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The Scientific World Journal
Volume 2013 (2013), Article ID 475643, 5 pages
New Result of Analytic Functions Related to Hurwitz Zeta Function
1Department of Mathematics, College of Sciences, University of Sharjah, Sharjah, United Arab Emirates
2School of Mathematical Sciences, Faculty of Science and Technology, Universiti Kebangsaan Malaysia, 43600 Bangi, Selangor Darul Ehsan, Malaysia
Received 25 September 2013; Accepted 11 November 2013
Academic Editors: L. Gosse, T. Li, and P. Wang
Copyright © 2013 F. Ghanim and M. Darus. 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.
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.
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 .
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.
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.
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 .
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
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.
Both authors read and approved the final paper.
The work here is supported by a special grant: DIP-2013-1.
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