ,(z)~ I [ f ( z ) -- f ( O ] x/1 -- uf(z) J = o

= O M ( z , ~ ; u ) + 1 z - - ~

Using the series development (0.7) and comparing equal powers of ~, we ob-

tain

L]

1 ~ j%(u)zj. (1.14) --zf ' (z)mk(u;f(z)) = ~ + j=o


Evidently, Fro(u, t) are polynomials of degree rn in t. Thus we have by (1.7)

. . . . . ~ - - r . u, w" (1.16) rnj(u; w) = x/1 - uw

We read off from (1.14) that

(1.17) -- z f '(z)mk(u ;f(z)) = z f ' ( z ) Fk (u'f~z))

f ( z ) ~/1 - uf(z)

1 ~ jcjk(u)zj. -- zk + j=o

The differential equation for the boundary arcs now becomes

( 1 . 1 8 ) H(z) = zf'(z) . 1_ ]~ 2jF/u, 1--~-] = real, }z I = 1 . f (z ) x/1 - u f (~ i =~ \ 'f(z)]

It follows from (1.14), (1.16) and (1.18) that H(z) has a pole at z = 0 with

local development

( 1 . 1 9 ) H(z) = s )~ s s 2jkCki(U)Z k j = o z7 +/=o ~=o

and is otherwise analytic in the unit disc except for a branch point at z - - r

since the square root vanishes there.

Consider now the function

(1.20) M ( z ) = ~ ( z - - r)~ - rz) H(z).

In the unit disc M(z) has a simple branch point at z = 0 and is real for [ z[ = 1.

By the Schwarz reflection principle, it can be continued into the entire complex

plane and has the form

(1.21) M(z)= s [ lj ]


and (1.18) goes over into the equation

(1.22)

(1) zf'(z) 1 ~ ).jFj u,f--~ = f(z) ~/1-- uf(z) j=o

1 ~ Elf_ j + ]jz,+t]. a/(z - r)(1 - rz) i =~

We also have, as a consequence of (1.19), (1.20) and (1.21)

(1.23)

1 j=O "~ + k~-- I ) 'jkcjk(u)Zk -- ~r ly

2. The Fnnc t iona l Corresponding to the Extremum Funct ion.

Our goal is to deduce the value of Re{p(u)} for the extremum function f(z) from the formulas (1.22) and (1.23). As a first step in this program we obtain

identities relating the sets of parameters {/j} and {2j}. The Legendre poly-

nomials play a significant role in this study.

Let

(2.1) x = ~ r + , x > l ,

and define the sequences {rim(x)} and {P,.(x)} by the generating functions

(2.2) .if1 - 2xz + z z = ~ 1-Im(X)Z" m=O

and

(2.3) 1 = ~ Pm(x)z", x/1 - 2xz + z 2 ,,=o

where the Pro(X), of course, are the Legendre polynomials.

By comparing powers of z in (1.23), we obtain the identities

(2.4) k ~ 2jCjk(U ) -- j=o

1 ~ [ijPj+k(X) + ljpk_j_x(X)] ~/~-r j=o

FURTHER GENERALIZATIONS OF THE G R U N S K Y INEQUALITIES 361

and

(2.5) lj = 4-----~ ~ 2#~p_j(X). p=j

Substituting (2.5) into (2.4), and noting that ~/-----~ is purely imaginary, we

obtain after a change of summation indices,

(2.6)

J

k Y. )qcj~(u) = ~2 Z Pj+~_p(x)rl.(x)2j - j = O j = O p=O

n j

~, ~, Pk+o-j-x(x)l-lv(x)2j. j=O p=O

We turn now to the problem of evaluating

(2.7) Re Coj(U)2j J

for the extremum function. We deduce from the series development (0.7)

(2.8) M(0, ( ;u) = ~ Cok(U)~ k. k=O

On the other hand, we have as a consequence of (1.3) the identity

(2.9) i f(r) ; dM(z, ~; u) f dw x/1 - u f (~ dz

-~z d z = j -- �9 w-S(O Z f(z) Z

Since f (r ) = I /u, it follows from (0.6) that

(2.10) M(r, ~; u) - M(z, (; u) = log - - - 1 ~ cj,(u)Fr - - r j , k = o

It then follows from (2.9) and (2.10) that

(2.11)

u-I f dw x/1 - uf(~) 1

w = log - z

f(z)

c ~(u)zJ~ k . ],k=O


In view of (1.15) we obtain by comparison of equal powers of ( in (2.11)

(2.12) ilt f dt F m u, --[ dt = 1 E Cjm(U)Z 1

f (z) 4 1 - - u t t m z m 1=o

for m > 1 and

(2.12') f!) x/1 ~- ut -t dt = log(1---~) - ~oCjO(U)Z~.

