A strengthened entropy power inequality for log-concave densities

TL;DR

This paper proves a strengthened version of Shannon's entropy-power inequality specifically for log-concave densities, using an extension of the Blachman–Stam argument to derive a sharp second derivative inequality of the entropy functional.

Contribution

It introduces a novel, sharper inequality for log-concave densities by extending classical methods to analyze the second derivative of entropy under the heat semigroup.

Findings

Established a sharper entropy-power inequality for log-concave densities.

Extended the Blachman–Stam argument to the second derivative of entropy.

Provided a new analytical tool for studying entropy in the context of log-concave distributions.

Abstract

We show that Shannon's entropy--power inequality admits a strengthened version in the case in which the densities are log-concave. In such a case, in fact, one can extend the Blachman--Stam argument to obtain a sharp inequality for the second derivative of Shannon's entropy functional with respect to the heat semigroup.