Convolution Properties of p-Valent Functions Associated with a Generalization of the Srivastava-Attiya Operator

Priyabrat Gochhayat

doi:10.1155/2013/676027

Abstract

Let 𝒜p denote the class of functions analytic in the open unit disc 𝕌 and given by the series f(z)=zp+∑n=p+1∞anzn. For f∈𝒜p, the transformation ℐp,δλ:𝒜p→𝒜p defined by ℐp,δλf(z)=zp+∑n=p+1∞((p+δ)/(n+δ))λanzn, (δ+p∈ℂ∖ℤ0-, λ∈ℂ; z∈𝕌), has been recently studied as fractional differintegral operator by Mishra and Gochhayat (2010). In the present paper, we observed that ℐp,δλ can also be viewed as a generalization of the Srivastava-Attiya operator. Convolution preserving properties for a class of multivalent analytic functions involving an adaptation of the popular Srivastava-Attiya transform are investigated.

Highlights

Introduction and PreliminariesLet A be the class of functions analytic in the open unit diskU := {z : z ∈ C, |z| < 1} . (1)Suppose that f and g are in A
If g is univalent in U, the reverse implication holds
We observed that Iλp,δ can be viewed as a generalization of the Srivastava-Attiya operator (take p = 1, λ = t, δ = a in (14)), suitable for the study of multivalent functions. transformation Iλp,δ generalizes several previously studied familiar operators

Summary

Introduction

Introduction and PreliminariesLet A be the class of functions analytic in the open unit diskU := {z : z ∈ C, |z| < 1} . (1)Suppose that f and g are in A. If there exists a function ω ∈ A, satisfying the conditions of the Schwarz lemma (i.e., ω(0) = 0 and |ω(z)| < 1) such that f (z) = g (ω (z)) (z ∈ U) . . .}) of A consisting of functions of the following form: With a view to define the Srivastava-Attiya transform we recall here a general Hurwitz-Lerch-Zeta function, which is defined in [2, 3] by the following series: Φ (z, t, a)

Results

Conclusion