On the entropy power inequality for the R\'enyi entropy of order [0,1]

TL;DR

This paper establishes sharp R\'enyi entropy power inequalities for log-concave vectors with parameters in (0,1), extending classical entropy inequalities using advanced inequalities and comparison results.

Contribution

It introduces new sharp R\'enyi entropy power inequalities for the range (0,1) for log-concave vectors, expanding the theoretical understanding of entropy inequalities.

Findings

Derived sharp R\'enyi entropy power inequalities for (0,1)

Extended classical entropy inequalities to R\'enyi entropy

Provided estimates sharp up to absolute constants

Abstract

Using a sharp version of the reverse Young inequality, and a R\'enyi entropy comparison result due to Fradelizi, Madiman, and Wang, the authors are able to derive R\'enyi entropy power inequalities for log-concave random vectors when R\'enyi parameters belong to $(0,1)$. Furthermore, the estimates are shown to be sharp up to absolute constants.