Inequalities for the Schwarzian derivative for subclasses of convex functions in the unit disc

Автор: Polatoglu Yasar, Caglar Mert, Sen Arzu

Журнал: Владикавказский математический журнал @vmj-ru

Статья в выпуске: 2 т.8, 2006 года.

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Nehari norm of the Schwarzian derivative of an analytic function is closely related to its univalence. The famous Nehari--Kraus theorem [3, 4] and Ahlfors--Weill theorem [1] are of fundamental importance in this direction. For a non-constant meromorphic function f on D the unite disc, the Schwarzian derivative S_f of f by is holomorphic at z_0\in D if and only if f is locally univalent at z_0. The aim of this paper is to give sharp estimates of the Nehari norm for the subclasses of convex functions in the unit disc.

Короткий адрес: https://sciup.org/14318185

IDR: 14318185

Текст научной статьи Inequalities for the Schwarzian derivative for subclasses of convex functions in the unit disc

Let ф(z) be an analytic function defined in D = {z : | z | < 1 } . If ф(z) satisfies the condition | ф(z) | 6 1 for all z E D, then it is called a Schwarzian function. The class of Schwarzian functions is denoted by Q * .

Next, let Q be the family of functions w(z) regular in D and satisfying the conditions w(0) = 0, | w(z) | < 1 for all z E D.

Further, for arbitrary fixed complex numbers A and B denote by P(A,B ) the family of functions

p(z) = 1 + P1Z + P2Z2 +---- regular in D and such that p(z) is in P(A, B) if and only if

1 + Aw ( z )

P(Z)   1 + Bw(z)

for some w(z) E Q and every z E D. This class was introduced by W. Janowski [2].

Finally, let C(A, B) denote the family of functions f (z) = z + a2 z2 +---- regular in D such that f (z) E C(A, B) if and only if

1+ f       P(z)(4)

for some p(z) E P (A, B ) and all z in D.

2.    Schwarzian Derivative Inequality For the Class C(A,B)

The following lemma which is due to Caratheodory [1] is fundamental for our present aim.

Lemma 1. If ф E Q* then uvm 1 -lф(z)l2

| ф ( z ) l 6 1 -|z| 2                                              (5)

for some complex number B such that | B | < 1 .

Lemma 2. The class Q * is invariant under rotations.

C It follows easily from the definition of Q * that the function ^ defined by ^(z) := e -ia ф(e ia z), ф(z) E Q * , z E D, 0 6 a 6 2n, satisfies

Kz) | = | e -ia ф(e ia z) | = | e -ia || ф(e ia z) | 6 1. B

Lemma 3. If ф(z') is an element of Q * , then

| ф ( z ) | 6

1 -| Ф(z) | 2

1 -| B | 2 |z| 2

C Using Lemma 1 and Lemma 2, and after simple calculations,we get

| B | < 1, 0 6 a 6 2n ^ etaBz E D ^

W <  1 - ^( e ia Bz ) | 2             1- \\ < B - Heia Bz ) | 2

' ф ( e B 1 - | e^Bz | 2     ^ | ф ( e Bz ) |6 1 - I B | 2 |z| 2 .           '

On the other hand, since e Bz D, the inequality (6) can be written in the form

| *' (z) | 6 Г-Ур^    ° (1 - | B | 2 1 z | 2 ) | *' ( z) | + |ФИ| 2 6 L B         P)

Theorem 1. If f belongs to the class C (A, B) then

| S f | 6

I

(A - B) | z | 2       | A+B | (A - B) | z | 2

(1 - B 2 | z | 2 ) 2           (1+B | z | ) 2

(1 - A)A | z | 2 ,

B = 0,

B = 0.

C Since f E C (A, B), we can write

1 + z7E )= p ( z )

1 + Aw(z)

1 + Bw(z)

From the equality (9) we have the following:

' f 00 _ p(z)-1      / f 00 V _ zp^fz^ - Cpfz)-^

f 0 = z ^ V 0 ) = z 2      

? £1 V = A-B h zW-w 1 - B(A - B) (W2

У f0 J (1+Bw) 2 |_ z 2 J (1+Bw) 2 у z 2

zw0 - z2

w   (A - B)zp0 - (A - Bp)(p - 1)

z2(A - Bp)2         1

w 2       (p - 1) 2

z 2     z 2 (A - Bp)21

Hence,

Sf = т f A

-

2 f У

1 (P - 1) 2

2    z 2

.

