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Using the methods of differential subordination and superordination, sufficient conditions involving the Schwarzian derivative of a normalized analytic functionf are obtained so that either

Volume 2008, Article ID 712328, 18 pages

doi:10.1155/2008/712328

Research Article

Subordination and Superordination on

Schwarzian Derivatives

Rosihan M Ali, 1 V Ravichandran, 2 and N Seenivasagan 3

1 School of Mathematical Sciences, Universiti Sains Malaysia (USM), 11800 Penang, Malaysia

2 Department of Mathematics, University of Delhi, Delhi 110 007, India

3 Department of Mathematics, Rajah Serfoji Government College, Thanjavur 613 005, India

Correspondence should be addressed to Rosihan M Ali,rosihan@cs.usm.my

Received 4 September 2008; Accepted 30 October 2008

Recommended by Paolo Ricci

Let the functionsq1be analytic and letq2be analytic univalent in the unit disk Using the methods

of differential subordination and superordination, sufficient conditions involving the Schwarzian derivative of a normalized analytic functionf are obtained so that either q1z ≺ zfz/fz ≺

q2z or q1z ≺ 1  zfz/fz ≺ q2z As applications, sufficient conditions are determined

relating the Schwarzian derivative to the starlikeness or convexity off.

Copyrightq 2008 Rosihan M Ali et al This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited

1 Introduction

LetHU be the class of functions analytic in U : {z ∈ C : |z| < 1} and Ha, n be the

subclass ofHU consisting of functions of the form fz  a  a n z n  a n1 z n1 · · · We will writeH ≡ H1, 1 Denote by A the subclass of H0, 1 consisting of normalized functions f

of the form

k2

a k z k z ∈ U. 1.1

LetS∗andK, respectively, be the familiar subclasses of A consisting of starlike and convex functions inU.

The Schwarzian derivative{f, z} of an analytic, locally univalent function f is defined

by

{f, z} : fz

fz



−1 2

fz

fz

2

Owa and Obradovi´c1 proved that if f ∈ A satisfies

R

 1 2



1zf fz z

2

 z2{f, z}



conditions:

R



1 zfz

fz



 αz2{f, z}



> 0 Rα ≥ 0,

R



1zf fz z

2

 z2{f, z}



> 0,

1.4

or

R



1zf fz z



e z2{f,z}

R



φ



1zf fz z , z2{f, z}; z



impliesf ∈ K Each of the conditions mentioned above readily followed by choosing an

appropriateφ Miller and Mocanu 2 also found conditions on φ : C3× U → C such that

Rφ zfz

zfz

fz , z2{f, z}; z



impliesf ∈ S∗ As applications, iff ∈ A satisfies either

R



α zfz

fz



 β



1zf fz z



zfz

fz



z2{f, z}



or

Rzfz

fz



1zf fz z  z2{f, z}



> −1

thenf ∈ S∗

is said to be superordinate to f, written fz ≺ Fz, if there exists a function w analytic in

U with w0  0 and |wz| < 1 z ∈ U, such that fz  Fwz If F is univalent, then fz ≺ Fz if and only if f0  F0 and fU ⊂ FU.

In this paper, sufficient conditions involving the Schwarzian derivatives are obtained for functionsf ∈ A to satisfy either

q1 z ≺ zfz

fz ≺ q2z or q1z ≺ 1  zfz

fz ≺ q2z, 1.10

where the functionsq1are analytic andq2 is analytic univalent inU InSection 2, a class of admissible functions is introduced Sufficient conditions on functions f ∈ A are obtained

so that zfz/fz is subordinated to a given analytic univalent function q in U As a

consequence, we obtained the result1.7 of Miller and Mocanu 2 relating the Schwarzian derivatives to the starlikeness of functionsf ∈ A.

