On the Schwarzian norm of harmonic mappings Текст научной статьи по специальности «Математика»

20

Probl. Anal. Issues Anal. Vol. 5(23), No. 2, 2016, pp. 20-32

DOI: 10.15393/j3.art.2016.3511

UDC 517.54

S. Yu. Graf

ON THE SCHWARZIAN NORM OF HARMONIC

MAPPINGS

Abstract. We obtain estimations of the pre-Schwarzian and Schwarzian derivatives in terms of the order of family in linear and affine invariant families L of sense preserving harmonic mappings of the unit disk D. As the converse result the order of family L is estimated in terms of suprema of Schwarzian and pre-Schwarzian norms over the family L. Main results are obtained by means of theory of linear invariant families.

Key words: harmonic mappings, linear invariant families, Schwarzian derivative

2010 Mathematical Subject Classification: 30C45, 30C55, 30C99

1. Preliminaries. Every sense preserving harmonic in the unit disk D = {z E C : |z| < 1} function f (z) admits representation f (z) = = h(z) + g(z), where h and g are analytic in D,

h(z) = ^ akzk, g(z) = ^ bkzk, (1)

k =0 k=1

the dilatation u(z) = g'(z)/h'(z) is analytic in D and |w(z)| < 1 for all

z E D.

The pre-Schwarzian and Schwarzian derivatives of a locally univalent analytic function h are given (cf. [1]) by

PhM = hi and Sh(z) = (.hg;)' - I (hf )2

respectively. The important role of the Schwarzian derivative in theory of univalent analytic functions is well known. The following theorem was proved by W. Krauss [2] and rediscovered later by Z. Nehari [3].

©Petrozavodsk State University, 2016 IMlIiHI

Theorem A. If h is a univalent analytic function in D, then

\Sh(z)\<

6

(1 -\z\2 )2

for any z E D.

The same inequality with the constant 2 instead of 6 in the numerator was found by Nehari as the sufficient condition of univalence of an analytic function. L. Ahlfors and G. Weill [4] expanded this result by establishing the condition under which a univalent analytic in D function has a quasiconformal extension onto the whole Riemann sphere.

In 2003 the notion of Schwarzian derivative was generalized by P. Duren, B. Osgood and M. Chuaqui [5] onto the case of locally univalent harmonic functions f = h + g in the disk D with dilatation u = g/h' = = q2, where q is some analytic function in D. Later R. Hernandez and M. J. Martin [6] proposed a modified definition of pre-Schwarzian Pf and Schwarzian Sf derivatives that are valid for the whole family of sense preserving harmonic mappings. This definitions preserve the main properties of classical Schwarzian derivative and are the following:

Pf (z) = Ph(z) -

Sf (z) = Sh(z) +

u'(z)u(z)

1 -Kz)\2'

w(z)

h"(z),/(z) 3 (zMz)

1 - \.(z)\Hh(z) w (z) - w (z)J - 2 T-R^

So as in the analytic case we have

Sf (z) = (Pf (z))z - 2(Pf (z))2 .

The pre-Schwarzian and Schwarzian derivatives of harmonic function have the chain rule property (cf., [6]) exactly in the same form as in the analytic case: if f is a sense preserving harmonic function and ^ is a locally univalent analytic function for which the composition f o ^ is defined, then

Pfo^(z)= Pf O P(z) ■ v'(z) + P^(z), Sf o^ (z) = Sf O V(z) ■ (v'(z))2 + S^(z).

2

Both pre-Schwarzian and Schwarzian derivatives are invariant under affine transformation of a harmonic function f: if A(w) = aw + bw + c, |a| > |b|, then

Pao/ (z) = Pf (z) and Sao/(z) = Sf (z). (3)

The pre-Schwarzian and Schwarzian norms of a harmonic function f are defined as

IIP/ II =sup |Pf (z)|(1 -|z|2),

ZGD

IIS/II =sup |S/(z)|(1 -|z|2)2.

ZGD

In the current paper we deal with estimations of this quantities and concerned functionals in the linear and affine invariant families of harmonic functions.

