Mittag-Leffler functions and convex ordering

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.