A converse to the neo-classical inequality with an application to the Mittag-Leffler function

TL;DR

This paper establishes new inequalities for the Mittag-Leffler function, including sub-additivity and super-additivity depending on the parameter, and explores its log-concavity and log-convexity properties.

Contribution

It introduces a converse to the neo-classical inequality and generalizes binomial inequalities, advancing understanding of Mittag-Leffler function properties.

Findings

Logarithm of Mittag-Leffler function is sub-additive for 0<α<1.

Logarithm of Mittag-Leffler function is super-additive for α>1.

Mittag-Leffler function is log-concave for 0<α<1 and log-convex for 1<α<2.

Abstract

We prove two inequalities for the Mittag-Leffler function, namely that the function $\log E_\alpha(x^\alpha)$ is sub-additive for $0<\alpha<1,$ and super-additive for $\alpha>1.$ These assertions follow from two new binomial inequalities, one of which is a converse to the neo-classical inequality. The proofs use a generalization of the binomial theorem due to Hara and Hino (Bull. London Math. Soc. 2010). For $0<\alpha<2,$ we also show that $E_\alpha(x^\alpha)$ is log-concave resp. log-convex, using analytic as well as probabilistic arguments.