Mittag-Leffler functions and convex ordering

2025/12/04 by Ferreira, Rui, Simon, Thomas
#Classical Analysis and ODEs (math.CA) #FOS: Mathematics #Probability (math.PR)

paper · doi:10.48550/arxiv.2512.04940

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↦ E_α(x^α) decreases on (0,2) for all x> 0, that the mapping α↦ E_α(-x^α) decreases on (0,1) for all x≥ 1 and that the mapping α↦ E_α(Γ(1+α)x) decreases on (0,1) for all x∈\mathbb R^∗. Analogous results are presented for the two parameter Mittag-Leffler functions E_{α, β} with β≥ α, with an emphasis on the extremal case β=α. Several applications of these results are discussed for Abelian integral equations and subdiffusions.

Citations

• Monotonicity in the parameter of the Mittag-Leffler function and determining the fractional exponent of the subdiffusion equation
• On the log-concavity of the Wright function
• On the local times of noise reinforced Bessel processes
• Inverse problem for determining the order of fractional derivative in mixed-type equations
• Some properties of the Kilbas-Saigo function
• An Inverse Potential Problem for Subdiffusion: Stability and Reconstruction
• Fractional extreme distributions
• Continuity of solutions of a class of fractional equations
• Mittag-Leffler functions and complete monotonicity
• Fractional Pearson diffusions

Related

• A Probabilistic Perspective on Feller, Pollard and the Complete Monotonicity of the Mittag-Leffler Function
• Starlikeness and convexity of integral operators involving Mittag-Leffler functions
• On Feller, Pollard and the Complete Monotonicity of the Mittag-Leffler Function E_α(-x)
• Mittag-Leffler functions and complete monotonicity
• On Some Special Properties of Mittag-Leffler Functions