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International Journal of Mathematics and Mathematical Sciences
Volume 2010, Article ID 275935, 12 pages
http://dx.doi.org/10.1155/2010/275935
Research Article

On Certain Classes of -Valent Functions by Using Complex-Order and Differential Subordination

1Department of Mathematics, Islamic Azad University, South Tehran Branch, Tehran, Iran
2Department of Mathematics and Institute for Mathematical Research, University Putra Malaysia (UPM), Serdang, Selangor 43400, Malaysia

Received 29 May 2010; Revised 24 September 2010; Accepted 16 October 2010

Academic Editor: Vladimir Mityushev

Copyright © 2010 Abdolreza Tehranchi and Adem Kılıçman. 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 the present paper is to study the -valent analytic functions in the unit disk and satisfy the differential subordinations where is an operator defined by Slgean and is a complex number. Further we define a new related integral operator and also study the Fekete-Szego problem by proving some interesting properties.

1. Introduction

Let be the class of analytic functions in . Let denote the class of all analytic functions in the form of where is Gaussian hypergeometric function defined by Note that it is easy to see that these functions are analytic in the unit disk ; for more details on hypergeometric functions , see [1, 2].

Definition 1.1. A function is said to be in the class , -valently starlike functions of order , if it satisfies . We write , the class of -valently starlike functions in .
Similarly, a function is said to be in the class , -valently convex of order , if it satisfies .
Let be analytic and . A function is in the class if The class and a corresponding convex class were defined by Ma and Minda in [3]. Similar results which are related to the convex class can also be obtained easily from the corresponding functions in . For example, (i)if and then the classes reduce to the usual classes of starlike and convex functions; (ii)if where , then the classes are reduced to the usual classes of starlike and convex functions of order ;(iii)if , where , then the classes are reduced to the class of Janowski starlike functions which is defined by (iv)if where and , then the classes reduce to the classes of strongly starlike and convex functions of order that consists of univalent functions satisfing or equivalently we have In the literature, there are several works and many researchers have been studying the related problems. For example, Obradović and Owa [4], Silverman [5], Obradowič and Tuneski [6], and Tuneski [7] have studied the properties of classes of functions which are defined in terms of the ratio of .

Definition 1.2. A function is said to be -valent Bazilevic of type and order if there exists a function such that for some and . We denote by , the subclass of consisting of all such functions. In particular, a function in is said to be -valently close-to-convex of order in .

Definition 1.3. Let and be analytic functions in , then we say is subordinate to and denoted by if there exists a Schwarz function , analytic in with and , such that . In particular, if the function is univalent in , the above subordination is equivalent to and . Also, we say that is superordinate to ; see [8].

Definition 1.4. Motivated by the multiplier transformation on , we define the operator ; by the following infinite series when then Sălăgean derivative operator is closely related to the operator ; see [9]. In [10], Uralegaddi and Somanatha also studied the case . The operator was studied recently by Cho and Srivastava [11] and Cho and Kim [12].

Definition 1.5. Differential operator, for each , we have where , and . In particular, if we have .

Definition 1.6. A function is said to be in the class if it satisfies the following subordination: and in this study we consider where and is a complex number; so we denote . Then we say that is superordinate to if satisfies the following: where is analytic in and .
Further we note that if By choosing , so , then . For , and , we have . But if and and , then , a class of Janowski starlike functions. If we put , then classes of strongly starlike. By Definition 1.2, if , univalent starlike, and and and if , then is a class Bazilevic functions of type and order .

2. Main Results

Theorem 2.1. Let the function be of the form (1.1). If some , and are complex numbers and then , where and for . The result is sharp.

Proof. Since the function in the theorem can be expressed in the form where and , and also we have for all , now, assume that the condition (2.1) holds true. We show that . Equivalently, we prove that where . But we have where denotes .
The last inequality is true by (2.1) and this completes the proof. The result is sharp for the functions defined in by for.

Remark 2.2. We observe that if , the converse of the above theorem needs not be true. For instance, consider the function defined by where thus accordingly , and . It is easily seen that and so that where are satisfying the conditions . This establishes our claim.

Theorem 2.3. If the function , then where and , and the estimate is sharp.

Proof. We have where is defined as in the Definition 1.3. Now we can write where . Now if we equalize the coefficients of the same power of in both sides, then we have where ’s are suitable constants. By multiplying each side of the above equation by its conjugate and letting , , we get so that Since and , we have and this completes the proof. Note that the estimate in (2.10) is sharp for the functions defined in ; when in (1.3), then where , and . We can choose .

Theorem 2.4 ([Fekete-Szego Problem]). Let the function , given by (2.2), be in the class and any complex number. Then

Proof. On using the coefficients of and , we get By using [13] that for every complex number , then we can write where

3. Integral Operator

Now, we introduce a new integral operator which is denoted by on functions belonging to as follows: and we verify the effect of this operator on (1.11); with a simple calculation, we have where . If we put in (1.11), then we obtain that is denoted by .

Now if we let be a class of functions analytic in and defined by (3.1) where . Then on using (3.1) and definition of subordination, we have the following theorem.

Theorem 3.1. if and only if

Proof. The conditions (3.3) and (3.1) give where . By putting and in (3.3), we obtain With a simple calculation on , we have Let be the class of functions analytic in defined by . We can write next theorem on using (3.3) and definition of subordination.

Theorem 3.2. The if and only if

Proof. Since then we have Now by making substitution and in (3.3), we obtain

Acknowledgments

The authors would like to thank referee(s) for the very useful comments that improved the quality of the paper very much. The second author also acknowledges that this research was partially supported by the University Putra Malaysia under the Research University Grant Scheme (RUGS) no. 05-01-09-0720RU.

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