We use now the identity (1.22) and (2.12), (2.12') to find

(2.13)

r

f ~ / ( ~ - r)(1 - r~)

-- ~.o,o,(- + ~. ~, ~ ~,~,~.~z,. J=1 jzJ J=O p=o

Taking the "part fini" of both sides of (2.13) at z = 0, we find that

(2.14) ni~,o_~Cpo(U)2p=p.f. ~ dr 1 ~, [ 11 ] ~- ~ + / ~ j �9

=o ~/1 - 2x~ + ~2 ~ / - - 7 j = o 0

3. F o r m u l a s for the C o e f f i c i e n t s .

It is a consequence of (2.6) that for the extremum function f(z) we have

(3.1) Re { ~ ~ Cjk(U)Aj2k}=Re {~= ~,[A.ik(X)2jAk+Bjk(X)2j~k]} k = l j = O k 1 1 = 0

where

] (3.2) kAjk(X) = ~ P]+k_p(X)Hp(x)

p=O

and

F U R T H E R G E N E R A L I Z A T I O N S OF THE G R U N S K Y I N E Q U A L I T I E S 363

J

(3.3) kB, k(X) = - ~ Pk+p-j- t (x)IIp(x) . p = O

It follows from (2.5) and (2.14) that

n el

where Aio(X ) and Bjo(X) are real except for Aoo(X) and independent of the 2's.

The latter fact, together with the symmetry of the elements cjk(u), allows us to

conclude that the coefficients Ajo and Bio are equal to Aoj and Boj, respectively.

Thus, (3.2) and (3.3) provide us with all of the coefficients except Aoo and Boo.

We conclude from (2.14) and (2.5) that

r

(3.5) Aoo(X) = _ p.f. f _dz_ = - l o g - 4 r ~/1 - 2x~ + T 2 1 - - r 2

and

r

(3.6) B~176 = ~/1 - 2xz + 3 2 - r) ' 0

or, in view of (2.1),

(3.5')

and

1, [1 - x2] Aoo(X) =

1, Ix + 1'~ (3.6') Boo(X) = x l o g / ~ p .

~ \ X - - l ]

4. Generating Function for Matrices.

The purpose of this section is to put the formulas for the coefficients Ajk(X )

and Big(X) into more convenient form. To this end, we consider the generating

function (0.6) for the particular function

364 R. N . P E D E R S O N A N D M. S C H I F F E R

(4.1) f ( z ) = z 1 1

( l + z ) ~ ' f ( r ) = u=2(--l+--x)"

Thus we define the coefficients ~..gjk(X ) by

~/1 2(1 + x)z ~ / ; 2(1 + x)~

(4.2) log - - ( 1 q - z ~ - - (1 +~)2 _

(1 + z) 2 + 1 (1 g r J ~z - r

gojk(x)zJr ~.

Differentiating both sides of (4.2) with respect to z and using (1.3) and (4.1),

we find that

(4.3) (1 - z)(1 + () ~/1-- 2x( + (2 1 _ • jgajk(X)Zi-l~k, ( Z - ~)(1 - z~) ~/~ Z-Zx z + z I z - ~ S=k.0

or, after rearranging,

(4.4) 1 1 ] x / ! - 2 x ( + ~ 2 1 ~o t~,.

' z - ~ - i - ~ ,/1 - S z + Z -~ - z - - ~ = s,~o j~~

We develop the left side of (4.4) into a power series in z and ~ obtaining

(4.5) z p - ~P 1 Z rI,(x)~'Pp(x) ErL(x)Pp(x)~'z ~= E jeaj~(x)zJ-~ k.

