An inequality for moments of log-concave functions on Gaussian random vectors

TL;DR

This paper establishes a precise moment inequality for log-concave and log-convex functions evaluated on Gaussian vectors, with applications to stability results in the logarithmic Sobolev inequality.

Contribution

It introduces a sharp moment inequality for log-concave and log-convex functions on Gaussian vectors, extending the understanding of their behavior and stability.

Findings

Proved a sharp moment inequality for log-concave functions on Gaussian vectors.

Applied the inequality to derive stability results for the logarithmic Sobolev inequality.

Enhanced the theoretical framework for analyzing functions on Gaussian spaces.

Abstract

We prove a sharp moment inequality for a log-concave or a log-convex function, on Gaussian random vectors. As an application we take a stability result for the classical logarithmic Sobolev inequality of L. Gross in the case where the function is log-concave.