A family L of sense preserving harmonic functions f = h + g, where h and g have form (1) with a0 = a1 — 1 = 0, is said to be linear invariant family (l.i.f.) if for any f E L

f) = f Sof r ELV *> = ^ E D,# E R

and L is called affine invariant family (a.i.f.) if the property

A [f ](z) = f (Z) te/(Z} e£V e G D £Lf ]() 1 + efz(0)

holds. The family L is called affine and linear invariant (a.l.i.f.) if it is l.i.f and a.i.f simultaneously. It is remarkable that many important properties of a.l.i.f. depend on supremum of the absolute value of the second coefficients a2 over the family L. The notion of order of family L was defined by T. Sheil-Small [7] as a = supjeL |a21 (exactly in the same form as in the analytic case, cf., [8]). This paper of Sheil-Small on linear and affine invariant families of harmonic functions attracted attention of many mathematicians (cf., [9, 10, 11]). In 2004 V. Starkov ([12], see also [10] for details) gave definition of new order of l.i.f. as

_ |a2 - bib21 a = suP -vT\2

fec 1 - |bi|2

such that a < a < a + 1/2. In 2007, the author [13] introduced the notion of specified order of a.l.i.f. L as

ao = sup |a2| f ec0

where L0 = {f E L : b1 = 0}. In [14], it was established that the new order a of l.i.f. L coincides with the specified order a0 of affine hull of L. Let fl0 = supfeLo |b21. So a0 < a < a0 + 1/2 and fl0 < 1/2 for any a.l.i.f.

At present the orders a and a0 are known only for several subclasses of the sense preserving univalent harmonic mappings f = h + g in D with h, g of the form (1), where a0 = ai — 1 = 0. Particularly for linear and affine invariant families Kh and Ch of convex and close-to-convex harmonic mappings the order a0 is proved to be 3/2 and 5/2 respectively (see [16] and [17]), while a = a0 + 1/2 and = 1/2 in both cases. Recall that the function f is called close-to-convex if the range D = f (D) under the mapping f is a close-to-convex domain i.e. C \ D can be represented as union of disjoint lines or half-lines intersecting only in the boundary of

2. Analogue of the Krauss theorem for harmonic functions.

The main results of the present paper are the estimations for pre-Schwar-zian and Schwarzian derivatives and their norms in the a.l.i.f. of locally univalent harmonic functions. We start with estimation of pre-Schwarzian derivative in the spirit of well-known inequality for |h''/h'| by L. Bieberbach for the class S of univalent conformal mappings h.

Theorem 1. For any locally univalent harmonic mapping f E L and for

all z G D

The estimation is sharp for example in a.l.i.f.s KH and CH of univalent convex and close-to-convex harmonic functions. Particularly

Proof. Let f = h + g G L and z G D be fixed. Using the property of linear invariance of L, define the linear transformation ^(Z) = (Z + z)/(1 + zZ) such that p(0) = z, P^(0) = -2z and consider F(Z) = H(Z) + G(Z) = = ](z) G L. Then in view of the chain rule (2) for pre-Schwarzian derivative we have

L (cf., [15]).

D.

(4)

||Pf||< 2(ao + 1).

Now, applying the affine invariance of L, let F0(Z) = H0(Z) + G0(Z) = = £[F](Z) G L, where e = -G(0) Then G0(0) = (G(0) + ëH'(0))/(l-

-H2) = 0 and hence F0 E L0 so wo(0) = G'0(0)/H0(0) = 0. Therefore, from (3), we conclude that Pf (0) = PFo (0) = H'0 (0) and

= H0 (o) +2z

Pf (z) 1 - IzI2 '

But IH0'(0)I < 2a0 by definition of specified order of a.l.i.f. L, and inequality (4) follows immediately from the last representation of P/ (z). The estimation of pre-Schwarzian norm of f follows easily from (4). To make sure that the inequality (4) is sharp consider for example the a.l.i.f. Ch of all univalent sense preserving close-to-convex harmonic functions. The well-known (cf., [16]) harmonic analogue of the Koebe function

_ z _ 1 z2 + 1 / 1 z2 + 1 ~3 \

K(z)= Hk(z) + Gk(z)= " J- +,1z +((5)

belongs to CH. It is easy to compute that H'K (0)/2 = 5/2 is equal to specified order of CH and (4) turns into equality for function K and any z = r E [0,1).

