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

Let f n (cid:2) z (cid:3) (cid:4) z (cid:5) (cid:2) nk (cid:4) 2 a k z k be the sequence of partial sums of the analytic function f (cid:2) z (cid:3) (cid:4) z (cid:5) (cid:2) ∞ k (cid:4) 2 a k z k . In this paper, we determine sharp lower bounds for (cid:0) { f (cid:2) z (cid:3) /f n (cid:2) z (cid:3) } , (cid:0) { f n (cid:2) z (cid:3) /f (cid:2) z (cid:3) } , (cid:0) { f (cid:3) (cid:2) z (cid:3) /f (cid:3) n (cid:2) z (cid:3) } , and (cid:0) { f (cid:3) n (cid:2) z (cid:3) /f (cid:3) (cid:2) z (cid:3) } . The usefulness of the main result not only provides the uniﬁcation of the results discussed in the literature but also generates certain new results.


Introduction and Preliminaries
Let A denote the class of functions of the form which are analytic and univalent in the open disc U {z : |z| < 1}.We also consider T a subclass of A introduced and studied by Silverman 1 , consisting of functions of the form Abstract and Applied Analysis For functions f ∈ A given by 1.1 and g ∈ A given by g z z ∞ k 2 b k z k , we define the Hadamard product or convolution of f and g by We recall here a general Hurwitz-Lerch zeta function Φ z, s, a defined in 2 by Φ z, s, a : 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 Motivated by  and making use of the generalized Srivastava-Attiya operator J m,η μ,b , we define the following new subclass of analytic functions with negative coefficients.
For λ ≥ 0, −1 ≤ γ < 1, and β ≥ 0, let P λ μ γ, β be the subclass of A consisting of functions of the form 1.1 and satisfying the analytic criterion where z ∈ U. Shortly we can state this condition by where 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 21-25 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 f z satisfy the condition 1.14 .Also, we will determine sharp lower bounds for Ê{f z /f n z }, Ê{f n z /f z }, Ê{f z /f n z }, and Ê{f n z /f z }.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 f z to belong to this generalized function class.
where, for convenience, and C m k b, μ , is given by 1.9 .
Proof.The proof of Lemma 1.1 is much akin to the proof of Theorem 1 obtained by Murugusundaramoorthy 17 , hence we omit the details.

Main Results
Theorem 2.1.If f of the form 1.1 satisfies the condition 1.14 , then where

2.2
The result 2.1 is sharp with the function given by

2.3
Proof.Define the function w z by

2.4
It suffices to show that |w z | ≤ 1.Now, from 2.4 we can write

2.5
Hence we obtain or, equivalently, From the condition 1.14 , it is sufficient to show that To see that the function given by 2.3 gives the sharp result, we observe that for z re iπ/n ,

2.11
We next determine bounds for f n z /f z .
Theorem 2.2.If f of the form 1.1 satisfies the condition 1.14 , then where ρ n 1 ≥ 1 − γ and

2.13
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 f z given by 2.3 .
We next turns to ratios involving derivatives.
Theorem 2.3.If f of the form 1.1 satisfies the condition 1.14 , then where ρ n 1 ≥ n 1 1 − γ and

2.19
The results are sharp with the function given by 2.3 .
Proof.We write where

2.21
Now From the condition 1.14 , it is sufficient to show that To prove the result 2.18 , define the function w z by where

2.26
Now Some other work related to partial sums and also related to zeta function can be seen in 26-29 for further views and ideas.

9 and
throughout this paper unless otherwise mentioned the parameters μ, b are constrained as b ∈ \ − 0 ; μ ∈ , η ≥ 2 and m > −1.It is of interest to note that J 1,2 μ,b is the Srivastava-Attiya operator 8 and J m,η 0,b is the well-known Choi-Saigo-Srivastava operator see 11, 12 .Suitably specializing the parameters m, η, μ, and b in J m,η μ,b f z , 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 .

2 . 27 ItRemark 2 . 4 .
suffices to show that the left hand side of 2.27 is bounded previously by the condition As a special case of the previous theorems, we can determine new sharp lower bounds for Ê{f z /f n z }, Ê{f n z /f z }, Ê{f z /f n z }, and Ê{f n z /f z } 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 η, m, μ, and b.