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