Partial Sums For Class Of Analytic Functions Defined By Integral Operator

In the present paper, we study the class of analytic functions involving generalized integral operator, which is defined by means of a general Hurwitz Lerch Zeta function denoted by , ( ) s b f z α I with negative coefficients. The aim of the paper is to obtain the coefficient estimates and also partial sums of its sequence , ( ) s b f z α I . | Univalent functions | uniformly starlike functions | Hadamard product | partial sums | fractional derivatives and fractional integrals | ® 2012 Ibnu Sina Institute. All rights reserved. http://dx.doi.org/10.11113/mjfas.v8n4.146

. Srivastava [3]) Let the function f be analytic in a simply connected domain of the z-plane containing the origin.The fractional derivative of f of order α is defined by 0 ( ) where the multiplicity of (z -t) α − is removed by requiring log(z -t) to be real when z -t 0 > .
Using Definition 1.1 and its known extensions involving fractional derivatives and fractional integrals, Owa and Srivastava [3] introduced the operator α : A -A Ω We can write this function as : ∑ By using Definitions (1.1) and (1.2), the authors [1] introduced the generalized integral operator , : A A as the follo wing: is Srivastav and A. Attiya integral operator [5].Finally, for different choices of s, b and α , several operators investigated earlier by other authors Cho and Kim [15], and Lin and Owa [16] are obtained.
By using our integral operator we introduce the following class of A .
A function A f ∈ is said to be in the class denoted by , ( , ),( 11), 0

Coefficient Estimates:
Before stating and proving our main results, we derive a sufficient condition giving the coefficient estimates for the function f to belong to the class , ( , ) s b SP α β γ .The result is contained in the following:

Theorem
A sufficient condition for a function f of the form (1) to be in , ( , ) Proof: the proof is complete.

partial sums
In this section we will examine the ratio of a function of the form (1) to its sequence of partial sums defined by when the coefficients of f are sufficiently small to satisfy the condition (2).We will determine sharp lower bounds for ( )

Theorem
Let f be given by ( 1) satisfying (2), then The results are sharp for every k with the function given by 1 , (1 ) We prove (3).Let f be given by (1)   satisfying ( 2), by sitting .
w z w z This will hold if we show that left-hand side of ( 6) is bounded above by a ≥ To see that ( ) gives sharp result, we observe that for i z re To prove the second part of this theorem, we write .
w z w z Making use of (2) to get (7).Finally, equality holds in (4) for the extremal function f given by ( 5).This completes the proof.

Theorem
Let f be given by ( 1) satisfying (2), then '( ) Re 1 ' ( ) 1 )( 1 The results are sharp for every k with the function give Proof: To prove the result (8), define thefunction w(z) by The proof is complete.
To prove the second part of this theorem, we write [ which gives (9).The bound in ( 9) is sharp for all k ∈ N with the extremal function (5).Special cases of Theorems 2.4, 2.5 by setting 0, s 0 and 0 α β = = = can be found in [11].

Corollary
Let f be given by (1) satisfying INTRODUCTIONLet A denote the class of all analytic functions in the open unit disk U {z :
γ reduces to the classes introduced and studied by various authors for example: