Starlikeness and convexity of integral operators involving Mittag-Leffler functions

TL;DR

This paper investigates the geometric properties, specifically starlikeness and convexity, of a class of integral operators involving normalized Mittag-Leffler functions, providing conditions for these properties in complex analysis.

Contribution

It establishes the order of starlikeness and convexity for a new class of integral operators involving Mittag-Leffler functions, extending geometric function theory.

Findings

Derived conditions for starlikeness of the integral operators

Derived conditions for convexity of the integral operators

Extended geometric analysis to Mittag-Leffler function-based operators

Abstract

In this paper, we shall find the order of starlikeness and convexity for integral operators \begin{equation*} \mathbb{F}_{\alpha _{j},\beta _{j},\lambda _{j},\zeta }(z)=\left\{ \zeta \int\limits_{0}^{z}t^{\zeta -1}\prod_{j=1}^{n}\left( \frac{\mathbb{E} _{\alpha _{j},\beta _{j}}(t)}{t}\right) ^{1/\lambda _{j}}dt\right\} ^{1/\zeta }, \end{equation*} where the functions $\mathbb{E}_{\alpha _{j},\beta _{j}}$ are the normalized Mittag-Leffler functions.