Abstract and Applied Analysis
Volume 2011, Article ID 849250, 9 pages
http://dx.doi.org/10.1155/2011/849250
Research Article

Partial Sums of Generalized Class of Analytic Functions Involving Hurwitz-Lerch Zeta Function

1School of Advanced Sciences, VIT University, Vellore 632014, India
2School of Mathematical Sciences, Faculty of Science and Technology, Universiti Kebangsaan Malaysia, Bangi 43600, Malaysia

Received 29 January 2011; Accepted 7 April 2011

Academic Editor: Yoshikazu Giga

Copyright © 2011 G. Murugusundaramoorthy et al. 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

Let be the sequence of partial sums of the analytic function . In this paper, we determine sharp lower bounds for and . The usefulness of the main result not only provides the unification of the results discussed in the literature but also generates certain new results.

1. Introduction and Preliminaries

Let denote the class of functions of the form which are analytic and univalent in the open disc . We also consider a subclass of introduced and studied by Silverman , consisting of functions of the form

For functions given by (1.1) and given by , we define the Hadamard product (or convolution) of and by We recall here a general Hurwitz-Lerch zeta function defined in  by (; , when ; , when ), where, as usual, , (); . Several interesting properties and characteristics of the Hurwitz-Lerch Zeta function can be found in the recent investigations by Choi and Srivastava , Ferreira and López , Garg et al. , Lin and Srivastava , Lin et al. , and others. Srivastava and Attiya  (see also Răducanu and Srivastava  and Prajapat and Goyal ) introduced and investigated the linear operator defined in terms of the Hadamard product by where, for convenience, We recall here the following relationships (given earlier by [9, 10]) which follow easily by using (1.1), (1.5), and (1.6): Motivated essentially by the Srivastava-Attiya operator , we introduce the generalized integral operator where and (throughout this paper unless otherwise mentioned) the parameters , are constrained as ; , and . It is of interest to note that is the Srivastava-Attiya operator  and is the well-known Choi-Saigo-Srivastava operator (see [11, 12]). Suitably specializing the parameters , , , and in , we can get various integral operators introduced by Alexander  and Bernardi . Further more, we get the Jung-Kim-Srivastava integral operator  closely related to some multiplier transformation studied by Flett .

Motivated by Murugusundaramoorthy  and making use of the generalized Srivastava-Attiya operator , we define the following new subclass of analytic functions with negative coefficients.

For , , and , let be the subclass of consisting of functions of the form (1.1) and satisfying the analytic criterion where . Shortly we can state this condition by where and .

Recently, Silverman  determined sharp lower bounds on the real part of the quotients between the normalized starlike or convex functions and their sequences of partial sums. In the present paper and by following the earlier work by Silverman  (see ) on partial sums of analytic functions, we study the ratio of a function of the form (1.1) to its sequence of partial sums of the form when the coefficients of satisfy the condition (1.14). Also, we will determine sharp lower bounds for , , , and . It is seen that this study not only gives as a particular case, the results of Silverman , but also gives rise to several new results.

Before stating and proving our main results, we derive a sufficient condition giving the coefficient estimates for functions to belong to this generalized function class.

Lemma 1.1. A function of the form (1.1) is in if where, for convenience, , , , and , is given by (1.9).

Proof. The proof of Lemma 1.1 is much akin to the proof of Theorem 1 obtained by Murugusundaramoorthy , hence we omit the details.

2. Main Results

Theorem 2.1. If of the form (1.1) satisfies the condition (1.14), then where The result (2.1) is sharp with the function given by

Proof. Define the function by
It suffices to show that . Now, from (2.4) we can write Hence we obtain Now if and only if or, equivalently,
From the condition (1.14), it is sufficient to show that which is equivalent to To see that the function given by (2.3) gives the sharp result, we observe that for ,

We next determine bounds for .

Theorem 2.2. If of the form (1.1) satisfies the condition (1.14), then where and The result (2.12) is sharp with the function given by (2.3).

Proof. We write where This last inequality is equivalent to We are making use of (1.14) to get (2.10). Finally, equality holds in (2.12) for the extremal function given by (2.3).

We next turns to ratios involving derivatives.

Theorem 2.3. If of the form (1.1) satisfies the condition (1.14), then where and The results are sharp with the function given by (2.3).

Proof. We write where Now if and only if From the condition (1.14), it is sufficient to show that which is equivalent to
To prove the result (2.18), define the function by where Now if and only if It suffices to show that the left hand side of (2.27) is bounded previously by the condition which is equivalent to

Remark 2.4. As a special case of the previous theorems, we can determine new sharp lower bounds for, , , and for various function classes involving the Alexander integral operator  and Bernardi integral operators , Jung-Kim-Srivastava integral operator  and Choi-Saigo-Srivastava operator (see [11, 12]) on specializing the values of , , , and .

Some other work related to partial sums and also related to zeta function can be seen in () for further views and ideas.

Acknowledgment

The third author is presently supported by MOHE: UKM-ST-06-FRGS0244-2010.

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