Entropic Isoperimetric and Cram\'er--Rao Inequalities for R\'enyi--Fisher Information

The de Bruijn identity states that Fisher information is equal to a half of the time-derivative of Shannon differential entropy along heat flow. In the same spirit, a generalized version of Fisher information, which we term the R\'enyi--Fisher information, is defined as a half of the time-derivative of R\'enyi differential entropy along heat flow. Based on this R\'enyi--Fisher information, we establish several sharp R\'enyi-entropic isoperimetric inequalities, which generalize the classic entropic isoperimetric inequality to the R\'enyi setting. Utilizing these isoperimetric inequalities, we extend the classical Cram\'er--Rao inequality from Fisher information to R\'enyi--Fisher information. We then use these generalized Cram\'er--Rao inequalities to determine the signs of derivatives of R\'enyi entropy along heat flow, strengthening existing results on the complete monotonicity of R\'enyi entropy. We lastly explore the implications of our R\'enyi-entropic isoperimetric inequalities for entropy power inequalities. We demonstrate that, unlike in the Shannon entropy case, the classic entropy power inequality does not admit a direct extension to R\'enyi entropy without introducing additional exponents or scaling factors. Furthermore, we establish a sharp R\'enyi entropy power inequality involving a scaling factor under the assumption that one of two independent random vectors is Gaussian.