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Abstract and Applied Analysis
Volume 2012 (2012), Article ID 383592, 10 pages
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.
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.
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 , , ,  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 , , , , if and only if there exists a , such that for where at . We denote by . Bazilevic shows that , for , . Note that
The generalized Salagean operator , , , is defined  as where
The Operator satisfies the following identity: Not that for , , Salagean differential operator .
Let we mean by , the solution of . Hence where . It is known  that , hence , and that
The class , is defined as if and only if . For , we get Salagean-type -starlike functions .
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.
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)For , , Salagean type close to convex functions, Blezu .(3)For , , , Kaplan .(4)For , , Abdul Halim  and , Opoola .
The following lemmas are needed to prove our results.
Lemma 2.1 (see ). 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 ). Let and , . Then .
For , let denote the largest positive number so that every is convex in . The following result is due to Al-Amiri .
Lemma 2.4. One has
Lemma 2.5 (see ). Let , with and . Let in , where Then for each , where stands for closed convex hull.
Lemma 2.7 (see ). Let be such that and let , , . Then
Lemma 2.8. One has
Lemma 2.9 (see ). Let , . Then where stands for coefficient majorization.
3. Main Results
Theorem 3.1. One has
Proof. Let . Then there exists , such that
Since , and , application of Lemma 2.3 gives
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
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)
Corollary 3.3 can be improved for , as follows.
Theorem 3.6. Let , . Then
Proof. We will use Ruscheweyh’s.method of proof . 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.
For the coefficient bounds of , Theorem 3.6 is not strong enough to settle this problem for , In 1962, Zamorski  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 , ,
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