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

Abstract

AbstractWe prove two inequalities for the Mittag-Leffler function, namely that the function $$\log E_\alpha (x^\alpha )$$ log E α ( x α ) is sub-additive for $$0<\alpha <1,$$ 0 < α < 1 , and super-additive for $$\alpha >1.$$ 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 $$0<\alpha <2,$$ 0 < α < 2 , we also show that $$E_\alpha (x^\alpha )$$ E α ( x α ) is log-concave resp. log-convex, using analytic as well as probabilistic arguments.