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 $\alpha \mapsto E_{\alpha }(\mathrm{\Gamma }(1+\alpha )x)$ decreases on $(0,1)$ for all $x\in {\mathbb{R}}^{\ast }\mathrm{.}$ 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 $\beta =\alpha \mathrm{.}$ Several applications of these results are discussed for Abelian integral equations and subdiffusions.