Recently, Miller and Mocanu 3 investigated certain first- and second-order dif-ferential superordinations, which is the dual problem to subordination Several authors have continued the investigation on superordination to obtain sandwich-type results4 20

In Section 3, superordination is investigated on a class of admissible functions Sufficient conditions involving the Schwarzian derivatives of functions f ∈ A are obtained so that

zfz/fz is superordinated to a given analytic subordinant q in U For q1analytic andq2

analytic univalent inU, sandwich-type results of the form

q1 z ≺ zf fzz ≺ q2z 1.11

are obtained This result extends earlier works by several authors

Section 4is devoted to finding sufficient conditions for functions f ∈ A to satisfy

q1 z ≺ 1  zfz

fz ≺ q2z. 1.12

As a consequence, we obtained the result1.6 of Miller and Mocanu 2

To state our results, we need the following preliminaries Denote byQ the set of all functionsq that are analytic and injective on U \ Eq, where

and are such thatqζ / 0 for ζ ∈ ∂U \ Eq Further, let the subclass of Q for which q0  a

be denoted byQa and Q1 ≡ Q1

positive integer The class of admissible functionsΨn Ω, q consists of those functions ψ :

C3× U → C that satisfy the admissibility condition

wheneverr  qζ, s  kζqζ, and

R

t



≥ kR ζqζ

qζ  1



z ∈ U, ζ ∈ ∂U \ Eq, and k ≥ n.

Definition 1.2see 3, Definition 3, page 817  Let Ω be a set in C, q ∈ Ha, n with qz / 0.

The class of admissible functionsΨ

n Ω, q consists of those functions ψ : C3× U → C that

satisfy the admissibility condition

wheneverr  qz, s  zqz/m, and

R

t



≤ m1Rzqz

qz  1



1Ω, q as ΨΩ, q

ψ



qz, zq mz;ζ



z ∈ U, ζ ∈ ∂U and m ≥ n.

Lemma 1.3 see 2, Theorem 2.3b, page 28  Let ψ ∈ Ψn Ω, q with q0  a If the analytic

function pz  a  a n z n  a n1 z n1  · · · satisfies

Lemma 1.4 see 3, Theorem 1, page 818  Let ψ ∈ Ψ

n Ω, q with q0  a If p ∈ Qa and

ψpz, zpz, z2pz; z is univalent in U, then

2 Subordination and starlikeness

We first define the following class of admissible functions that are required in our first result

consists of those functionsφ : C3× U → C that satisfy the admissibility condition

whenever

qζ qζ / 0,

R



2w  u2− 1  3v − u2

2v − u



≥ kR ζqζ

qζ  1



,

2.2

z ∈ U, ζ ∈ ∂U \ Eq, and k ≥ 1.

Theorem 2.2 Let f ∈ A with fzfz/z / 0 If φ ∈ Φ S Ω, q and



φ zfz

zfz

fz , z2{f, z}; z





then

zfz

A simple calculation yields

1zf fz z  pz  zp pzz 2.6

Further computations show that

z2{f, z}  zpz  z pz2pz−3

2

zpz

pz

2

1− p2z

Define the transformation fromC3toC3by

2

s

r

2

1− r2

Let

ψr, s, t; z  φu, v, w; z  φ



r, r  s r , s  t r −3

2

s

r

2

1− r2

2 ;z



. 2.9 The proof will make use ofLemma 1.3 Using2.5, 2.6, and 2.7, from 2.9 we obtain

ψpz, zpz, z2pz; z φ zfz

zfz

fz , z2{f, z}; z



. 2.10 Hence2.3 becomes

A computation using2.8 yields

t

s 1 

2w  u2− 1  3v − u2

Thus the admissibility condition for φ ∈ Φ S Ω, q in Definition 2.1 is equivalent to the admissibility condition forψ as given inDefinition 1.1 Henceψ ∈ ΨΩ, q and byLemma 1.3,

pz ≺ qz or

zfz

IfΩ / C is a simply connected domain, then Ω  hU for some conformal mapping

is an immediate consequence ofTheorem 2.2

Theorem 2.3 Let φ ∈ Φ S h, q If f ∈ A with fzfz/z / 0 satisfies

φ zfz

zfz

fz , z2{f, z}; z



≺ hz, 2.14

then

zfz

Following similar arguments as in 2, Theorem 2.3d, page 30 ,Theorem 2.3 can be extended to the following theorem where the behavior ofq on ∂U is not known.

