Abstract

By using a linear operator, we obtain some new results for a normalized analytic function f defined by means of the Hadamard product of Hurwitz zeta function. A class related to this function will be introduced and the properties will be discussed.

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. For example, the Gamma function is meromorphic in the whole complex plane; see [1, 2].

In the present paper, we will derive some properties of univalent functions defined by means of the Hadamard product of Hurwitz Zeta function; a class related to this function will be introduced and the properties of the liner operator will be discussed.

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: (, ; when ; when ).

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 (10) that

In order to prove our main results, we recall the following lemma according to Yang [13].

Lemma 1. Let be analytic functions in with for . If where , and then The bound in (14) is the best possible.

3. Main Results

We begin with the following theorem.

Theorem 2. Let , for and suppose that where Then The bound in (17) is the best possible.

Proof. Define the function by Then, clearly analytic function in with for . It follows from (18) and (11) that by making use of the familiar identity (11) in (19), we obtain or, equivalent, Applying Lemma 1, with , we get the required result.

Letting in Theorem 2, we have the following.

Corollary 3. Let for and suppose that where Then The bound in (24) is the best possible.

Letting in Corollary 3, we have the following.

Corollary 4. Let and for and suppose that Then is starlike in .

Theorem 5. Let , for and suppose that where Then The bound in (28) is the best possible.

Proof. Define the function by Then, clearly analytic function in with for . It follows from (29) that by making use of the familiar identity (11) in (30), we get or, equivalent Applying Lemma 1, with , we get the required result.

Letting in Theorem 5, we have

Corollary 6. Let for and suppose that where Then
The bound in (35) is the best possible.

Letting , , and in Corollary 6, we have the following.

Corollary 7. Let for and suppose that Then The result is sharp.

Theorem 8. Let , for and suppose that where Then The bound in (40) is the best possible.

Proof. Define the function by Then, clearly analytic function in with for . Also by a simple computation and by making use of the familiar identity (11), we find from (41) that Applying Lemma 1, with , we get the required result.

Letting in Theorem 8, we have the following.

Corollary 9. Let , for and suppose that where Then The bound in (45) is the best possible.

Letting , , and in Corollary 9, we have the following.

Corollary 10. Let for and suppose that Then The result is sharp.

Conflict of Interests

The authors declare that they have no competing interests.

Authors’ Contribution

Both authors read and approved the final paper.

Acknowledgment

The work here is supported by a special grant: DIP-2013-1.