Abstract and Applied Analysis

Volume 2012 (2012), Article ID 150571, 10 pages

doi:10.1155/2012/150571

Research Article

On Subclass of -Uniformly Convex Functions of Complex Order Involving Multiplier Transformations

Waggas Galib Atshan and Ali Hamza Abada

Department of Mathematics, College of Computer Science and Mathematics, University of Al-Qadisiya, Diwaniya, Iraq

Received 30 December 2011; Revised 28 February 2012; Accepted 13 March 2012

Academic Editor: Ondřej Došlý

Copyright © 2012 Waggas Galib Atshan and Ali Hamza Abada. 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.

Abstract.

We introduce a subclass of -uniformly convex functions of order with negative coefficients by using the multiplier transformations in the open unit disk . We obtain coefficient estimates, radii of convexity and close-to-convexity, extreme points, and integral means inequalities for the function that belongs to the class .

1. Introduction

Let denote the class of functions of the form:

which are analytic and univalent in the open unit disk (see [1]). Also denote by the subclass of consisting of functions of the form:

For any integer , we define the multiplier transformations (see [2, 3]) of functions by

where .

A function is said to be in the class (-uniformly starlike Functions of order ) if it satisfies the condition:

and is said to be in the class (-uniformly convex Functions of order ) if it satisfies the condition:

Indeed it follows from (1.4) and (1.5) that

The interesting geometric properties of these function classes were extensively studied by Kanas et al., in [4, 5], motivated by Altintas et al. [6], Murugusundaramoorthy and Srivastava [7], and Murugusundaramoorthy and Magesh [8, 9], Atshan and Kulkarni [10] and Atshan and Buti [11].

Now, we define a new subclass of uniformly convex functions of complex order.

For , , , we let be the class of functions satisfying (1.2) with the analytic criterion:

where is given by (1.3).

2. Main Results

First, we obtain the necessary and sufficient condition for functions in the class .

Theorem 2.1. The necessary and sufficient condition for of the form of (1.2) to be in the class is

where , , .

Proof. Suppose that (2.1) is true for . Then

if

that is, if

Conversely, assume that , then

Letting along the real axis, we have

Hence, by maximum modulus theorem, the simple computation leads to the desired inequality

which completes the proof.

Corollary 2.2. Let the function defined by (1.2) belong to . Then,

where , , with equality for

3. Radii of Convexity and Close-to-Convexity

We obtain the radii of convexity and close-to-convexity results for functions in the class in the following theorems.

Theorem 3.1. Let . Then is convex of order in the disk , where

Proof. Let . Then by Theorem 2.1, we have

For , we need to show that

and we have to show that

Hence,

This is enough to consider

Therefore,

Setting in (3.7), we get the radius of convexity, which completes the proof of Theorem 3.1.

Theorem 3.2. Let . Then is close-to-convex of order in the disk , where

Proof. Let . Then by Theorem 2.1, we have

For , we need to show that

and we have to show that

Hence,

This is enough to consider

Therefore,

Setting in (3.14), we get the radius of close-to-convexity, which completes the proof of Theorem 3.2.

4. Extreme Points

The extreme points of the class are given by the following theorem.

Theorem 4.1. Let

for .

Then, if and only if it can be expressed in the form:

where and

Proof. Suppose that can be expressed as in (4.2). Our goal is to show that . By (4.2), we have that

Now,

Thus, .

Conversely, assume that . Since

we can set

Then,

This completes the proof of Theorem 4.1.

5. Integral Means

In order to find the integral means inequality and to verify the Silverman Conjuncture [12] for , we need the following definition of subordination and subordination result according to Littlewood [13].

Definition 5.1 (see [13]). Let and be analytic in . Then, we say that the function is subordinate to if there exists a Schwarz function , analytic in with , such that . We denote this subordination or . In particular, if the function is univalent in , the above subordination is equivalent to , .

Lemma 5.2 (see [13]). If the functions and are analytic in with , then

Applying Theorem 2.1 with the extremal function and Lemma 5.2, we prove the following theorem.

Theorem 5.3. Let . If and are nondecreasing sequences, then, for and , one has

where

Proof. Let of the form of (1.2) and

then we must show that

By Lemma 5.2, it suffices to show that

Setting

from (5.7) and (2.1) we obtain

This completes the proof of Theorem 5.3.

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