~t't-fl>0 Z - - ( 1 - - Z ( j , k = O

Comparing the coefficients of z i-~(k on both sides, we find that

k k

(4.6) JPik(X) = ~ H~(x)Pi+k-,(x) - E H, (x)Pi -k- I+, (X) . �9 = 0 ~ = 0

By comparing (4.6) with (3.2) and (3.3), we see that

(4.7) J g) iR(X) = jAjk(X) + jBlk(X)"

It therefore follows from (4.4) and (4.5) that we have the identities

(4.8) 1 I~/~i--2x~ ~-~--2 1] =

-" - ~ - 2 ~ + ~ 2 jAik(x)z 1- l~k

j , k = O


and

(4.9) 1 4 1 - 2x~ + ~2

. . . . . . . . . . . . . ~, jBjk(X)Zi- t~k. 1 - z ~ n/ l _ 2xz + z 2 j.k=O

Let

(4.10) w(z) = nil - 2xz + z z

and form

(4.11) A(z, ~) = log 1 - x(z + 0 + z~ - w(z)w(O

(z - 0 z x/x z - 1

This is a symmetric function in z and ( and we find easily

OA 1_!__ tw(O \ (4.12) = - 1 1 "

O z z - ~ \ w ( z ) .1 '

thus, by comparing with (4.8), we see that (4.11) generates the coefficients

Ark f o r j + k r

(4.13) log 1 - x(z + 0 + z~ - w(z)w(O = ~ Ajk(X)zI~k "

- ( z - 0 2 ~ / ~ - - 1 j,k=o

That (4.13) generates Aoo(X ) given by (3.5') is verified by direct computation.

Similarly, we can deduce that

(4.14) B(z, 0 = log w(z)w(O + x(1 + z O - z - ~ _ ~ Bjk(X)ZJ~k

( 1 - zO~/x ~ - 1 s , ~ = o

as a consequence of

aB 1 w(O (4.15) Oz - 1 - ~z w(z)

and (3.6').

366 R. N. PEDERSON A N D M. SCHIFFER

We shall now derive convenient formulas for the coefficients Ajk and Bjk

in terms of the Legendre polynomials. Multiplying (4.8) by z, we obtain

(4.16) z [ 1 - 2 x ~ + ~ z _ ] = ~ / z - ~ x / l - 2 x z + z z ~/i - 2x~+ ~2 1 j,k Ajk(X)zJ~k"

By adding (4.16) to its value with z and ~ interchanged, using the symmetry

of the coefficients Ajk(X), we get

(4.17) 1 - z~ - 1 = ~ ( j + k)Ajk(x)zJ~ k. x/1 - 2xz + z 2 ~/1 - 2x~ + ~2 j,k=0

It follows that

(4.18) ( j + k)Ajk(X) = Pj(X)Pk(X) -- P j - I ( X ) P k - I ( x ) , j + k ~ O.

In a similar fashion we obtain from (4.9) the identity

(4.19) z - ~ = _ ~ ( j _ k)Bjk(x)zj~k n/ l _ 2xz + zZn/ l _ 2x~ + ~2 j,k=o

which implies that

(4.20) (j - - k ) B i k ( X ) = P j ( x ) P k _ l ( x ) - - P j _ x (X)PR(X ) .

In order to compute the diagonal elements Bjj(x), we set z = ~ in (4.9) to

obtain

1 _ ~ jBjk(X)Z.i+k_ k (4.21) 1 - - Z 2 J,k=O

Since the left side of (4.21) is an even function, we have

2k

(4.22) • jBj,2R-~ (x) = O. J = O

It follows that


k - I 2k

(4.23) kBkk(X) = -- ~, jBj ,2k- j (x) -- • jBj ,2k- j (X) . j=O j = k + l

Replacing the index in the second sum on the right by 2k - j , und using the

symmetry of the coefficients, we obtain

k - I

(4.24) Bkk(X ) = --2 ~Z Bj,2k_j(X ). j=O

It then follows from (4.20) that

I , - 1 P j ( x ) P z k - j - ~(x) - P j_ t ( x ) P 2 k - j ( x ) (4 .25) B~(x) = Z k # O.

1 = o (k - j ) '

Since Pk(1) = 1, it follows from (4.18), (4.20) and (4.25) that

(4.26) Ajk(1) = 0 for j , k > 1,

1 (4.27) Ajo(1) = _ for j => 1,

J

1 (4.28) Bjk(1) = j Ojk for j , k > 1,

and

1 (4.29) Bi0(1 ) = J for j ~ I .

