A Generalization of the Concavity of Renyi Entropy Power

Recently, Savare-Toscani proved that the Renyi entropy power of general probability densities solving the p-nonlinear heat equation in R n is a concave function of time under certain conditions of three parameters n , p , mu , which extends Costa's concavity inequality for Shannon's entropy power to the Renyi entropy power. In this paper, we give a condition f ( n , p , mu ) of n , p , mu under which the concavity of the Renyi entropy power is valid. The condition f ( n , p , mu ) contains Savare-Toscani's condition as a special case and much more cases. Precisely, the points ( n , p , mu ) satisfying Savare-Toscani's condition consist of a two-dimensional subset of R (3) , and the points satisfying the condition f ( n , p , mu ) consist a three-dimensional subset of R (3) . Furthermore, f ( n , p , mu ) gives the necessary and sufficient condition in a certain sense. Finally, the conditions are obtained with a systematic approach.