Stochastic analysis of Beckner's and related functional inequalities

Beckner's inequality is a family of inequalities that interpolates the two fundamental functional inequalities, the logarithmic Sobolev and Poincar\'e's inequalities. It is parametrized by exponent $p\in (1,2]$ and it implies the logarithmic Sobolev inequality as $p\to 1$ and agrees with Poincar\'e's inequality when $p=2$. In this paper, employing a stochastic method, we prove an improvement of Beckner's inequality under the Gaussian measure when $4/3\le p<2$; in particular, when $p=3/2$, the error bound is expressed in terms of the entropy functional. A similar reasoning to the derivation of the improvement also enables us to obtain a H\"older-type inequality that holds among the entropy, variance and related functionals.