The concavity of R\`enyi entropy power

We associate to the p-th R\'enyi entropy a definition of entropy power, which is the natural extension of Shannon's entropy power and exhibits a nice behaviour along solutions to the p-nonlinear heat equation in $R^n$. We show that the R\'enyi entropy power of general probability densities solving such equations is always a concave function of time, whereas it has a linear behaviour in correspondence to the Barenblatt source-type solutions. We then shown that the p-th R\'enyi entropy power of a probability density which solves the nonlinear diffusion of order p, is a concave function of time. This result extends Costa's concavity inequality for Shannon's entropy power to R\'enyi entropies.