A Quantitative Log-Sobolev Inequality for a Two Parameter Family of Functions

We prove a sharp, dimension-free stability result for the classical logarithmic Sobolev inequality for a two parameter family of functions. Roughly speaking, our family consists of a certain class of log C^1,1 functions. Moreover, we show how to enlarge this space at the expense of the dimensionless constant and the sharp exponent. As an application, we obtain new bounds on the entropy.