# Radii of starlikeness and convexity of $q-$Mittag--Leffler functions

## TL;DR

This paper investigates the radii of starlikeness and convexity of the $q$-Mittag--Leffler functions using Hadamard factorization and Euler-Rayleigh inequalities, providing bounds for these radii in the complex plane.

## Contribution

It introduces new bounds for the radii of starlikeness and convexity of $q$-Mittag--Leffler functions via Hadamard factorization and Euler-Rayleigh inequalities.

## Key findings

- Derived tight bounds for radii of starlikeness.
- Applied Hadamard factorization to analyze function properties.
- Utilized Laguerre-Pólya class in the analysis.

## Abstract

In this paper we deal with the radii of starlikeness and convexity of the $q-$Mittag--Leffler function for three different kinds of normalization by making use of their Hadamard factorization in such a way that the resulting functions are analytic in the unit disk of the complex plane. By applying Euler-Rayleigh inequalities for the first positive zeros of these functions tight lower and upper bounds for the radii of starlikeness of these functions are obtained. The Laguerre-P\'olya class of real entire functions plays a pivotal role in this investigation.