Abstract and Applied Analysis

Volume 2012 (2012), Article ID 837913, 10 pages

doi:10.1155/2012/837913

Research Article

On Certain Sufficiency Criteria for -Valent Meromorphic Spiralike Functions

Muhammad Arif

Department of Mathematics, Abdul Wali Khan University Mardan, Mardan, Pakistan

Received 10 June 2012; Accepted 6 August 2012

Academic Editor: Allan Peterson

Copyright © 2012 Muhammad Arif. 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.

We consider some subclasses of meromorphic multivalent functions and obtain certain simple sufficiency criteria for the functions belonging to these classes. We also study the mapping properties of these classes under an integral operator.

1. Introduction

Let denote the class of functions of the form

which are analytic and -valent in the punctured unit disk . Also let and denote the subclasses of consisting of all functions which are defined, respectively, by

We note that for and , the above classes reduce to the well-known subclasses of consisting of meromorphic multivalent functions which are, respectively, starlike and convex of order  . For the detail on the subject of meromorphic spiral-like functions and related topics, we refer the work of Liu and Srivastava [1], Goyal and Prajapat [2], Raina and Srivastava [3], Xu and Yang [4], and Spacek [5] and Robertson [6].

Analogous to the subclass of for meromorphic univalent functions studied by Wang et al. [7] and Nehari and Netanyahu [8], we define a subclass of consisting of functions satisfying

For more details of the above classes see also [9, 10].

Motivated from the work of Frasin [11], we introduce the following integral operator of multivalent meromorphic functions

For , (1.4) reduces to the integral operator introduced and studied by Mohammed and Darus [12, 13]. Similar integral operators for different classes of analytic, univalent, and multivalent functons in the open unit disk are studied by various authors, see [14–19].

In this paper, first, we find sufficient conditions for the classes and and then study some mapping properties of the integral operator given by (1.4).

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

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

Lemma 1.1 (see [20]). If with and satisfies the condition

then

Lemma 1.2 (see [21]). Let be a set in the complex plane and suppose that is a mapping from to which satisfies for , and for all real such that . If is analytic in and for all , then .

2. Some Properties of the Classes and

Theorem 2.1. If satisfies

then .

Proof. Let us set a function by

for . Then clearly (2.2) shows that .

Differentiating (2.2) logarithmically, we have

which gives

Thus using (2.1), we have

Hence, using Lemma 1.1, we have .

From (2.3), we can write

Since , it implies that . Therefore, we get

or

and this implies that .

If we take , we obtain the following result.

Corollary 2.2. If satisfies

then .

Theorem 2.3. If satisfies

then .

Proof. Let us set

Also let

Then clearly and . Now

Differentiating logarithmically and then simple computation gives us

Therefore, by using Lemma 1.1, we have

which implies that . Since

therefore

Since , so

or

It follows that .

Theorem 2.4. If satisfies

then , where , and

Proof. Let us set

Then is analytic in with .

Taking logarithmic differentiation of (2.22) and then by simple computation, we obtain

with

Now for all real and satisfying , we have

Reputing the values of , , , and and then taking real part, we obtain

where , , and are given in (2.21).

Let . Then and , for all real and satisfying , . By using Lemma 1.2, we have , that is .

If we put , we obtain the following result.

Corollary 2.5. If satisfies

then , where , .

Theorem 2.6. For , let and satisfy (2.9). If

then , where .

Proof. From (1.4), we obtain

Differentiating again logarithmically and then by simple computation, we get

or, equivalently we can write

Now taking real part on both sides, we obtain

This further implies that

Let

Clearly we have

Then by using (2.28) and Corollary 2.2, we obtain

Therefore with .

Making use of (2.27) and Corollary 2.5, one can prove the following result.

Theorem 2.7. For , let and satisfy (2.27). If

then , where .

Acknowledgment

The author would like to thank Prof. Dr. Ihsan Ali, Vice Chancellor Abdul Wali Khan University Mardan for providing excellent research facilities and financial support.

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