To prove the sharpness of the inequality (4) in the class KH of convex univalent sense preserving harmonic functions it is enough to note that in this case a0 = 3/2 (cf., [16]) and for the function

L(z) = Re f + ¿Imf., z ,2) E KH (6)

1 - z) V(! - z)

its pre-Schwarzian equals 3 at origin, i.e. coincides with 2«o • The proof is complete. □

Note that for a.l.i.f. Kh of univalent convex harmonic mappings our estimation of the pre-Schwarzian norm (||P/1| < 5) coincides with the estimation obtained recently by Hernández and Martin [6].

Now we are going to prove the estimation of the Schwarzian derivative of harmonic functions. As in the analytic case the key role in studying of ||S/1| plays the estimation of the Fekete-Szego functional |a3 — a^|. We will obtain its upper bound via the logarithmic coefficients of harmonic function. Let f = h + g G L0. Define locally univalent analytic function Fe(z) = h(z) + eg(z) in D, where e is arbitrary such that |e| < 1. Let

to

(z)=log F'£ (z)^ ck (e)z

k=i

k

We will designate ck (e) as logarithmic coefficients of f.

Theorem 2. Let a locally univalent harmonic mapping f G L0 and let |e| < 1. Then

|ck (e)| < min Ck (e, r) < Ck e,

re(Q,i) \ V k +1

k- 1

(7)

where

Ck (e, r) =

2

ao + eoi£!n

1+rjej

krk 2 (1 — r2 )

ao + A^

(8)

Proof. For a function f E L0 and arbitrary e, |e| < 1, we define analytic function as above. Then (z) = hh/(Z)+eg'(Z)>>. The sharp estimation of the modulus of this ratio in the a.l.i.f. L was obtained in [11] (the proof is available in [15]) and yields the inequality

—

2z

1 — |z|2

<

2

1 — |z|2

ao + во

|z| + |e|

1 + |z||e|

Then for any fixed r E (0,1) and any Z, |Z| < 1, in view of the maximum principle for analytic functions we have

1 r2

Z Ф (rZ ) — r

< ao + во

r + |e| 1 + r|e|

Consider the analytic function in D, defined by

*(Z ) =

where

1 r2

Z ф'е (rZ ) — r ao + во

r + |e| 1 + r|e|

-i

E dk Zk

k=o

dk = kck (e)r

k- 11 — r2

ao + eo

r + |e| 1 + r|e|

for k > 1 in terms of logarithmic coefficients of f and

do = —r ao + eo

r + |e| 1 + r|e|

r

r

Function ^ meets the inequality )I < 1 and hence as a consequence of the well-known Schwarz lemma (cf., [18], pp. 319-320) we obtain

IdkI < 1 - N2, k > 1.

From there taking expressions for dk into account we have

Ick(e)I < Ck(e,r)

for any r E (0,1), where Ck are given by (8). To finish the proof we have to note that the function r1-k(1 - r2)-1 takes its minimal value when r = sj(k - 1)/(k + 1) and therefore,

Ick (£)I< Ck ^V^TT

which is the required inequality. □

As a consequence we obtain the estimation of Fekete-Szego functional for family L0.

Corollary. Let a locally univalent harmonic mapping f E L0. Then |a3 - a2| < 1 (a0 + min C2(0,r)J <

n J (9)

a2 1 ( K n 1 (9)

Proof. Let f = h + g E L0 and h(z) = z + a2z2 + a3z3 + ■ ■ ■. Applying inequality (7) with e = 0 and k = 2, we obtain

3

Ic2(0)I < 2 (^3a0 + ^0 -

2 V v/3a + ^0/

On the other hand it is easy to verify that in the current case the logarithmic coefficient c2 (0) = 3a3 - 2a2. Therefore

We obtain the desired estimation (9) from the last inequality because Ia21 < a0 for any function f E L0. □

As an immediate consequence of the Corollary 1 we obtain an estimation of the Schwarzian norm in a.l.i.f. L.

