The Scientific World Journal

Volume 2014 (2014), Article ID 541371, 7 pages

http://dx.doi.org/10.1155/2014/541371

Research Article

On Certain Subclass of Meromorphic Spirallike Functions Involving the Hypergeometric Function

Lei Shi and Zhi-Gang Wang

School of Mathematics and Statistics, Anyang Normal University, Anyang, Henan 455000, China

Received 23 March 2014; Revised 11 May 2014; Accepted 12 May 2014; Published 25 May 2014

Academic Editor: Jin-Lin Liu

Copyright © 2014 Lei Shi and Zhi-Gang Wang. 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 introduce and investigate a new subclass of meromorphic spirallike functions. Such results as integral representations, convolution properties, and coefficient estimates are proved. The results presented here would provide extensions of those given in earlier works. Several other results are also obtained.

1. Introduction

Let denote the class of functions of the form

which are analytic in the punctured open unit disk:

Let , , where is given by (1) and is defined by

Then the Hadamard product (or convolution) of and is defined by

Let denote the class of functions given by

which are analytic in and satisfy the condition

For which is real with , , we denote by and the subclasses of which are defined, respectively, by

By setting in (7), we get the well-known subclasses of consisting of meromorphic functions which are starlike and convex of order , respectively. For some recent investigations on meromorphic spirallike functions and related topics, see, for example, the earlier works [1–4] and the references cited therein.

For , Wang et al. [5] and Nehari and Netanyahu [6] introduced and studied the subclass of consisting of functions satisfying

Let be the class of functions of the form

which are analytic in . A function is said to be in the class if it satisfies the condition

The function class is introduced and studied recently by Orhan et al. [7]. An analogous of the class has been studied by Murugusundaramoorthy [8].

For complex parameters and the generalized hypergeometric function is defined by

where denotes the set of all positive integers and is the Pochhammer symbol defined by

For a function , we consider a linear operator (which is a meromorphically modified version of the familiar Dziok-Srivastava linear operator [9, 10]:

where , .

From the definition of the operator , it is easy to observe that

where

is a positive number for all .

Recently, Aouf [11], Liu and Srivastava [12], and Raina and Srivastava [13] obtained many interesting results involving the linear operator . In particular, for

we obtain the following linear operator:

which was introduced and investigated earlier by Liu and Srivastava [14] and was further studied in a subsequent investigation by Srivastava et al. [15]. It should also be remarked that the linear operator is a generalization of other linear operators considered in many earlier investigations (see, e.g., [16–18]).

Using the operator , we introduce the following class of meromorphic functions.

Definition 1. For , , and , let denote a subclass of consisting of functions satisfying the condition that

where is given by (13).

We note that, for , , , , and , the class becomes the class .

In the present paper, we aim at proving some interesting properties such as integral representations, convolution properties, and coefficient estimates for the class .

The following lemma will be required in our investigation.

Lemma 2. Suppose that the sequence is defined by

Then

Proof. From (19), we have

Combining (21), we find that

Thus, for , we deduce from (22) that

This completes the proof of Lemma 2.

2. Main Results

We begin by proving the following integral representation for the class .

Theorem 3. Let . Then

where is analytic in with and .

Proof. Suppose that and

We know that , which implies

where is analytic in with and . We find from (26) that

which follows

Integrating both sides of (28) yields

From (29), we obtain

Thus, the assertion (24) of Theorem 3 follows directly from (30).

Next, we derive a convolution property for the class .

Theorem 4. Let and . Then if and only if

Proof. From the definition (18), we know that if and only if

which is equivalent to

On the other hand, we find from (14) that

Combining (33) and (34), we get assertion (31) of Theorem 4.

Now, we discuss the coefficient estimates for functions in the class .

Theorem 5. Suppose that . Then

Proof. Let . Then there exists such that

It follows from (36) that

Combining (1) and (37), we have

Evaluating the coefficient of in both sides of (38) yields

By observing the fact that for , we find from (39) that

Now we define the sequence as follows:

In order to prove that

we use the principle of mathematical induction. Note that

Therefore, assume that

Combining (41) and (42), we get

Hence, by the principle of mathematical induction, we have

as desired. By means of Lemma 2 and (42), we know that (20) holds. Combining (47) and (20), we readily get the coefficient estimates asserted by Theorem 5.

Remark 6. By setting , , , , and in Theorem 5, we get the corresponding result due to Wang et al. [5].

In what follows, we present some sufficient conditions for functions belonging to the class .

Theorem 7. Let be a real number with . If satisfies the condition

then provided that

Proof. From (48), it follows that

where is analytic in with and . Thus, we have

provided that . This completes the proof of Theorem 7.

If we take in Theorem 7, we obtain the following result.

Corollary 8. If satisfies the inequality

then .

Theorem 9. If a function given by (1) satisfies the inequality

then it belongs to the class .

Proof. In virtue of Corollary 8, it suffices to show that condition (52) holds. We observe that

The last expression is bounded by , if

which is equivalent to

This completes the proof of Theorem 9.

Conflict of Interests

The authors declare that there is no conflict of interests regarding the publication of this paper.

Acknowledgments

The present investigation was supported by the National Natural Science Foundation under Grant nos. 11301008 and 11226088, the Foundation for Excellent Youth Teachers of Colleges and Universities of Henan Province under Grant no. 2013GGJS-146, and the Natural Science Foundation of Educational Committee of Henan Province under Grant no. 14B110012 of China.

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