Geometric Properties of a Certain Class of Mittag–Leffler-Type Functions

Hari M Srivastava,Sourav Das,Khaled Mehrez,Anish Kumar

doi:10.3390/fractalfract6020054

Hari M Srivastava, Sourav Das + Show 2 more

Open Access

https://doi.org/10.3390/fractalfract6020054

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Abstract

The main objective of this paper is to establish some sufficient conditions so that a class of normalized Mittag–Leffler-type functions satisfies several geometric properties such as starlikeness, convexity, close-to-convexity, and uniform convexity inside the unit disk. Moreover, pre-starlikeness and k-uniform convexity are discussed for these functions. Some sufficient conditions are also derived so that these functions belong to the Hardy spaces Hp and H∞. Furthermore, the inclusion properties of some modified Mittag–Leffler-type functions are discussed. The various results, which are established in this paper, are presumably new, and their importance is illustrated by several interesting consequences and examples. Some potential directions for analogous further research on the subject of the present investigation are indicated in the concluding section.

Highlights

Introduction and MotivationGeometric Function Theory is one of the important branches of complex analysis.It deals with the geometric properties of analytic functions
The main foundation of Geometric Function Theory is the theory of univalent functions, but a number of new associated areas have emerged and led to various strong results and applications
Several researchers have constructed some new classes of functions involving fractional q-calculus operators, which are analytic in the unit disk and have established several interesting results with applications

Summary

Introduction and Motivation

Geometric Function Theory is one of the important branches of complex analysis. It deals with the geometric properties of analytic functions. We recall the class of starlike functions of order α(0 5 α < 1), denoted by S ∗ (α), which is defined as follows: zg (z). An analytic function g(z) in A is said to be convex in D, if g(z) is a univalent function in D with g(D) as a convex domain in C We denote this class of convex functions by K, which can be described as follows: zg00 (z). We remark in passing that Prabhakar [26] considered a singular integral equation involving a three-parameter Mittag–Leffler-type function in its kernel, which happens to be a special case of the general Wright function Eα,β (φ; z) in (6) when φ(n) =.

A Set of Useful Lemmas

Starlikeness and Convexity of Normalized Mittag–Leffler-Type Functions

Hardy Space of the Mittag–Leffler-Type Functions

Inclusion Properties

Pre-Starlikeness and k-Uniform Convexity

Concluding Remarks and Observations