Theorem 2.4 Let h and q be univalent in U with q0  1, and set q ρ z  qρz and h ρ z 

i φ ∈ Φ S h, q ρ for some ρ ∈ 0, 1, or

ii there exists ρ0 ∈ 0, 1 such that φ ∈ Φ S h ρ , q ρ for all ρ ∈ ρ0, 1.

If f ∈ A with fzfz/z / 0 satisfies 2.14, then

zfz

The next theorem yields the best dominant of the differential subordination 2.14

Theorem 2.5 Let h be univalent in U, and φ : C3× U → C Suppose that the differential equation

φ



qz, qz  zq qzz , zqz  z qz2qz−3

2

zqz

qz

2

 1− q2z

2 ;z



 hz 2.17

has a solution q with q0  1 and one of the following conditions is satisfied:

1 q ∈ Q1and φ ∈ Φ S h, q ,

2 q is univalent in U and φ ∈ Φ S h, q ρ for some ρ ∈ 0, 1, or

3 q is univalent in U and there exists ρ0 ∈ 0, 1 such that φ ∈ Φ S h ρ , q ρ for all ρ ∈ ρ0, 1.

If f ∈ A with fzfz/z / 0 satisfies 2.14, then

zfz

and q is the best dominant.

a dominant from Theorems2.3and2.4 Sinceq satisfies 2.17, it is also a solution of 2.14, and thereforeq will be dominated by all dominants Hence q is the best dominant.

We will applyTheorem 2.2to two specific cases First, letqz  1  Mz, M > 0.

Theorem 2.6 Let Ω be a set in C, and φ : C3× U → C satisfy the admissibility condition

φ



1 Me iθ , L; z



whenever z ∈ U, θ ∈ R, with

R

2L 1 Me iθ2

− 1 e −iθ  M 3k2M2



≥ 2k2M 2.20

for all real θ and k ≥ 1.

If f ∈ A with fzfz/z / 0 satisfies

φ zfz

fz , 1 

zfz

fz , z2{f, z}; z



then



zf fzz− 1

it belongs to the class of admissible functionsΦS Ω, 1  Mz The result follows immediately

fromTheorem 2.2

In the special caseΩ  qU  {ω : |ω − 1| < M}, the conclusion ofTheorem 2.6can be written as



φ zf fzz , 1  zf fz z , z2{f, z}; z



− 1

 < M ⇒zf fzz− 1

 < M. 2.23

φ2 u, v, w; z : v/u,  0 < M ≤ 2 satisfy the admissibility condition 2.19 and hence

Theorem 2.6yields



1 − α zf fzz  α



1 zf fz z



− 1

 < M ⇒zf fzz − 1

 < M α ≥ 2M − 1 ≥ 0,



1 zf zfz/fzz/fz− 1



 < M ⇒zf fzz − 1

 < M 0 < M ≤ 2.

2.24

By considering the functionφu, v, w; z : uv−1λu−1 with 0 < M ≤ 1, λ2−M ≥

0, it follows again fromTheorem 2.6that



z2fz fz  λ zfz



 ≤ M2  λ − M ⇒zf fzz− 1

 < M. 2.25 This above implication was obtained in21, Corollary 2, page 583

A second application ofTheorem 2.2is to the caseqU being the half-plane qU 

{w : Rw > 0} : Δ.

