i[(^J-i0^) ], zeD

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a symmetric generating function for the series of its derivatives is constructed, ... in D in terms of the series of the derivatives of the Schwarzian derivative.
ON THE DERIVATIVES OF THE SCHWARZIAN DERIVATIVE OF A UNIVALENT FUNCTION AND THEIR SYMMETRIC GENERATING FUNCTION REUVEN HARMEL1N ABSTRACT

For every analytic function in a domain D in the complete complex plane, bounded in a certain norm, a symmetric generating function for the series of its derivatives is constructed, bounded in another norm. New conditions are deduced for univalence and quasiconformal extendability of a meromorphic function in D in terms of the series of the derivatives of the Schwarzian derivative. In particular an inequality satisfied by Grunsky coefficients of univalent functions in the unit disc is derived.

1. Introduction Let D be a simply-connected domain in the extended complex plane C, the boundary dD of which consists of more than one point, so that the Poincare metric pD(z) in D is well defined by (1.1)

pD(/(2))|/'(z)|==(l-MT\

where /(z) is a conformal mapping of the unit disc U onto D. For every meromorphic function /(z) in D we define M ™ « / n

d

,

/(Z)/(P

/W(C)

= I (n-l)^n(f,z)(C-z)"-2,

zeD,\C-z\„(/> z)> f° r n ^ 2, were discovered. LEMMA

A [1, Lemma; 7, Lemma 1]. / / / is univalent in the unit disc U, then

(1.5) Received 16 March, 1982. J. Loiu/oit Math. Soc. (2), 27 (1983), 489-499

490

REUVEN HARMEL1N

and if f has a fi-quasiconformal extension into C\U, then

n= 2

LEMMA

B [8, Lemma 2]. Let g(z) =

-, with ad —be = 1. Then cz-\-d

(1.6) THEOREM

C [7, Lemma 2], / / / is univalent in a domain D {of any connectivity),

then

£ {n-i)d{z,dD)ln\il,n{f,z)\2

(1.7)

^ 1, z e D ,

n= 2

and if f has a n-quasiconformal extension into C \ D , then

£ (n-l)d(z,dD)2"\Uf,z)\2 ^ IN 2 ,

(1.7)

zeD.

n= 2

THEOREM

(1.8)

D [1, Theorem; 8, Lemma 5]. / / / is univalent in U, then

I (n-1) Y

n-2 , k-2

(-£)""*(!-|CI 2 ) k «K(/,0

ici < i ,

and if f has a n-quasiconformal extension into C\U, the upper bound for the left-hand safe in (1.8) is ||fHiTHEOREM

(1.9)

E [8, Theorems 1 and 1'].

/ / / is univalent in U, then

(l-|z| 2 n«M/,z)| ^ Pn-2(\z\) = t ( " ~ f ) - 7 = , \

K Z

and if f has a n-quasiconformal extension into C\U, should be multiplied by \\n\\x.

J

\z\ < 1,

/ k l

then the right-hand side o/(1.9)

The object of this note is to prove analogous results about the sequence {*,(/, z)}«.a defined by

This work is based on part of a Ph.D. Thesis, written at the Technion, Israel Institute of Technology, under the supervision of Professor Uri Srebro. The author is grateful to Professor Srebro for his guidance and support.

ON THE DERIVATIVES OF THE SCHWARZIAN DERIVATIVE

491

2. The generating function It is well known that the Schwarzian derivative {/, z}, of a locally univalent meromorphic function / in D, is an element of the complex Banach space B2(D) of analytic functions (z) in D for which the norm (2.1)

ll^lb

D

^ sup

PD(Z)~

|^>(Z)|

zeD

is finite. Throughout this note let Dj and D2 be a pair of complementary simplyconnected domains in C, with a common boundary C = dD^. In [5] Bers showed that if ^ G L C 0 ( D 1 ) (that is ^ is a complex-valued bounded measurable function supported in D t ) then 1

(2.2)

(f>(z) --

n is an element of B2{D2), and moreover, if C = dDt is a quasicircle, then Im^Ri2* = B2(D2). In view of formula (1.2) we formally define (2.3)

T{-z,w)= £ ( n - l ) t f > n ( z ) ( w - z r 2 ,

z s D 2 , | w - z | < d(2, C),

n=2

for all e B2(D2), where

6.U) = EXAMPLE.

(2.4)

Let a>2(( ; z) = (C — z)~ 4 for a fixed C e D j . Then

T(o;2;z,W)= £ (n-l)(C-z)- n - 2 (w-zy 2

for z e£>2 and |vv —z| < d(z, C). Hence, if cj> = SOJ^'/i for some \i e ^ ( D J , then (2.5)

for ZED2 and |w — z\ < d(z, C). Notice that 9W*^ is a symmetric function in its two variables z and w, and is analytic over all D2xD2. Both properties are confirmed by the following lemma. LEMMA

(2.6)

1. (i) If cj) is analytic in D2, then

7-(0;z,w) = —5-_

(w-z)-

492

REUVEN HARMELIN

(ii) If> e L00(DJ then (2.5')

