Self-Similar Solutions of Rényi's Entropy and the Concavity of Its Entropy Power

We study the class of self-similar probability density functions with finite mean and variance, which maximize Rényi’s entropy. The investigation is restricted in the Schwartz space S(R) and in the space of l-differentiable compactly supported functions C l c(R). Interestingly, the solutions of this optimization problem do not coincide with the solutions of the usual porous medium equation with a Dirac point source, as occurs in the optimization of Shannon’s entropy. We also study the concavity of the entropy power in R with respect to time using two different methods. The first one takes advantage of the solutions determined earlier, while the second one is based on a setting that could be used for Riemannian manifolds.