Journal of Applied Mathematics

Volume 2013 (2013), Article ID 396484, 6 pages

http://dx.doi.org/10.1155/2013/396484

Research Article

New Criteria for Meromorphic Multivalent Alpha-Convex Functions

Muhammad Arif,¹ Maslina Darus,² Fazal Ghani,¹ and Saeed Islam¹

¹Department of Mathematics, Abdul Wali Khan University Mardan, Khyber Pakhtunkhwa, Pakistan

²School of Mathematical Sciences, Faculty of Science and Technology, Universiti Kebangsaan Malaysia, 43600 Bangi, Selangor, Malaysia

Received 9 March 2013; Revised 14 July 2013; Accepted 15 July 2013

Academic Editor: Zhihua Zhang

Copyright © 2013 Muhammad Arif et al. 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 obtain sufficient condition for the class of meromorphic -valent alpha convex functions of order and then to study mapping properties of the newly defined integral operators. Many known results appeared as special consequences of our work.

1. Introduction

Let denote the class of meromorphic functions normalized by

which are analytic and -valent in the punctured unit disk . In particular, , , and . For which is real with , , , and , , we denote by , , and , the subclasses of consisting of all meromorphic -valent functions of the form (1) which are defined, respectively, by

Making , in (2), we get the well-known subclasses of consisting of meromorphic-valent functions which are starlike, convex, and alpha convex of order , respectively. For details of the classes defined by (2) and related topics, we refer the raeder to the work of Aouf and Hossen [1], Aouf and Srivastava [2], Ali and Ravichandran [3], Goyal and Prajapat [4], Joshi and Srivastava [5], Liu and Srivastava [6], Raina and Srivastava [7], Xu and Yang [8], and Owa et al. [9].

For , Wang et al. [10] and Nehari and Netanyahu [11] introduced and studied the subclass of consisting of functions satisfying

We now extended this concept to define a subclass of consisting of functions of the form (1) satisfying

For and in (4), we obtain the classes and of , respectively, studied by Arif  [12]; also see [13, 14].

Integral operators for different classes of analytic, univalent, and multivalent functions in the open unit disk are studied by various authors; see [15–21]. We now define the following general integral operator of meromorphic-valent functions:

For , we obtain the integral operator studied recently in [22, 23], and, further for , we obtain the integral operator introduced and studied by Mohammed and Darus [24].

Sufficient conditions were studied by various authors for different subclasses of analytic and multivalent functions; for some of the related work see [25–27]. The object of the present paper is to obtain sufficient conditions for the class and then study mapping properties of the integral operator given by (5). We also consider some special cases of our results which lead to various interesting corollaries and relevance of some of these results with other known results which are also mentioned.

We will assume throughout our discussion, unless otherwise stated, that is real with , , ,, , for , , , and

To obtain our main results, we need the following lemmas.

Lemma 1 (see [27]). If with and satisfies the condition

then

Lemma 2 (see [28]). Let satisfy the following condition:

If the function is analytic in and

then

2. Sufficiency Criteria for the Class

In this section we establish a new sufficiency criteria for the subclass of .

Theorem 3. If satisfies

then , where is given by (6).

Proof. Let us set a function by

for . Then clearly (13) shows that .

Logarithmic differentiating of (13) gives

which further implies

Thus using (12), we get

Therefore by Lemma 1, we have .

From (14), we can write

Since , it implies that . Therefore, we get

or

And therefore .

By taking and in Theorem 3, we obtain Corollaries 4 and 5, respectively, proved by Arif  [12].

Corollary 4. If satisfies

then .

Corollary 5. If satisfies

then .

Remarks. We note that by simple computation (13) gives

By taking suitable meromorphic starlike function for in (22) such as , which satisfies the inequality of Lemma 1, we can conclude that the functionof (22) is the subclass of a meromorphic function.

3. Some Properties of the Integral Operator

In this section, we discuss some mapping properties of the integral operator .

Theorem 6. For , let and satisfy (20). If

then with , and is given by (5).

Proof. From (5), we obtain

Dividing both sides by , we have

Differentiating again logarithmically, we have

Now by simple computation, we get

or equivalently we have

By taking real part on both sides, we obtain

which further implies that

Let

Clearly we have

Then by using (23) and Theorem 3 with , we obtain

Therefore with .

Making in Theorem 6, we have the following.

Corollary 7. For , let and satisfy (20). If

then with .

Theorem 8. For , let . If

then , where with .

Proof. From (26), we obtain

Let

such that is analytic in with . Then (36) can be written as

Taking real part on both sides, we have

where we have used (35) and the assumption that . Let us put

Then, for such that , we have

Thus using Lemma 2, we conclude that , which is equivalent to

That is, .

Acknowledgment

The work presented here was partially supported by LRGS/TD/2011/UKM/ICT/03/02.

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