Differential Subordination Results for Fractional Integral Associated with Generalized Mittag-Leffler Function

Abbas Kareem Wanas,Alaa Hussein Khaleel

doi:10.34198/ejms.1219.143155

Abbas Kareem Wanas, Alaa Hussein Khaleel

Open Access

https://doi.org/10.34198/ejms.1219.143155

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Abstract

In this work, by making use of fractional integral, we define a certain class of holomorphic functions defined by generalized Mittag-Leffler function in the open unit disk U. Also, we establish some results for this class related to integral representation, inclusion relationship and argument estimate.

Highlights

Let A indicate the family of all functions f of the form: ∑∞f (z) = z + anzn, (1.1)n=2 which are holomorphic in the open unit disk U = {z ∈ C : z < 1}.For functions f given by (1.1) and g ∈ A given by g(z) = z + bn zn, n=2Received: January 22, 2019; Accepted: February 11, 20192010 Mathematics Subject Classification: 30C45.Keywords and phrases: holomorphic functions, generalized Mittag-Leffler function, fractional integral, subordination, integral representation.Abbas Kareem Wanas and Alaa Hussein Khaleel the Hadamard product f ∗ g of f and g is defined by ( f ∗ g ) (z) = z + anbn zn = (g ∗ f ) (z). k=2
We establish some results for this class related to integral representation, inclusion relationship and argument estimate
N=2 which are holomorphic in the open unit disk U = {z ∈ C : z < 1}

Summary

Introduction

Let A indicate the family of all functions f of the form:. n=2 which are holomorphic in the open unit disk U = {z ∈ C : z < 1}. Keywords and phrases: holomorphic functions, generalized Mittag-Leffler function, fractional integral, subordination, integral representation. The fractional integral of order λ, (λ > 0) is defined for a function f by. Let T be the class of functions h of the form: h(z) = 1 + hn zn , n =1 which are analytic and convex univalent in U and satisfy the condition: Re{h(z)} > 0, (z ∈ U ). If q is analytic in U with q(0) = 1, the subordination q(z) zq′( z ) uq(z) +. If q is analytic in U and q(0) = h(0), the subordination q(z) + T (z) zq′(z) ≺ h(z) implies that q(z) ≺ h(z).

Main Results

Suppose that

An application of