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 [3–7].
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.