Theorem 3. Let a locally univalent harmonic mapping f E L. Then

||Sf || < 2( a2 + min £2(0, r)) < 2a2 + 3( V3ao + Ad — ^^-

V re(0,1) J \ V3ao + Ao

Proof. Let f = h+g E L and z E D are fixed. As in the proof of Theorem 1 we will use the property of linear invariance of L. For the Mobius function p(Z) = (Z + z)/(1 + zZ), we consider F(Z) = H(Z) + G(Z) = = L^ [f](z) E L. Then in view of the chain rule (2) for the Schwarzian derivative and the well known fact that S^ = 0 we have

Sf (z) = Sf (0)

(1 - |z|2)2"

Using the affine invariance of L for e = — G' (0) we define F0(Z) = H0 (Z) + +Go (Z) = Ae [F](Z) G L0 such that G'o(0) = 0. Therefore from (3) we conclude that

5v (0)=.v. (0)=.H,«0)=^—3 ( m )2=6(^3—Ai),

where Ak are the Taylor coefficients of function H0. Finally applying the inequality (9) we are able to estimate the Schwarzian of f :

|Sf (z)|(1 — |z|2)2 = |Sf0(0)| = 6IA3 — A21 < 2 («0 + mï^ ^2(0,r)

The estimation of the Schwarzian norm of f follows easily from the last expression by definition. □

Theorem 3 allows us to obtain estimations of the Schwarzian norms in a.l.i.f.s whose orders are known. For example if f belongs to the family Ch of close-to-convex univalent harmonic mappings (remind that a0 = 5/2 in this case) then ||Sf || < 26.368.

Note that for the class Kh of convex harmonic mappings estimation of the Schwarzian norm ||Sf || < 6 was obtained by Hernández and Martin [6].

3. Estimation of order of a.l.i.f. As a converse problem let us consider the question of estimating the orders a or a0 of a.l.i.f. L if

the estimations of Schwarzian or pre-Schwarzian norms are known for the family L.

Theorem 4. Let L be a.l.i.f. of harmonic functions and there exists a constant C such that

sup IP/(z)(1 -IzI2)I< C + 2IzI.

f eL

Then a0 < §, a < C+1.

Estimations are sharp for example in the a.l.i.f. Kh of convex harmonic mappings.

Particularly, if sup/eL IIP/ II < D, then a0 < y, a < ^i1.

Proof. If there exists a finite C such that IP/(z)(1 - IzI2)I < C + 21zI for any f E L and all z E D then, particularly, IP/(0)I < C. In view of definition of Pf this means that

„ t u' (0) 2 a2 - 61- V ;

C

1 -Iu(0)I2

for any f E L with dilatation u and 61 = f z (0). Therefore,

I"21 < 2 (c+N1^ )■ ("»

Obviously I61I < 1. Applying invariant form of the Schwarz lemma (cf. [18], pp. 319-320) to the dilatation u we deduce that I u'(0)I /(1-1 u(0) I 2) < 1 and hence I a2 I < (C + 1)/2 for any f E L. This inequality gives us the desired estimation of order a of L.

In the case when f E L0 we have 61 =0 and estimation of the specified order a0 easily follows from (10).

The sharpness of the result of the theorem in class KH of convex harmonic mappings can be verified directly. Indeed, from Theorem 1 it follows that the best possible value of constant C is equal to 2a0 = 3 in a.l.i.f. KH. On the other hand, for function L given by (6) the second coefficient a2 = 3/2 while 61 = 0. So (10) turns into equality. Therefore the order and specified order of a.l.i.f. KH should not be greater than 2 and 3/2, respectively, and that represent the facts. □

To obtain the similar result in terms of the Schwarzian norm we first have to estimate the Fekete-Szego functional I a3 - a2 I by supremum of I I S/ I I over a.l.i.f. L.

Theorem 5. Let L be a.l.i.f. of harmonic functions and there exists a constant C such that

sup ||Sf || < C. f ec

Then |a3 — a21 < C/6 for any f E L0 and

|«3 — a21 <

2. C + 2ao + 3/2

6

for any f G L.

Proof. First, if the condition of Theorem 5 is satisfied and f G L0, then Sf (0) = 6(a3 — a2) and statement of theorem is clear. Now, let f G L. Then there exists a function f0 = h0 + g0 G L0 such that

f (z) = fo(z) + bifo (z).

