An inequality for moments of log-concave functions on Gaussian random   vectors
  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. 
Nikos Dafnis, Grigoris Paouris
math.PR