Abstract and Applied Analysis

VolumeΒ 2011, Article IDΒ 849250, 9 pages

http://dx.doi.org/10.1155/2011/849250

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

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

Received 29 January 2011; Accepted 7 April 2011

Academic Editor: YoshikazuΒ Giga

Copyright Β© 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 [1], 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 [2] 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 [3], Ferreira and LΓ³pez [4], Garg et al. [5], Lin and Srivastava [6], Lin et al. [7], and others. Srivastava and Attiya [8] (see also RΔducanu and Srivastava [9] and Prajapat and Goyal [10]) 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 [8], 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 [8] 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 [13] and Bernardi [14]. Further more, we get the Jung-Kim-Srivastava integral operator [15] closely related to some multiplier transformation studied by Flett [16].

Motivated by Murugusundaramoorthy [17β19] 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 [20] 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 [20] (see [21β25]) 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 [20], 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,
**
β, , , and , is given by (1.9). *

*Proof. *The proof of Lemma 1.1 is much akin to the proof of Theoremβ1 obtained by Murugusundaramoorthy [17], 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 [13] and Bernardi integral operators [14], Jung-Kim-Srivastava integral operator [15] 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 ([26β29]) for further views and ideas.*

*Acknowledgment*

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

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