T(9K[?V ',z,w) = (5R^)(z, w),

(z, w) eD2xD2

Proof. The proof of (2.6) is based on the identity

0

0 0

Thus

w —z u

6 0

0

For the proof of (ii) use the identity

f(w-P(C J (oi-C

l 6

^

6((D-Z)2(O)-W)2'

which implies that

,

,f(w-O(C-z)

;z, w) = 6

(w-z) 3

D|

In [12, 9] it was shown that whenever /(z) is a meromorphic univalent function in D, then the function Sf(z, (), defined in (1.2), is an element of the complex Banach space Bfx{D) of the symmetric analytic functions $(z,() in DxD which are bounded in the norm (2.7)

Uhv.o = sup

1

ji Jj|0(z,

ON THE DERIVATIVES OF THE SCHWARZ1AN DERIVATIVE

493

Lemma 1 and the next lemma will imply the analogous result for

(2.8)

7}(z, w) = T(0 2 (/, 0 ; z, w) = f (W ~ 0(C " Z) {/,

n= 2

LEMMA

2. Let /ie L00(/),). Then

(2-9)

V

Proo/. Assuming that /i has a compact support in C, we can write (z),

n

(z, w) e D 2 x D 2 )

c

where w(C) = A*(C)/(C — w ) 2 a n d -^w is the Hilbert transform of the 1} (C)-function co with a compact support. The L2-isometry property of Jf implies that

D,

But by the results of Bergman and Schiffer [4], Burbea [6], and Beardon and Gehring [3], we have

where KDl(z, w) is the Bergman kernel function of D2. Hence

{

1 CC

-n

JJ

11/2

|9W^(z, w ) | 2 d x ^ IJ

/i

^

f

PP2

\~*

V

II^O)|IL2

^ ||)

VTT

D2

Now, if /i has a non-compact support, let g be any Mobius transformation such that 0(Dj) is compact, and let v e LCC(^(D1)) be such that v(g(CJ)g'(t)g'(Q x = for C e Dj. Then v has a compact support and IMI^ = \\n\\x. Next, using the identity (2.10)

(g{z)-g(n))2 =

(z-W)2g'(z)g'{w),

which holds for all Mobius transformations g, we deduce that

494

REUVEN HARMELIN

v(()g'(z)g'(W)

I ff v(g(C))|g'(C)iy(*)g'(w)

JJ

=

11

(z, w)

(£-z)2(f_w)2

But an easy computation shows that the mapping

(f> i

• (j)(g{z),

g(w))g'(z)g'(w)

is B2*00-norm-preserving, and therefore (2.11) and (2.9') yield that lt«»cVll2.co;D2 = H2W9VH2,oo;92) ^ ML =

ML-

We can now prove our main result. THEOREM 1. Let C be a quasicircle. Then there is a constant K = K(C) > 1, depending only on C, such that

(2.12)

l|T0||2iOC:D2 ^ K(C)\\(j)\\2,D2,

0 6Ba(D2).

In particular if C is the unit circle, then K(C) ^ 3. Proof. Bers [5] proved that if C is a quasicircle, then Im2R^ = B2(D2) and, moreover, the operator aR :£»(£>!) >B2(D2) has right inverse (2.13)

Ag>:B2(D2)30.

• 3(C- MC))2 ^

HHO) e L« (D,),

where /i: Dj -»• D 2 is an appropriate quasiconformal reflection at C, such that (see [2]) (2.14)

HA'Moo < JC(C)||fl|2ifl2,

4>eB2(D2),

for some constant K(C) > 1 depending only on C. Therefore (2.5') yields that (2.15)

T = T o 2R(C2) o Ag> = SRC* o A(c2).

Thus (2.9), (2.14) and (2.15) imply (2.12). Notice that if C is the unit circle, we may choose h{£) = 1/C in (2.13) and obtain (2.14) with K{C) ^ 3.

ON THE DERIVATIVES OF THE SCHWARZ1AN DERIVATIVE

495

In what follows we use the notation

(cf. [9, Section III]), for every hyperbolic simply-connected domain, where

COROLLARY

(2.16)

1. Let C be a quasicircle in C. / / / is univalent in D2, then

\\Tf\\2,x;D2 < K(C)||2(/)IU < K(C)(p(D2) + p(f{D2))) < 2K(C)

where K(C) is the constant of Theorem 1. / / / has a ix-quasiconformal extension into £>,, then (2.17)

}2

and in particular, if D2 is the unit disc U, then

(2.17) Proof The left inequality in (2.16) follows from (2.12) and the definition of 7} in (2.8). The right one is obtained using the identity 2(f, z) = ^2(f, z) = Sf(z, z) and the chain HM/)ll2
(3-7') (ii) Let g(z) =

(3.8) Proof

g(w))g'(z)g'(W).

-, with ad —be = 1. Then cz + d

n(f,z)\2^9, n= 2

z e D ,

ON THE DERIVATIVES OF THE SCHWARZ1AN DERIVATIVE

and if f has a fi-quasiconformal extension into C\D,

THEOREM

(3.10)

£ (ii-

THEOREM

zsD.

4. / / / is univalent in the unit disc U, then

L l fc = 2 / c - 2

and iff has a fi-quasiconformal extension into C\U, side 0/(3.10) is replaced by ^

(3.11)

then

£ (n-l)d(z,dD)2"\