Theorem 2.8 Let Ω be a set in C and let the function φ : C3× U → C satisfy the admissibility

condition

for all z ∈ U and for all real ρ, τ, ξ and η with

21  3ρ2, ρη ≥ 0. 2.27

Let f ∈ A with fzfz/z / 0 If

φ zfz

fz , 1 

zfz

fz , z2{f, z}; z



then f ∈ S∗.

we obtain

2 , ζ2qζ  1  ρ21 − iρ

2 , 2.29 whereρ : cotθ/2 Note that

Rζqζ

qζ  1



 0 ζ / 1. 2.30

We next describe the class of admissible functions ΦS Ω, 1  z/1 − z in

Definition 2.1 Forζ / 1,

u  qζ : iρ, v  qζ  kζq qζζ  i



ρ  k1  ρ2

2ρ



: iτ, w  ξ  iη 2.31 with

R2w  u2− 1  3v − u2

2v − u



 2ρη

Thus the admissibility condition for functions inΦS Ω, 1  z/1 − z is equivalent to 2.26, whenceφ ∈ Φ S Ω, 1  z/1 − z FromTheorem 2.2, we deduce thatf ∈ S∗

functionsΦS hU, Δ as Φ SΔ , the following result is a restatement of 1.7, which is an immediate consequence ofTheorem 2.8

Corollary 2.9 see 2, Theorem 4.6a, page 244  Let φ ∈ ΦS Δ If f ∈ A with fzfz/z / 0

satisfies

R



φ zfz

zfz

fz , z2{f, z}; z



then f ∈ S∗.

3 Superordination and starlikeness

Now we will give the dual result ofTheorem 2.2for differential superordination

Φ

S Ω, q consists of those functions φ : C3× U → C that satisfy the admissibility condition

whenever

mqz

qz / 0, zqz / 0,

R



2w  u2− 1  3v − u2

2v − u



≤ m1Rzqz

qz  1



,

3.2

z ∈ U, ζ ∈ ∂U and m ≥ 1.

Theorem 3.2 Let φ ∈ Φ

S Ω, q , and f ∈ A with fzfz/z / 0 If zfz/fz ∈ Q1 and φzfz/fz, 1  zfz/fz, z2{f, z}; z is univalent in U, then

Ω ⊂



φ zfz

zfz

fz , z2{f, z}; z





3.3

implies

ψr, s, t; z  φ



r, r  s r , s  t r  3

2

s

r

2

 1− r2

2 ;z



 φu, v, w; z, 3.5 equations2.10 and 3.3 yield

t

s 1 

2w  u2− 1  3v − u2

the admissibility condition forφ ∈ Φ

S Ω, q is equivalent to the admissibility condition for ψ

as given inDefinition 1.2 Henceψ ∈ ΨΩ, q , and byLemma 1.4,qz ≺ pz or

IfΩ / C is a simply connected domain, then Ω  hU for some conformal mapping h

S hU, q as Φ

S h, q ,Theorem 3.2can be written in the following form

Theorem 3.3 Let q ∈ H, h be analytic in U and φ ∈ Φ

S h, q If f ∈ A, fzfz/z / 0,

zfz/fz ∈ Q1and φzfz/fz, 1  zfz/fz, z2{f, z}; z is univalent in U, then

hz ≺ φ zfz

zfz

fz , z2{f, z}; z



3.9

implies

Theorems 3.2 and 3.3 can only be used to obtain subordinants of differential superordinations of the form3.3 or 3.9 The following theorem proves the existence of the best subordinant of3.9 for an appropriate φ.

Theorem 3.4 Let h be analytic in U and φ : C3× U → C Suppose that the differential equation

φ



qz, qz  zq qzz , zqz  z qz2qz−3

2

zqz

qz

2

 1− q2z

2 ;z



 hz 3.11

has a solution q ∈ Q1 Let φ ∈ Φ

S h, q , and f ∈ A with fzfz/z / 0 If zfz/fz ∈ Q1and

φ zfz

zfz

fz , z2{f, z}; z



3.12

is univalent in U, then

hz ≺ φ zfz

zfz

fz , z2{f, z}; z



3.13

qz ≺ zfz

and q is the best subordinant.

