Improving Beckner's bound via Hermite functions

Abstract

We obtain an improvement of the Beckner inequality $\parallel f{\parallel }_2^2-\parallel f{\parallel }_p^2\le \left(2-p\right)\parallel \nabla f{\parallel }_2^2$ valid for $p\in \left[1,2\right]$ and the Gaussian measure. Our improvement is essential for the intermediate case $p\in \left(1,2\right)$, and moreover, we find the natural extension of the inequality for any real $p$.