Mittag-Leffler functions and convex ordering

Abstract

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