# Radii problems for the generalized Mittag-Leffler functions

## TL;DR

This paper investigates the radii of various geometric properties of the generalized Mittag-Leffler functions, providing explicit bounds based on their Hadamard factorization and roots of functional equations.

## Contribution

It introduces new radii bounds for generalized Mittag-Leffler functions related to convexity and starlikeness, using Hadamard factorization and functional equations.

## Key findings

- Derived radii as smallest positive roots of functional equations
- Established bounds for η-uniformly convexity and related properties
- Analyzed three different normalizations of the Mittag-Leffler function

## Abstract

In this paper our aim is to investigate the radii of $\eta-$uniformly convexity, $\alpha-$convexity, $\eta-$parabolic starlikeness and strong starlikeness of order $\rho$ of the generalized Mittag-Leffler function for three different kinds of normalization by using their Hadamard factorization in such a way that the resulting functions are analytic. The basic tool of this study is the Mittag-Leffler function in series. Also we have shown that the obtained radii are the smallest positive roots of some functional equations.