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 +} A^{w ( 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 ^ e^taBz E D ^

W <  ¹ - ^^{( e} i^a B^{z ) | 2}             1- \\ < B ^- H^ei^{a 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 ^iα 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 + z}7E )= ^{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 ^{) z}p 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 = -в (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):

4в 2

\ S f \ 6 (1 - в ² \ z \ ² ) ² ‘