Abstract and Applied Analysis

Volume 2013 (2013), Article ID 763756, 7 pages

http://dx.doi.org/10.1155/2013/763756

Research Article

A Study of a Certain Subclass of Hurwitz-Lerch-Zeta Function Related to a Linear Operator

F. Ghanim

Department of Mathematics, College of Sciences, University of Sharjah, Sharjah, UAE

Received 8 July 2013; Accepted 23 August 2013

Academic Editor: Mohamed Amal Aouf

Copyright © 2013 F. Ghanim. 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.

By using a linear operator with Hurwitz-Lerch-Zeta function, which is defined here by means of the Hadamard product (or convolution), the author investigates interesting properties of certain subclasses of meromorphically univalent functions in the punctured unit disk .

1. Introduction

A meromorphic function is a single-valued function that is analytic in all but possibly a discrete subset of its domain, and at those singularities, it must go to infinity like a polynomial (i.e., these exceptional points must be poles and not essential singularities). A simpler definition states that a meromorphic function is a function of the form

where and are entire functions with (see [1, page 64]). A meromorphic function therefore may only have finite-order, isolated poles and zeros and no essential singularities in its domain. A meromorphic function with an infinite number of poles is exemplified by on the punctured disk . An equivalent definition of a meromorphic function is a complex analytic map to the Riemann sphere. Another definition for a meromorphic function is the following (see [2]).

Definition 1. A function on an open set is meromorphic if there exists a discrete set of points such that is holomorphic on and has poles at each . Furthermore, is meromorphic in the extended complex plane if   is either meromorphic or holomorphic at . In this case, we say that has a pole or is holomorphic at infinity.

Example 2. The Gamma function is meromorphic in the whole complex plane.

Example 3. All rational functions such as

are meromorphic on the whole complex plane.

Example 4. The functions and are meromorphic on the whole complex plane.

Example 5. The function is defined in the whole complex plane except for the origin . However, is not a pole of this function, rather an essential singularity. Thus, this function is not meromorphic in the whole complex plane.

Example 6. The complex logarithm function is not meromorphic on the whole complex plane, as it cannot be defined on the whole complex plane except for an isolated set of points.

The aim of this paper is to investigate interesting properties of certain subclasses of meromorphically univalent functions with linear operator which are defined here by means of the Hadamard product (or convolution) in the punctured unit disk .

2. Preliminaries

Let denote the class of meromorphic functions normalized by

which are analytic in the punctured unit disk . For , we denote by and the subclasses of consisting of all meromorphic functions which are, respectively, starlike of order and convex of order in .

For functions defined by

we denote the Hadamard product (or convolution) of and by

Let us define the function by

for , and , where is the Pochhammer symbol. We note that

where

is the well-known Gaussian hypergeometric function.

We recall here a general Hurwitz-Lerch-Zeta function, which is defined in [3, 4] by the following series:

.

Important special cases of the function include, for example, the Riemann Zeta function , the Hurwitz Zeta function , the Lerch Zeta function , , and the polylogarithm . Recent results on can be found in the expositions [5, 6]. By making use of the following normalized function we define

.

Corresponding to the functions and using the Hadamard product for , we define a new linear operator on by the following series:

.

The meromorphic functions with the generalized hypergeometric functions were considered recently by many others; see, for example, [7–12].

It follows from (11) that

We denote by the class of all functions such that

where satisfies the following condition:

where and are real numbers such that , , and with .

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

Lemma 7 (see [13]). Let be a set in the complex plane and let the function satisfy the condition for all real . If is analytic in with and , , then .

Lemma 8 (see [14]). If is analytic in with , and if with , then , implies , where is given by

which is increasing function of and . The estimate is sharp in the sense that the bound cannot be improved.

For real or complex numbers , , , , the Gauss hypergeometric function is defined by

One notes that the above series converges absolutely for and hence represents an analytic function in the unit disc (see, for details, [15, Chapter 14]).

Each of the identities (asserted by Lemma 9) is fairly well known (cf., e.g., [15, Chapter 14]).

Lemma 9. For real or complex parameters , , , , then

,

3. Main Results

Theorem 10. Let , and . Then

, where the function satisfies condition (14).

Proof. Let , and we define the function by

Then is analytic in and . If we set

then by the hypothesis . Differentiating (19) and using the identity (12), we have

Let us define the function by

Using (21) and the fact that , we obtain

Now for all real , we have

Hence for each , . Thus by Lemma 7, we have , and hence

This proves Theorem 10.

Corollary 11. Let the functions and be in , and let satisfy condition (14). If , , and

, , then

Proof. We have

Since making use of (27) and (19) (for ), we deduce that

Corollary 12. Let with and . If satisfies the following condition:

, then

Further, if , , and satisfies

then

Proof. The results (19) and (20) follow by putting in Theorem 10 and Corollary 11, respectively.

Remark 13. (i) Putting and in Corollary 12, we have that

implies

(ii) For with , , and in Corollary 12, we have that

implies

Choosing and appropriately in Corollary 12, we obtain the following results.

(iii) For , , and in Corollary 12, we have that

implies

(iv) For with , , , and in Corollary 12, we have that

implies

(v) Replacing by in the result (ii), we have that

implies

(vi) For with , , , and in Corollary 12, we have that

implies

Theorem 14. Let with and . If satisfies the following condition:

then

where

Proof. Let

Then is analytic in with . Differentiating with respect to and using the identity (12), we obtain

so that by the hypothesis (47), we have

In view Lemma 8, this implies that

where

Putting , we have

Using (17)-(18), we obtain

Thus the proof of Theorem 14 is complete.

Corollary 15. Let , with . If satisfies

,  then

where

Proof. The result follows by using the identity

Remark 16. (i) We note that if , , and in Corollary 15, that is,

then (32) implies that

whereas if satisfies condition (61); then, by using Theorem 14, we have

which is better than (61).

(ii) We observe that if satisfying and

Then, from Theorem 10 (for ), we have that

implies

whenever

Let ; then, from (66), we have that implies

whenever

In the following theorem, we will extend the above results as follows.

Theorem 17. Suppose that the functions and are in , and suppose that satisfies condition (14). If

,  then

Proof. Let

Then is analytic in with . Putting

we observe that by hypothesis , in . A simple computation shows that

where

Using the hypothesis (70), we obtain

Now, for all real , we have

This shows that for each .

Hence by Lemma 7, we have that this proves (71). The proof of (72) follows by using (71) and (72) in the identity:

This completes the proof of Theorem 17.

Remark 18. (i) Putting , , and in Theorem 17 for all , we obtain that

implies and

(ii) For , , and in Theorem 17 for all , we have that

implies

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