Combining Theorems2.3and3.3, we obtain the following sandwich-type theorem

Corollary 3.5 Let h1and q1 be analytic functions in U, let h1 be an analytic univalent function in U, q2∈ Q1with q1 0  q20  1 and φ ∈ Φ S h2, q2 ∩ Φ

S h1, q1 Let f ∈ A with fzfz/z / 0.

If zfz/fz ∈ H ∩ Q1and φzfz/fz, 1  zfz/fz, z2{f, z}; z is univalent in U, then

h1 z ≺ φ zfz

zfz

fz , z2{f, z}; z



≺ h2z 3.15

implies

q1 z ≺ zfz

fz ≺ q2z. 3.16

4 Schwarzian derivatives and convexity

We introduce the following class of admissible functions

consists of those functionsφ : C2× U → C that satisfy the admissibility condition

φ



qζ, kζqζ 1− q2ζ

2 ;z



z ∈ U, ζ ∈ ∂U \ Eq, and k ≥ 1.

Theorem 4.2 Let φ ∈ Φ Sc Ω, q , and f ∈ A with fz / 0 If



φ



1zf fz z , z2{f, z}; z





then

1zf fz z ≺ qz. 4.3

Proof Define the function p by

Clearlyp ∈ A, and a simple calculation yields

z2{f, z}  zpz 1− p2z

Define the transformation fromC2toC2by

Let

ψr, s; z  φu, v; z  φ



r, s  1− r2

2 ;z



The proof will make use ofLemma 1.3 Using4.4 and 4.5, from 4.7, we obtain

ψpz, zpz; z φ



1zf fz z , z2{f, z}; z



Hence4.2 becomes

From 4.7, we see that the admissibility condition for φ ∈ Φ Sc Ω, q is equivalent to the

admissibility condition forψ as given inDefinition 1.1 Henceψ ∈ ΨΩ, q and byLemma 1.3,

pz ≺ qz or

1zf fz z ≺ qz. 4.10

We will denote byΦSc h, q the class Φ Sc hU, q , where h is the conformal mapping

established, which we state without proof

Theorem 4.3 Let φ ∈ Φ Sc h, q If f ∈ A with fz / 0 satisfies

φ



1zf fz z , z2{f, z}; z



1zfz

fz ≺ qz. 4.12

We extendTheorem 4.3to the case where the behavior ofq on ∂U is not known.

Theorem 4.4 Let Ω ⊂ C and let q be univalent in U with q0  1 Let φ ∈ Φ Sc h, q ρ for some

ρ ∈ 0, 1 where q ρ z  qρz If f ∈ A with fz / 0 satisfies 4.2, then 4.12 holds.

Theorem 4.5 Let Ω be a set in C, qz  1  Mz, M > 0, and φ : C2× U → C satisfy

φ



1 Me iθ ,2k − 1 − Meiθ

2 Me iθ;z



φ



1zfz

fz , z2{f, z}; z



then



zf fz z



In the special caseΩ  qU  {ω : |ω − 1| < M},Theorem 4.5gives the following: let



φ1 Me iθ ,2k − 1 − Meiθ

2 Me iθ;z



− 1

 ≥ M 4.16

wheneverz ∈ U, θ ∈ R, and k ≥ 1; if f ∈ A with fz / 0 satisfies



φ1zf fz z , z2{f, z}; z



− 1

 < M, 4.17

then



zf fz z



Example 4.6 If 0 < M < 2, and f ∈ A with fz / 0 satisfies



zf fz z  z2{f, z}



 < M, 4.19

Proof Define the function p by

Clearlyp ∈ A, and a simple calculation yields...

 φu, v, w; z, 3.5 equations2.10 and 3.3 yield

t

s... Δ.

Theorem 2.8 Let Ω be a set in C and let the function φ : C3×