5. The Expl ic i t Inequal i ty .

By collecting the results of the previous section, we obtain the following

sharp

Theorem 5.1. Let f normalized by (0.1) be analyt ic and univalent

in the unit disc and set u = 1/ f ( re~) , 0 < r < 1, x = ~ r + . I f the co-

efficients c ik(u) are defined by the generat ing funct ion (0.7), then for any

set of complex numbers (2 o, 2 D . " , 2.) we have


/ n } (5.1) Re Z Cjk(U)2j2k -- ~ Im(2~)

~j,k = 0

I ~ e-i(j+k)'~tj~k e- l (J-k)~j~k]} < Re [Ajk(x) + Sjk(X) . ~j,k = 0

The matrices (Ajk) and (Big) are defined by the generating funct ions (4.13)

and (4.14) or by the explicit formulas (3.5'), (3.6'), (4.18), (4.20) and (4.25).

Proof . The proof for the case ~ = 0 is an immediate consequence of

the previous sections. In order to dispose of the general case, we apply the

special case to the function

(5.2) f * ( z ) = e- i ' f (e i 'z) = z + ~ e l(n-l~ a,z ~. n=2

Clearly, with

(5.3) U* ~ 1 = el~u f* ( r )

whence

(5.5) c~k(u*) = CjR(U)e l~i+k)~ , C*o(U*)= Coo(U) + i~

Hence, by applying the special case (~ = 0) to f* ( z ) at z = r with the param-

eters (2o, e-'i~;q, ..., e-in~2n), we obtain the general inequality (5.1).

6. The Cond i t i on p'(u) = O.

The reader will observe that the inequality (5.1) gives information about

the points u which can be assumed by a univalent function at a point

z = re ~. Conversely, given a value u, one can obtain information about the

(5.4) M*(z,~; u*) = ~ Cjk(u)et~J+k)'zt~k + i~, j,k=O

we have, in view of (0.6) and (0.7),


points z = re ~ for which the value u is assumed by some univalent function.

Garabedian and Schiffer were successful in proving the local Bieberbach

theorem by imposing restraints which precluded the possibility of the ex-

tremum function attaining a given value u in the interior of the unit disc.

A reasonable question to ask is whether similar constraints imply that the

extremum function for the inequality (5.1) must attain a value u on a

specified subset of the unit disc. The condition

(6.1) p ' (u) = 0

is again relevant in answering this question. We differentiate the identity

(0.7) and use the identity (1.4) to find

(6.2) c;~(u)zJr = 1 _ ~ _ _ 1

j , k = O U x/1 _ u f ( z ) a l l u f (~j

Thus, if we define d.(u) by

(6.3) 1 - ~ d, (u)z" . x/1 - u f ( z ) ~=o

we arrive at

(6.4)

h e n c e

(6.5)

The condition

c j , ( u ) = !dj(u)d~(u);

1 ~ . jd j (u ) . p'(u) = u j

(6.6) p'(u) = 0

together with an additional constraint will lead to a generalization of (5.1)

which includes the inequality of Garabedian and Schiffer E83 as a special

case.

370 R . N . PEDERSON AND M. SCHIFFER

7. Properties of the Bound r

Let us define the right side of the inequality (5.1) by

(7.1) g(x ; a) = Re ~ Ajk(x)e- i(j +k)at2j2k ~,j,k=O

+ ~ Bjk(x)e-iO-k)'2~.lk, j,k=O

We then have

(7.2) Og 1 ( 1 ) a8 o-7=~ 1-~- ~-x"

By differentiating the identities (4.18) and (4.20), we find that

(7.3) ( j + k)Ajk(X ) = Pj(X)Pk(X) + Pj(x)P'k(x)

- P~_ ~(x)P,_ l(x) - e j _ ~(x)P~_ l (x) ,

and

(7.4)

X ----

j + k # O

(x 2 -- 1)P'_t(x ) = nP,(x) - nxP ,_ t (x )

(x 2 _ ~ { x P j(X)Pk(X) - P j - t(x)Pk(x) - P.i(X)Pk- t (X) + xP 1_ t(x)ek_ I (x)},

(7.5)

and

(7.6)

imply that

(7.7) 1

A' j k ( x ) =

and

(x 2 - 1)P',(x) = nxP,,(x) - uP,,_ t(x)

( j -- k)Bjk(x ) = P}(x)P k_ t(x) + Pj(x)P~_ t(x)

- P'j_ ~(x)Pk(x) - P j_ ~(x)e~(x), j - k ~ o.

