A Generalization of the Concavity of Rényi Entropy Power

Recently, Savaré-Toscani proved that the Rényi entropy power of general probability densities solving the p-nonlinear heat equation in [Formula: see text] is a concave function of time under certain conditions of three parameters [Formula: see text] , which extends Costa’s concavity inequality for Shannon’s entropy power to the Rényi entropy power. In this paper, we give a condition [Formula: see text] of [Formula: see text] under which the concavity of the Rényi entropy power is valid. The condition [Formula: see text] contains Savaré-Toscani’s condition as a special case and much more cases. Precisely, the points [Formula: see text] satisfying Savaré-Toscani’s condition consist of a two-dimensional subset of [Formula: see text] , and the points satisfying the condition [Formula: see text] consist a three-dimensional subset of [Formula: see text] . Furthermore, [Formula: see text] gives the necessary and sufficient condition in a certain sense. Finally, the conditions are obtained with a systematic approach.