Crame´r-Rao and moment-entropy inequalities for Renyi entropy and generalized Fisher information

The moment-entropy inequality shows that a continuous random variable with given second moment and maximal Shannon entropy must be Gaussian. Stam´s inequality shows that a continuous random variable with given Fisher information and minimal Shannon entropy must also be Gaussian. The Crame´r-Rao inequality is a direct consequence of these two inequalities. In this paper, the inequalities above are extended to Renyi entropy, p^th moment, and generalized Fisher information. Generalized Gaussian random densities are introduced and shown to be the extremal densities for the new inequalities. An extension of the Crame´r-Rao inequality is derived as a consequence of these moment and Fisher information inequalities.