On the log-concavity of the Wright function

Abstract

We investigate the log-concavity on the half-line of the Wright function $\varphi (-\alpha ,\beta ,-x),$ in the probabilistic setting $\alpha \in (0,1)$ and $\beta \ge 0.$ Applications are given to the construction of generalized entropies associated to the corresponding Mittag-Leffler function. A natural conjecture for the equivalence between the log-concavity of the Wright function and the existence of such generalized entropies is formulated. The problem is solved for $\beta \ge \alpha$ and in the classical case $\beta =1-\alpha$ of the Mittag-Leffler distribution, which exhibits a certain critical parameter ${\alpha }_{\ast }=\mathrm{0.771667...}$ defined implicitly on the Gamma function and characterizing the log-concavity. We also prove that the probabilistic Wright functions are always unimodal, and that they are multiplicatively strongly unimodal if and only if $\beta \ge \alpha$ or $\alpha \le 1/2$ and $\beta =0.$

To appear in Constructive Approximation