So ak = ak + b1bk, where |61| < 1, ak are Taylor coefficients of the analytic part of f while ak, &k are the coefficients of analytic and co-analytic parts of f0, respectively.

Then the Fekete-Szego functional of f takes the form

|a3 — a21 = |a3 — (a°)2 + 61 (&0 — 2a°b° — h (&0)2) | . (11)

To estimate the last quantity we will evaluate 63 via derivative of dilatation w0 = g0/h0 of function f0 at origin:

„ = (g0/;h0 — g0hO77) h0 — 2h0' (g0'h0 — goh0) ^ =-

(h0)3

But g0 (0) = 0 for f0 G L0 and h'0 (0) = 1. Therefore

4'(0)=663 — 8a°60. (12)

Note that analytic function w0(z) = 262>z + 2-W(0)z2 + ■ ■ ■ satisfies conditions of the Schwarz lemma, so |w0(z)| < |z| and function u(z) = = w0(z)/z = 2&0 + 2-1<y(0)z + ■ ■ ■ satisfies the invariant form of the Schwarz lemma. Hence |u'(0)| < 1 — |u(0)|2 and

1^0'(0)|< 2 (l — 4|60|2) . (13)

Now we will use equality (12) together with estimation (13) in order to obtain an upper bound for Ia3 - a21 in (11):

|a3 - a22 | <

- №

+ I61I

1 u0'(0) - 2a060 - 61 (60)2

6

3

<

1 1 2 2 < -C +-Iu''(0)I + - |a01|62| + |6012 < 6 6 3

.01|60| + ^|2 < ^C + ^ (1 - 16012 + 2 |a2| 162

1

— I 6

1 1 3

< 6C +3 a0 +4

because I a§ I < a0 for f0 E L0 and function y(t) = 1-12 + 2a0t is increasing for 0 < t < < 1/2 and a0 > 1/2. Here we have to mark out that the order a0 of any a.l.i.f. of a harmonic sense preserving functions is not less than 1/2. Indeed, it is known (cf., [8]) that order a of any l.i.f. (even in the analytic case) is not less than 1, and a0 > a - 1/2 as was mentioned above. □

Corollary. Let L be a.l.i.f. of harmonic functions and there exists a constant C such that

sup IIS/II < C.

f eL

Then a < iW|E2C.

Proof. First of all note that the set of analytic parts h of harmonic functions f E L forms a linear invariant family U of analytic functions. Ch. Pommerenke [8] proved that in any l.i.f. U of analytic functions h the order a satisfies the inequality

a2 - 1 < 3 sup |a3 - a22 |.

heu

From Theorem 5 we deduce that

sup |a3 - a22 | = sup |a3 - a22 | <

2 C + 2a + 3/2

heu

f eL

6

because specified order a0 < a in any a.l.i.f. L. Therefore

2 7 1

a2 - a - T - - C < 0. 4 2 <

Solution of this quadratic inequality gives us the desired estimation of order a in a.l.i.f. £. □

Note that relations (12) and (13) allow us to obtain the sharp estimation of |b3| in the family L0 with specified order a0 > 1. Really, if f G L0 with dilatation u then coefficients of f satisfy the conditions

N =

u" (0) + 8a2&2

6

< 1 - 4|b212 + 4|a2||b21 " 3

Taking the maximum of the last ratio over all |b2| < 1/2 and |a21 < < a0, a0 > 1, we obtain

.. . ^ 1 - 4|b2|2 + 4ao|b2K 2 |b31 < -3- < 3 ao.

Inequality is sharp. Equality occurs on functions (5) and (6) with a0 = 5/2 and 3/2 for subfamilies CH and respectively.

Acknowledgment. This work was supported by the the Strategic Development Program of Petrozavodsk State University.

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Received October 19, 2016.

In revised form, December 11, 2016. Accepted December 13, 2016.

Tver State University

33, Zheliabova st., 170100 Tver, Russia;

Petrozavodsk State University

33, Lenina st., 185910 Petrozavodsk, Russia

E-mail: Sergey.Graf@tversu.ru