These, together with the familiar identities between Legendre polynomials


1 (7.8) B'jk(X) = (X 2 - - 1) {xPJ(X)Pk- l ( x ) - Pj- I(x)Pk- l ( x ) - - PJ(X)Pk(x)

+ xP,_ i(x)Pk(x)}.

The reader will observe that (7.7) and (7.8) now hold for all values o f j and k.

Now define

(7.9) p(x,e ~) = ~ 2kPk(x)e -tk~ , k=O

(7.10) a(x,e ~) = ~ 2kPk_x(x)e -sk~, k = l

and

(7.11) Q(x, e I') = p(x, e i~) + tr(x, ei").

As a consequence of (7.9) and (7.10), we have

(7.12) dx 1 Re{xp z - 2ap + xa 2 + 2xp~ - l p [ 2 - 1 12} ( x 2 - 1)

1 Re(x(~ + o.)2 _ ] ~ _1_ crl2 } ( x 2 - 1)

1 1 Q ) 2 (Ira Q)2. (x + 1) (Re (x - 1)

i

By differentiating (7.1) with respect to a and using (7.9) and (7.10), we obtain

(7.13) _~8_ = im{p 2 _ a2 + pa - ~tr} = - I m { Q 2} = - 2 Re{Q}Im{Q}. da

In the case where ;t o is real, we are now able to cast Theorem 5.1 into the

following elegant form:

T h e o r e m 7.1. I f in addition to the hypotheses of Theorem 5.1 it is

assumed that :to is real, then


(7.14) .~

Re{p(u)} < ~ Xk k =1 k �9 Rei'Q(t' ei~'J2 )]

1

/ (t - 1) Imi'Q(t'e'~)]2] at"

P r o o L have

(7.15)

Re{p(u)} < 8(y, ct) +

It follows from (5.1), (7.11) and (7.12) that for any y > 1 we

x

f { ReEQ te Y

Im t ia 2 1 1) l-Q( ,e )] }at. ( t -

For real 2 o, it follows from (3.5') and (3.6') that

(7.16) 22Aoo(Y) + 22Boo(Y) = ~ log (y + 1)2 4

~0 as y ~ 1.

By (4.20)-(4.24) we see that

n 2

(7.17) lim {8(y, c0 2 _2oAoo(Y )_2~Boo(y)} = ~ 12k] y"*l k = ! k

The proof is completed by letting y ~ 1.

The reader will observe that when x = 1 (hence r = 1), the inequality

(7.14) agrees with the inequality of Garabedian and Schiffer 1"9]. It is reasonable

t ~ ask whether for any r and ct the second term on the right of (7.14) can be

negative and hence yield an improved inequality. The following theorem

shows that the right side of (7.14) achieves its minimum on the boundary

x = l .

Theo rem 7.2. The funct ion 8(x, ct) cannot achieve an interior maxi -

m u m or m i n i m u m in the set 1 < x <_ Xo, 0 <_ ~t <_ 2~.

Proof . The function Q (and hence 8) is clearly an analytic function of

the real variables (x, ~). Let # be the smallest integer such that all derivatives

of Re(Q} and Im{Q} of order < # vanish at (Xo,%) but that either Re(Q}

or Im(Q} has at least one nonvanishing derivative of order # at (Xo, %). It is

an easy consequence of (7.12) and (7.13) that the first nonvanishing derivative


of ~ at (Xo, C~o) is of order 2p + l . It follows that 8 cannot have an interior

maximum or minimum. That the minimum is achieved for x = 1 follows

from (7.12), (7.13) and the fact that cf(l,~) is constant when 2o is real.

The above theorem is valid whether or not 20 is real. However, if 2 o is not

real, 8 is unbounded from below at x = 1.

We are now in a position to deduce a generalization of the special case

corresponding to v = 0 of the inequality (1.78) of Garabedian and Schiffer

[8] from Theorem 7.1.

Corollary 7.3. Suppose that f ( z ) is analytic and univalent in the unit

disc and does not assume the value l/u for [z I < r o. Then

(7.18) Re{p(u)} < max 8(Xo, a).

