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Abstract and Applied Analysis
Volume 2012 (2012), Article ID 383592, 10 pages
http://dx.doi.org/10.1155/2012/383592
Research Article

-Bazilevic Functions

Department of Mathematical Sciences, College of Science, Princess Nora bint Abdul Rahman University, P.O. Box 4384, Riyadh 11491, Saudi Arabia

Received 31 December 2011; Revised 2 March 2012; Accepted 3 March 2012

Academic Editor: Khalida Inayat Noor

Copyright © 2012 F. M. Al-Oboudi. 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

The aim of this paper is to define and study a class of Bazilevic functions using the generalized Salagean operator. Some properties of this class are investigated: inclusion relation, some convolution properties, coefficient bounds, and other interesting results.

1. Introduction

Let be the set of analytic functions in the open unit disc . Let be the set of functions , with , and let be the set of functions , with . Let be the class of functions , which are univalent in . Denote by , , the class of starlike (convex)(close-to-convex) functions of order . Note that when , then , let . A function , where belongs to the Kaplan class , , , [1] if for and .

The Dual of is defined as where denotes Hadamard product (convolution).

A set is called a test set for (denoted by ) if . Note that if , then .

Denote by the class of functions , such that , in , and let . Note that for , and that , , if, and only if, .

For and , define the class as Note that , , .

A function , is called prestarlike of order , , (denoted by ) if and only if , or if and only if

Let , , , denote the class of Bazilevic functions in , introduced by Bazilevic [2], , , , if and only if there exists a , such that for where at . We denote by . Bazilevic shows that , for , . Note that

For further information, see [37].

The generalized Salagean operator , , , is defined [8] as where

The Operator satisfies the following identity: Not that for , , Salagean differential operator [9].

Let we mean by , the solution of . Hence where . It is known [10] that , hence , and that

The class , is defined as if and only if . For , we get Salagean-type -starlike functions [9].

The operator is now called “Al-Oboudi Operator” and has been extensively studied latly, [5, 11, 12].

In this paper we define and study a class of Bazilevic functions using the operator and study some of its basic properties, inclusion relation, convolution properties coefficient bounds, and other interesting results.

2. Definition and Preliminaries

In this section, the class of -Bazilevic functions , , where , is defined and some preliminary lemmas are given.

2.1. Definition

Let . Then , , , if and only if there exists a , such that where the power is chosen as a principal one.

Denote by the class of functions , where .

Using (1.10), we see that , from which the following special cases are clear.

2.1.1. Special Cases
(1)For , , , Bazilevic [2].(2)For , , Salagean type close to convex functions, Blezu [13].(3)For , , , Kaplan [14].(4)For , , Abdul Halim [15] and , Opoola [16].
2.2. Lemmas

The following lemmas are needed to prove our results.

Lemma 2.1 (see [10]). Let and , . Then for

Lemma 2.2. If , , then .

Proof. Since , we will show that . Now , implies Hence From (1.13), we get the required result.

Lemma 2.3 (see [1]). Let and , . Then .

For , let denote the largest positive number so that every is convex in . The following result is due to Al-Amiri [17].

Lemma 2.4. One has

Lemma 2.5 (see [18]). Let , with and . Let in , where Then for each , where stands for closed convex hull.

Remark 2.6. In [1], it was shown that condition (2.6) is satisfied for all in whenever is in and is in .

Lemma 2.7 (see [10]). Let be such that and let , , . Then

From (1.12) and (1.13), we immediately have;

Lemma 2.8. One has

Lemma 2.9 (see [10]). Let , . Then where stands for coefficient majorization.

3. Main Results

Theorem 3.1. One has

Proof. Let . Then there exists , such that Hence Since , and , application of Lemma 2.3 gives hence Using Lemma 2.2 we deduce that .
As a consequence of (3.1) we immediately have the following.Corollary 3.2. One has

Corollary 3.3. If , , , then, for

Proof. Since , there exists a or such that using (1.3). From (3.6), we conclude that which implies that Applying Lemma 2.3 to (3.9), we get the result.

Theorem 3.4. Let . Then

Proof. From (2.1), we see that Since , and , then which is the required result.

In the following we prove the converse of Theorem 3.1, for .

Theorem 3.5. Let , . Then in , where is given by (2.5)

Proof. implies (2.1), where .
Now Using Lemma 2.4, we see that in , for , where is given by (2.5).
From Remark 2.6, we conclude
Applying Lemma 2.5, we deduce hence in , as required.

Corollary 3.3 can be improved for , as follows.

Theorem 3.6. Let , . Then

Proof. We will use Ruscheweyh’s.method of proof [10]. implies (3.8), where , .
Let , where and .
Then and . This implies
Using Lemma 2.7, we get
Hence To prove that , we have to show that , or equivalently .
Since , then from (1.5) From Lemma 2.8, (1.13), and (3.22), we see that . Using (3.21), we obtain . From (1.1) we get the required result.

Remark 3.7. For , Theorem 3.6 and other stronger results depending on , are proved by Sheil-Small [7].

For the coefficient bounds of , Theorem 3.6 is not strong enough to settle this problem for , In 1962, Zamorski [19] proved the Bieberbach conjecture for , when , in the following we prove this result for , using the extreme points of Kaplan class .

Theorem 3.8. For , ,

Proof. From (3.9) and Lemma 2.9, we get using (2.10). Raising both sides of (3.24) to the th power, where , we get the required result.

Remark 3.9. For , we get the result of Zamorski [19], and the result of Sheil-Small [7], from which we get the idea of proof.

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