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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 C1,1 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.

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