P r o o f . Let F,o denote the image, under f , of the circle I zl = ro" Then

Re{p(u)} is a harmonic function of 1/u in the exterior of F,o and assumes

the value - oo or 0 at u = 0 according to whether 20 r 0 or 20 = 0. In either

case Re{p(u)} assumes its maximum, in the exterior of F, o, on F,o. But then

u = 1/f(ro ei~) for some 0~. The conclusion follows immediately from Theorem

5.1.

8. Properties of Q(x, ~).

Our previous results indicate the desirability of determining the location

of the roots of

(8.1) Q(x,r = ~ 2kPk(x)~ k + ~ Ikek_,(x)~ -k. k = O k = l

Garabedian and Schiffer proved that if

(8.2) Q ( 1 , 0 = o

has all of its roots on the unit circumference, then

(8.3) Q(x, e i~ = 0

374 R . N . PEDERSON AND M. SCHIFFER

has no real solutions 0 for any x > 1. The proof was based on a lemma of

Berg [1] and Lewy [15] concerning univalent mappings which satisfy certain

elliptic partial differential equations. In this section we shall derive an inde-

pendent proof of this result which yields the stronger assertion that (8.2) is

equivalent to the statement that

(8.4) Q(x , ( ) = 0

has all of its roots in the interior of the unit disc. The first step in our proof is

Theorem 8.1. I f for some Xo >= 1, Q(Xo,() = 0 has all o f its roots in

the interior of the unit disc, then so does Q(x ,O = O for each x > x o.

(8.5)

and

Proof. Observe that

R(x, 0 - (Qr ~) = k[2kPk(x)~ k - i~kpk_ t(X)~ -k] k = I

(8.6) Qx(x,() = ~ [2kP~(x)( k +'~kP'k-t(X)(-k]. k = l

Substitute the identities (7.5) and (7.6) into (8.6) to obtain

(8.7) n

_ 1 k ~1 k [2k(xPk(x ) -- Pk- 1 (X))~ k + Ik(Pk(X) -- XPk- I(X))(- k]. O~(x,O ( x 2 - 1 ) =

It then follows from (8.5) and (8.7) that

x (8.8) Qx(x,O - (x z _ l ~ R ( x ' ( ) + (x 2 _ 1) R x, .

Now consider

re(x) = rain ]Q(x,~)l. I~l__. t


If the roots of Q(x, ~) lie inside of the unit circumference, then this minimum

is positive. Moreover, since I o<x, o l - , oo .s Ir ~ , it must be assumed

on the unit circle. If the minimum is achieved at ~(x), Ir l = 1, we have

re(x) = [ Q(x,r (8.9)

and

(8.10)

A computation shows that

(8.11)

Q;(x, r ~ o.

I Q~(x, ~o) ~---x l~176 Re{~xl~ = Re [ Q~,~o) }

- 1) Q~ ,~o) + (x~- 1)

by (8.5) and (8.8).

We have

(8.12) ~oQr x, ~o) = ff--~ Q(x, r(o) I" ='

and hence, since by the minimum proper tyQr= real,

x 0 l~ I Q[ + Re (8.13) logle(x,r -- (x 2 - 1) 0r 2(x 2 - 1) Q-~

01QI*} 0r "

since a lQ [~/0r >- 0, it follows that

I 1 (8.14) log lQI _-_ 12_1 1 2 - 1

_ 1 ~ l o g l Q I (x + 1)

As a consequence of (8.10) and (8.12), the right side of (8.14) is positive. Hence

I Q(x,(o)] increases with x. It remains to show that m(x) increases with x.


Now either [ Q(x, e ~) 12 is constant in 0t for x = Xo or, since it is a real analytic

function, it has at most a finite number of minima. If it is constant, then

m(Xo)= Ia(xo, e'~)[ for every ~, and it follows from (8.14) that I~(x,e'~)l increases with x for each 0t; hence re(x) does also. If IQ(x,e '~) 12 is not con-

stant, then at each absolute minimum point its first nonvanishing derivative

with respect to ~ is of even order and positive. This fact together with (8.14)

implies that I Q(x, e ~) I increases with x for each ~ in a neighborhood of each

absolute minimum point. It follows that re(x) increases with x at Xo.

The previous considerations imply that once all of the roots of Q(x, ~) = 0

enter the interior of the unit disc, they can never return to the boundary.

This completes the proof of Theorem 8.1.

