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

Abstract

We prove two inequalities for the Mittag-Leffler function, namely that the function log Eα(xα) is sub-additive for 0 < α< 1 , and super-additive for α> 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 < α< 2 , we also show that Eα(xα) is log-concave resp. log-convex, using analytic as well as probabilistic arguments.