Gaussian Extremality for Derivatives of Differential Entropy under the Additive Gaussian Noise Flow

Let Z be a standard Gaussian random variable, X be independent of Z, and t be a strictly positive scalar. For the derivatives in t of the differential entropy of X+√tZ, McKean noticed that Gaussian X achieves the extreme for the first and second derivatives, and he conjectured that this holds for general orders of derivatives. Here we show that, when the probability density function of X+√tZ is log-concave, this conjecture holds for orders up to at least five. We also recover Toscani's result on the non-negativity of the third derivative of the entropy power of X+√tZ for log-concave densities, using a much simpler argument.