On Certain Subclass of Meromorphic Spirallike Functions Involving the Hypergeometric Function

Abstract

We introduce and investigate a new subclass [superscript]Mm1[/superscript] (θ,λ,η) 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.

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Academic Editor:Jin-Lin Liu

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

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

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.

1. Introduction

Let M denote the class of functions f of the form [figure omitted; refer to PDF] which are analytic in the punctured open unit disk: [figure omitted; refer to PDF]

Let f , g∈M , where f is given by (1) and g is defined by [figure omitted; refer to PDF] Then the Hadamard product (or convolution) f*g of f and g is defined by [figure omitted; refer to PDF]

Let P denote the class of functions p given by [figure omitted; refer to PDF] which are analytic in U and satisfy the condition [figure omitted; refer to PDF]

For θ which is real with |θ|<π/2 , 0...4;γ<1 , we denote by M^S* (θ,γ) and M^K* (θ,γ) the subclasses of f∈M which are defined, respectively, by [figure omitted; refer to PDF] By setting θ=0 in (7), we get the well-known subclasses of f∈M consisting of meromorphic functions which are starlike and convex of order γ (0...4;γ<1) , 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 η>1 , Wang et al. [5] and Nehari and Netanyahu [6] introduced and studied the subclass M(η) of M consisting of functions f satisfying [figure omitted; refer to PDF]

Let A be the class of functions of the form [figure omitted; refer to PDF] which are analytic in U . A function f∈A is said to be in the class ^S* (θ,δ,γ) if it satisfies the condition [figure omitted; refer to PDF] The function class ^S* (θ,δ,γ) is introduced and studied recently by Orhan et al. [7]. An analogous of the class ^S* (θ,δ,γ) has been studied by Murugusundaramoorthy [8].

For complex parameters _α1 ,...,_αl and _β1 ,...,_βm (_βj ...0;0,-1,-2,...; j=1,2,...,m) the generalized hypergeometric function _lFm (z) is defined by [figure omitted; refer to PDF] where N denotes the set of all positive integers and (a_)k is the Pochhammer symbol defined by [figure omitted; refer to PDF]

For a function f∈M , we consider a linear operator (which is a meromorphically modified version of the familiar Dziok-Srivastava linear operator [9, 10]: [figure omitted; refer to PDF] where _αs >0 , _βt >0 (s=1,...,l; t=1,...,m; l...4;m+1; l,m∈_N0 ) .

From the definition of the operator ^Hml [α,β]f , it is easy to observe that [figure omitted; refer to PDF] where [figure omitted; refer to PDF] is a positive number for all n∈N .

Recently, Aouf [11], Liu and Srivastava [12], and Raina and Srivastava [13] obtained many interesting results involving the linear operator ^Hml [α,β] . In particular, for [figure omitted; refer to PDF] we obtain the following linear operator: [figure omitted; refer to PDF] 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 ^Hml [α,β] is a generalization of other linear operators considered in many earlier investigations (see, e.g., [16-18]).

Using the operator ^Hml [α,β]f , we introduce the following class of meromorphic functions.

Definition 1.

For |θ|<π/2 , 0...4;λ<1/2 , and η>1 , let ^Mml (θ,λ,η) denote a subclass of M consisting of functions satisfying the condition that [figure omitted; refer to PDF] where ^Hml [α,β]f is given by (13).

We note that, for l=2 , m=1 , _α1 =_α2 =1 , _β1 =1 , and θ=λ=0 , the class ^M12 (0,0,η) becomes the class M(η) .

In the present paper, we aim at proving some interesting properties such as integral representations, convolution properties, and coefficient estimates for the class ^Mml (θ,λ,η) .

The following lemma will be required in our investigation.

Lemma 2.

Suppose that the sequence ^{{_An }n=1∞} is defined by [figure omitted; refer to PDF] Then [figure omitted; refer to PDF]

Proof.

