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A Generalization of the Concavity of Rényi Entropy Power

Authors:	Laigang Guo Chun-Ming Yuan Xiao-Shan Gao

Affiliation:	1.Laboratory of Mathematics and Complex Systems (Ministry of Education), School of Mathematical Sciences, Beijing Normal University, Beijing 100875, China;2.KLMM, Academy of Mathematics and Systems Science, Chinese Academy of Sciences, Beijing 100190, China;3.University of Chinese Academy of Sciences, Beijing 100049, China

Abstract:	Recently, Savaré-Toscani proved that the Rényi 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 $n, p, μ$ , which extends Costa’s concavity inequality for Shannon’s entropy power to the Rényi entropy power. In this paper, we give a condition $Φ (n, p, μ)$ of $n, p, μ$ under which the concavity of the Rényi entropy power is valid. The condition $Φ (n, p, μ)$ contains Savaré-Toscani’s condition as a special case and much more cases. Precisely, the points $(n, p, μ)$ satisfying Savaré-Toscani’s condition consist of a two-dimensional subset of $R^{3}$ , and the points satisfying the condition $Φ (n, p, μ)$ consist a three-dimensional subset of $R^{3}$ . Furthermore, $Φ (n, p, μ)$ gives the necessary and sufficient condition in a certain sense. Finally, the conditions are obtained with a systematic approach.

Keywords:	Ré nyi entropy, entropy power inequality, nonlinear heat equation

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