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The Scientific World Journal

Volume 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

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 [14] 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., [1618]).

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.

References

  1. M. K. Aouf and B. A. Frasin, “Properties of some families of meromorphic multivalent functions involving certain linear operator,” Filomat, vol. 24, no. 3, pp. 35–54, 2010. View at Publisher · View at Google Scholar · View at Scopus
  2. M. Arif, J. Sokół, and M. Ayza, “Suffcient condition for functions to be in a class of meromorphic multivalent sakaguchi type spirallike functions,” Acta Mathematica Scientia, vol. 34, no. 2, pp. 575–578, 2014. View at Google Scholar
  3. L. Shi, Z.-G. Wang, and J.-P. Yi, “A new class of meromorphic functions associated with spirallike functions,” Journal of Applied Mathematics, vol. 2012, Article ID 494917, 12 pages, 2012. View at Publisher · View at Google Scholar
  4. L. Shi, Z.-G. Wang, and M.-H. Zeng, “Some subclasses of multivalent spirallike meromorphic functions,” Journal of Inequalities and Applications, vol. 336, 12 pages, 2013. View at Publisher · View at Google Scholar
  5. Z.-G. Wang, Y. Sun, and Z.-H. Zhang, “Certain classes of meromorphic multivalent functions,” Computers and Mathematics with Applications, vol. 58, no. 7, pp. 1408–1417, 2009. View at Publisher · View at Google Scholar · View at Scopus
  6. Z. Nehari and E. Netanyahu, “On the coefficients of meromorphic schlicht functions,” Proceedings of the American Mathematical Society, vol. 8, no. 1, pp. 15–23, 1957. View at Google Scholar
  7. H. Orhan, D. Răducanu, M. C. Çağlar, and M. Bayram, “Coefficient estimates and other properties for a class of spirallike functions associated with a differential operator,” Abstract and Applied Analysis, vol. 2013, Article ID 415319, 7 pages, 2013. View at Publisher · View at Google Scholar
  8. G. Murugusundaramoorthy, “Subordination results for spiral-like functions associated with the Srivastava-Attiya operator,” Integral Transforms and Special Functions, vol. 23, no. 2, pp. 97–103, 2012. View at Publisher · View at Google Scholar · View at Scopus
  9. J. Dziok and H. M. Srivastava, “Classes of analytic functions associated with the generalized hypergeometric function,” Applied Mathematics and Computation, vol. 103, no. 1, pp. 1–13, 1999. View at Google Scholar · View at Scopus
  10. J. Dziok and H. M. Srivastava, “Certain subclasses of analytic functions associated with the generalized hypergeometric function,” Integral Transforms and Special Functions, vol. 14, no. 1, pp. 7–18, 2003. View at Publisher · View at Google Scholar · View at Scopus
  11. M. K. Aouf, “Certain subclasses of meromorphically multivalent functions associated with generalized hypergeometric function,” Computers and Mathematics with Applications, vol. 55, no. 3, pp. 494–509, 2008. View at Publisher · View at Google Scholar · View at Scopus
  12. J.-L. Liu and H. M. Srivastava, “Classes of meromorphically multivalent functions associated with the generalized hypergeometric function,” Mathematical and Computer Modelling, vol. 39, no. 1, pp. 21–34, 2004. View at Publisher · View at Google Scholar · View at Scopus
  13. R. K. Raina and H. M. Srivastava, “A new class of meromorphically multivalent functions with applications to generalized hypergeometric functions,” Mathematical and Computer Modelling, vol. 43, no. 3-4, pp. 350–356, 2006. View at Publisher · View at Google Scholar · View at Scopus
  14. J.-L. Liu and H. M. Srivastava, “A linear operator and associated families of meromorphically multivalent functions,” Journal of Mathematical Analysis and Applications, vol. 259, no. 2, pp. 566–581, 2001. View at Publisher · View at Google Scholar · View at Scopus
  15. H. M. Srivastava, D.-G. Yang, and N.-E. Xu, “Some subclasses of meromorphically multivalent functions associated with a linear operator,” Applied Mathematics and Computation, vol. 195, no. 1, pp. 11–23, 2008. View at Publisher · View at Google Scholar · View at Scopus
  16. N. E. Cho and K. I. Noor, “Inclusion properties for certain classes of meromorphic functions associated with the Choi-Saigo-Srivastava operator,” Journal of Mathematical Analysis and Applications, vol. 320, no. 2, pp. 779–786, 2006. View at Publisher · View at Google Scholar · View at Scopus
  17. J.-L. Liu and S. Owa, “Some families of meromorphic multivalent functions involving certain linear operator,” Indian Journal of Mathematics, vol. 46, pp. 47–62, 2004. View at Google Scholar
  18. K. Piejko and J. Sokół, “Subclasses of meromorphic functions associated with the Cho-Kwon-Srivastava operator,” Journal of Mathematical Analysis and Applications, vol. 337, no. 2, pp. 1261–1266, 2008. View at Publisher · View at Google Scholar · View at Scopus