From (19), we have [figure omitted; refer to PDF] Combining (21), we find that [figure omitted; refer to PDF] Thus, for n...5;2 , we deduce from (22) that [figure omitted; refer to PDF] This completes the proof of Lemma 2.

2. Main Results

We begin by proving the following integral representation for the class ^Mml (θ,λ,η) .

Theorem 3.

Let f∈^Mml (θ,λ,η) . Then [figure omitted; refer to PDF] where ω is analytic in U with ω(0)=0 and |ω(z)|<1 .

Proof.

Suppose that f∈^Mml (θ,λ,η) and [figure omitted; refer to PDF] We know that τ∈P , which implies [figure omitted; refer to PDF] where ω is analytic in U with ω(0)=0 and |ω(z)|<1 . We find from (26) that [figure omitted; refer to PDF] which follows [figure omitted; refer to PDF] Integrating both sides of (28) yields [figure omitted; refer to PDF] From (29), we obtain [figure omitted; refer to PDF] Thus, the assertion (24) of Theorem 3 follows directly from (30).

Next, we derive a convolution property for the class ^Mml (θ,λ,η) .

Theorem 4.

Let ξ∈C and |ξ|=1 . Then f∈^Mml (θ,λ,η) if and only if [figure omitted; refer to PDF]

Proof.

From the definition (18), we know that f∈^Mml (θ,λ,η) if and only if [figure omitted; refer to PDF] which is equivalent to [figure omitted; refer to PDF] On the other hand, we find from (14) that [figure omitted; refer to PDF] Combining (33) and (34), we get assertion (31) of Theorem 4.

Now, we discuss the coefficient estimates for functions in the class ^Mml (θ,λ,η) .

Theorem 5.

Suppose that f∈^Mml (θ,λ,η) . Then [figure omitted; refer to PDF]

Proof.

Let f∈^Mml (θ,λ,η) . Then there exists τ∈P such that [figure omitted; refer to PDF] It follows from (36) that [figure omitted; refer to PDF] Combining (1) and (37), we have [figure omitted; refer to PDF] Evaluating the coefficient of ^zn in both sides of (38) yields [figure omitted; refer to PDF] By observing the fact that |_τn |...4;2 for n∈N , we find from (39) that [figure omitted; refer to PDF] Now we define the sequence {_An^}n=1∞ as follows: [figure omitted; refer to PDF] In order to prove that [figure omitted; refer to PDF] we use the principle of mathematical induction. Note that [figure omitted; refer to PDF] Therefore, assume that [figure omitted; refer to PDF] Combining (41) and (42), we get [figure omitted; refer to PDF] Hence, by the principle of mathematical induction, we have [figure omitted; refer to PDF] 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 θ=0 , l=2 , m=1 , _α1 =_α2 =_β1 =1 , and λ=0 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 ^Mml (θ,λ,η) .

Theorem 7.

Let ζ be a real number with 0...4;ζ<1 . If f∈M satisfies the condition [figure omitted; refer to PDF] then f∈^Mml (θ,λ,η) provided that [figure omitted; refer to PDF]

Proof.

From (48), it follows that [figure omitted; refer to PDF] where ω is analytic in U with ω(0)=0 and |ω(z)|<1 . Thus, we have [figure omitted; refer to PDF] provided that cos...θ...5;(1-ζ)/(η-1) . This completes the proof of Theorem 7.

If we take ζ=1-(η-1)cos...θ in Theorem 7, we obtain the following result.

Corollary 8.

If f∈M satisfies the inequality [figure omitted; refer to PDF] then f∈^Mml (θ,λ,η) .

Theorem 9.

If a function f∈M given by (1) satisfies the inequality [figure omitted; refer to PDF] then it belongs to the class ^Mml (θ,λ,η) .

Proof.

In virtue of Corollary 8, it suffices to show that condition (52) holds. We observe that [figure omitted; refer to PDF] The last expression is bounded by (η-1)cos...θ , if [figure omitted; refer to PDF] which is equivalent to [figure omitted; refer to PDF] This completes the proof of Theorem 9.

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.

Conflict of Interests

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

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