We next prove

Theorem 8.2. I f in a neighborhood of Xo, ((x) is a simple root of

Q(x, ((x)) = 0 satisfying Ir l -- 1, then

( 8 . 1 5 ) d 1

_1 < l~162 l =< ( x o + l ) (Xo 1) = ~x=~o

Proof . Differentiating the relation Q(x,((x)) = O, we obtain

1 d( Qx(x,O ( 8 . 1 6 ) - - - = dx ~Qdx, O"

Taking real parts of both sides of (8.16) and using (8.5) and (8.8), we have

(8.17) Xo 1 Re/JR(x o, (o)]2/" ,

The conclusion (8.15) is immediate.

We are now in a position to obtain a particularly simple proof of the follow-

ing theorem, the sufficiency being due to Garabedian and Schiffer [-9, pp.

21-24-].

Theorem 8.3. I f 20 is real, then a sufficient condition that Q(x,e l~ = 0

have all of its real solutions 0 lying on the line x = 1 is that


(8.18) Q(I,~) -- 0

have all o f its roots on the uni t per iphery .

Proof . Consider first the sufficiency in the case that the roots of (8.18)

are simple. It then follows from Theorem 8.2 that in a deleted neighborhood

of x = 1 (x > 1) Q(x, ~) = 0 has all of its roots in the interior of the unit disc.

As a consequence of Theorem 8.1 they can never return to the unit circum-

ference.

If Q(1,() has not necessarily distinct roots

~1, ' " ,~ . , /~1,'",/~.,

then the complex numbers

elktek, e-~ktflk, k = 1 ,2 , ' . ' , n ,

are distinct for all sufficiently small positive t. Define the complex numbers

,2.k(t ) by the identity

(8.19) ~-" fi (~ - eU"o~j)(~ - 6 -ik' flj) = k = l

, ~ k ( t ) ~ ~ . k = - - n

A symmetry argument implies that 2k(t ) = /~_k(t)'. NOW consider the poly-

nomials

n

(8.20) Q,(x,~) = ~ ~.k(t)Pk(x)~ k + ~ 2k(t)Pk-l(X)~ -k �9 k = 0 k = l

Since the Legendre polynomials satisfy Pk(1) = 1, we may deduce from (8.19)

that Qt(1, 0 has simple roots of absolute value one. It follows from Rolle's

theorem that each of the functions

d k (8.21) (i(~--() Q,(1,() = 0

has simple roots, all lying on the unit circumference. The argument of the

previous paragraph then implies that


(8.22) d k

has all of its roots in the interior of the unit disc for each x > 1. Hence, by

letting t - , 0, we conclude that the roots of

are in the closed unit disc for each x _~ 1. Suppose that for some Xo > 1,

Qo(x, ~) = 0 has a root ~o of absolute value one. Let its multiplicity be k.

Then ~o is a simple root of

'

(8.24) i( -d~l Qo(xo,() = O.

But then (8.15) implies that for some x < Xo, (8.24) has a root of absolute

value greater than one, which is impossible. This completes the proof.

We now complement Theorem 8.3 by proving

Theorem 8.4. I f 2 o is real, then a necessary and sufficient condition

that Q(x, e i~ = 0 have all of its real solutions 0 on the line x = 1 is that

Q(x,~) = 0 have all of its roots in the interior of the unit disc for x > 1.

P r o o f . The proof of the sufficiency is obvious. In order to prove the

necessity, we suppose that for some Xo > 1, Q(x, ~) = 0 has a root of absolute

value greater than one. Since the Legendre polynomial Pn(x) is positive for

x > 1, the condition Q(x, ~) = 0 may be written

I t follows that for large x all of the roots lie in the interior of the disc. But

then, by continuity, there must be a root on the unit circumference for some

x > x o . This is a contradiction.

9. Another Grunsky Type Inequality.

In this section we obtain a generalization of another inequality of Gara-

bedian and Schiffer [9, p. 19].


T h e o r e m 9.1. Suppose that f ( z ) is normalized, analytic and univalent

in the unit disc and that u is a complex number satisfying

(9.1) p'(u) = O.

