Abstract
Recently, Savaré-Toscani proved that the Rényi entropy power of general probability densities solving the p-nonlinear heat equation in is a concave function of time under certain conditions of three parameters , which extends Costa’s concavity inequality for Shannon’s entropy power to the Rényi entropy power. In this paper, we give a condition of under which the concavity of the Rényi entropy power is valid. The condition contains Savaré-Toscani’s condition as a special case and much more cases. Precisely, the points satisfying Savaré-Toscani’s condition consist of a two-dimensional subset of , and the points satisfying the condition consist a three-dimensional subset of . Furthermore, gives the necessary and sufficient condition in a certain sense. Finally, the conditions are obtained with a systematic approach.
Highlights
In 1948, Claude Elwood Shannon [1] first introduced his mathematical theory of information
Savaré-Toscani [22] proved that the concavity of entropy power is a property which is not restricted to the Shannon entropy power (3) in connection with the heat Equation (4), but it holds for the p-th Rényi entropy power (2)
We show that the points satisfying T(1, 2) consist of a three-dimensional subset of
Summary
In 1948, Claude Elwood Shannon [1] first introduced his mathematical theory of information. Related to EPI, Costa [14] proved that the Shannon entropy power N (u) = 2πe e is a concave function in t; that is, (d/dt) N (u) ≥ 0 and (d /d t) N (u) ≤ 0. Savaré-Toscani [22] proved that the concavity of entropy power is a property which is not restricted to the Shannon entropy power (3) in connection with the heat Equation (4), but it holds for the p-th Rényi entropy power (2). We give a generalization for the concavity of the p-th Rényi entropy power (CREP).
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