Mittag-Leffler functions and convex ordering

TL;DR

This paper studies how the Mittag-Leffler functions change with their parameters using convex ordering, revealing monotonicity properties and applications to integral equations and subdiffusions.

Contribution

It establishes new monotonicity results for Mittag-Leffler functions with respect to their parameters using convex ordering techniques.

Findings

The function $E_eta(x^eta)$ decreases on (0,2) for all $x>0$.

The function $E_eta(-x^eta)$ decreases on (0,1) for all $x extgreater 1$.

Applications include Abelian integral equations and subdiffusions.

Abstract

The monotonicity of the Mittag-Leffler function $E_{\alpha}$ with respect to the parameter $\alpha$ is investigated, via some convex ordering properties for related random variables. In particular, it is shown that the mapping $\alpha\mapsto E_\alpha(x^\alpha)$ decreases on $(0,2)$ for all $x> 0$, that the mapping $\alpha\mapsto E_\alpha(-x^\alpha)$ decreases on $(0,1)$ for all $x\ge 1$ and that the mapping $\alpha\mapsto E_\alpha(\Gamma(1+\alpha)x)$ decreases on $(0,1)$ for all $x\in{\mathbb R}^\ast.$ Analogous results are presented for the two parameter Mittag-Leffler functions $E_{\alpha, \beta}$ with $\beta\ge \alpha,$ with an emphasis on the extremal case $\beta =\alpha.$ Several applications of these results are discussed for Abelian integral equations and subdiffusions.