I f x o > 1 and Q(xo,~) has all of its roots in the interior of the unit disc or

if x o = 1,2 o is real and Q(1,() has all of its roots on the unit circle, then

(9.2) Re{p(u)} < max g(x o, ~). r

P r o o f . If (9.1) is satisfied, then either u - t is an omitted value of the

restriction o f f to ] z [ < ro, or u = 1/f(re '~) for some r < r o . in the first case

the conclusion follows from Corollary 7.3. In the second case we consider

the extremum function f for the class of all univalent functions satisfying

this condition. By following the variational technique, see [9, pp. 18-20],

it is shown that

(9.3) Re{p(u)} < Re p = ~f(x, fl), x = ~ a +

where pe ~p, p < ro, is the point at which the maximum is achieved. The

condition (9.1) then implies that

08 06' (9.4) . . . . 0. op a/~

But by (7.12) and (7.13) this is equivalent to

(9.5) Q(y,e ' p ) = 0 , y = ~ p + �9

An appeal to Theorems (8.1) and (8.3) then shows that p > ro. The conclusion

of Theorem 9.1 then again follows from Corollary 7.3.

REFERENCES

I. P. W. Berg, On univalent mappings by solutions of linear elliptic partial differential equations, Trans. Amer. Math. $oe., 84 (1957), 310-318.

2. S. Bergman and M. Schiffer, Kernel functions and conformal mapping, Compositto Math., $ (1951), 205-249.


3. L. Bieberbach, Ober die Koeffizienten derjenigen Potenzreihen, welche eine schlichte Abbildung des Einheitskreises vermitteln, Preuss. Akad. Wiss. Berlin, Sitzungsberichte 1916, 940-955.

4. E. Bombieri, On the local maximum property of the Koebe function, Invent. Math., 4 (1967), 26~7.

5. Z. Charzynski and M. Schiffer, A new proof of the Bieberbach conjecture for the fourth coefficient, Arch. Rational Mech. Anal., 5 (1960), 187-193.

6. P. R. Garabedian, Inequalities for the fifth coefficient, Comm. Pure Appl. Math., 19 (1966), 199-214.

7. - - - , An extension of Grunsky's inequalities bearing on the Bieberbach conjecture, J. d'Analyse Math., 18 (1967), 81-97.

8. - - and M. Schiffer, A proof of the Bieberbach conjecture for the fourth coefficient, J. Rational Mech. Anal., 4 (1955), 427-465.

9 . - and M. Schiffer, The local maximum theorem for the coefficients of univalent functions, Arch. Rational Mech. AnaL, 26 (1967), 1-32.

10. P. R. Garabedian, G. G. Ross, and M. Schiffer, On the Bieberbach conjecture for even n, J. Math. Mech., 14 (1965), 975-989.

11. H. Grunsky, Koeffizientenbedingungen for schlicht abbildende meromorphe Funktionen, Math. Z., 45 (1939), 29-61.

12. J. A. Hummel, and M. Schiffer, Coefficient inequalities for Bieberbach-Eilenberg functions, Arch. Rational Mech. Anal., 32 (1966), 87-99.

13. J. A. Jenkins, Some area theorems and a special coefficient theorem, Illinois J. Math., 8 (1964), 88-99.

14. - - , On certain coefficients of univalent functions. Analytic functions, Princeton 1960, 159-194.

15. H. Lewy, On the non-vanishing of the Jacobian in certain one-to-one mappings, Bull. Amer. Math. Soc., 42 (1936), 689-692.

16. K. Loewner, Untersuchungen tiber schlichte konforme Abbildungen des Einheits- kreises, Math. Z., 89 (1923), 103-121.

17. J. M. Mihlin, The area method in the theory of univalent functions, Soviet Math. DokL, 5 (1964), 78-81.

18. Z. Nehari, Inequalities for the coefficients of univalent functions, Arch. Rational, Mech. Anal., 34 (1969), 301-330.

19. M. Ozawa, On the Bieberbach conjecture for the sixth coefficient, Kodai Math. Sere. Rep., 21 (1969), 97-128.

20. R. Pederson, A proof of the Bieberbach conjecture for the sixth coefficient, Arch. Rational Mech. Anal., 31 (1968), 331-351.

21. ~ On unitary properties of Grunsky's matrix, Arch. Rational Mech. Anal., 29 (1968), 370-377.

22. M. Schiffer, Extremum problems and variational methods in conformal mapping Proc. Internat. Congress Math. 1958, 211-231.

CARNEGIE-MELLON UNIVERSITY Prrl'sat.rRG~, PENNSYLVA~_A, U.S.A.

AND STANFORD UNIVERSITY

STANFORD, CALIFORNIA, U.S.A.

further generalizations of the grunsky inequalities

Documents