B [( A - B)zp 0 - (A - Bp)(p - 1

-

A + B I |2

A-B | P - 11

On the other hand, we have

| p(z) - 1 | -

| z | 2 | A - p(z) | 2 = (1 B 2 | z | 2 ) p(z)

-

1 - AB | z | 2

1 - B 2 | z | 2

-

(A - B ) 2 | z | 2 (1 - B 2 | z | 2 ) ,

and

( w(z) E Q, ф(z) E Q * ^ w(z) = zф(z) ^ Ф 0 (z) = zw ( z ) I(1 - B 2 | z | 2 ) | ф 0 (z) | + | ф(z) | 2 6 1 ^ (1 - B 2 | z | 2 ) | zw l lzl - wiz)

-

z 2

z 2

,

■               ■                  1.

Considering (10)–(16) together yields (8). B

3. Special Cases

For special values of A and B , we obtain the following inequalities. (1) From Lemma 2 and the equality (11) we have

I ( A - B ) zp 0 ( z ) - ( p ( z ) - 1)( A - Bp ( z )) 1 6 ,(A    B.^z' 2 .

(1 - B | z | )

In this case we have the following inequalities.

(1a) A = 1, B = - 1:

12 zp 0 ( z ) + 1 - ( p ( z )) 2 1 6 (1 -|ZZ| 2 ) 2

This inequalities was proved by M. S. Robertson ([5]).

(1b) A = 1 - 2a (0 6 a < 1), B = - 1:

|2(1 - a)zp,(z) - (p(z) - 1)((1 - 2a) + p(z)) 1 6 4(1—a)2*z2| .

(1 - | z | )

(1c) A = 1, B = ^M - 1 (M> 1 ):

(2 - mm} zp ( z ) - ( p ( z ) - 1) (1 - (1 - M} p ( z))

(2 — i)2H2

-

(Mm - 1) WT

(1d) A = в , B = (0 6 в <  1):

1 zp 0 ( z ) + 1 - ( p ( z )) 2 1 6 (1^в21_.

(2) A = 1, B = - 1:

4 | z | 2

| S f 1 6 (1 - | z | 2 ) 2

This equality was obtained by M. S. Robertson [5].

  • (3)    A = 1 - 2a, B = - 1:

4(1 - a)^2 _ 4a (1 - a ) \ z \ 2 ' f | 6 (1 -\ z \ 2 ) 2         (1 -\ z \ ) 2

  • (4)    a = 1, в = M - 1 (M >  2 ):

    | S f | 6


(2 - MV \ z \ 2          Mm (2 - M^ \ z \ 2

-

(1 - ( M - 1) 2 1 z 1 2)2     (1 + (MM - 1)1 z I f

  • (5)    A = в, B = - в (0 6 в < 1):

2

\ S f \ 6 (1 - в 2 \ z \ 2 ) 2

Список литературы Inequalities for the Schwarzian derivative for subclasses of convex functions in the unit disc

  • Duren P. L. Univalent Functions.-New York: Springer, 1983.
  • Janowski W. Some Extremal Problems for Certain Families of Analytic Functions. I//Annales Polinici Math.-1973.-V. 28.-P. 297-326.
  • Kraus W. Uber den Zusammanhang Charakteristiken Eines Einfach Zusammeshangenden Bereiches mit der Kreisabbildung//Mitt. Math. Sem. Giessen.-1932.-V. 21.-P. 1-28.
  • Nehari Z. The Schwarzian Derivative and Schlicht Functions//Bull. Amer. Math. Soc.-1949.-V. 55.-P. 445-551.
  • Robertson M. S. Univalent Functions for which zf'(z) is spirallike//Michigan Math. J.-1969.-V. 16.-P. 97-101.
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