Mittag-Leffler functions and convex ordering
Rui Ferreira, Thomas Simon
TL;DR
The paper investigates how Mittag-Leffler functions vary with the fractional parameter α using convex ordering of associated random variables. By deriving integral representations and leveraging kernel monotonicity, two- and multi-parameter cases are analyzed to prove monotonicity results for E_α(x^α), E_α(−x^α), E_α(Γ(1+α)x), and E_{α,β}, including the extremal β=α scenario. The work provides probabilistic proofs via MGFs, product/Beta factorizations, and intersection arguments, yielding sharp bounds and applications to Abelian integral equations and subdiffusions. These results refine known bounds and establish new monotonicity properties with potential impact on inverse problems and stochastic modeling of anomalous diffusion. Overall, the paper demonstrates how convex ordering provides a robust framework for comparing Mittag-Leffler-related transforms across parameter regimes.
Abstract
The monotonicity of the Mittag-Leffler function $E_α$ with respect to the parameter $α$ is investigated, via some convex ordering properties for related random variables. In particular, it is shown that the mapping $α\mapsto E_α(x^α)$ decreases on $(0,2)$ for all $x> 0$, that the mapping $α\mapsto E_α(-x^α)$ decreases on $(0,1)$ for all $x\ge 1$ and that the mapping $α\mapsto E_α(Γ(1+α)x)$ decreases on $(0,1)$ for all $x\in{\mathbb R}^\ast.$ Analogous results are presented for the two parameter Mittag-Leffler functions $E_{α, β}$ with $β\ge α,$ with an emphasis on the extremal case $β=α.$ Several applications of these results are discussed for Abelian integral equations and subdiffusions.