The de Bruijn identity states that Fisher information is equal to twice the time-derivative of Shannon differential entropy along heat flow. In the same spirit, a generalized version of Fisher information, termed the Rényi–Fisher information, was introduced by Jizba, Dunningham, and Prokš [Entropy, 2021], which is defined as twice the time-derivative of Rényi differential entropy along heat flow. Based on this Rényi–Fisher information, we establish several sharp Rényi-entropic isoperimetric inequalities, which generalize the classic entropic isoperimetric inequality to the Rényi setting. Utilizing these isoperimetric inequalities, we extend the classical Cramér–Rao inequality from Fisher information to Rényi–Fisher information. We then use these generalized Cramér–Rao inequalities to determine the signs of derivatives of Rényi entropy along heat flow, strengthening existing results on the complete monotonicity of Rényi entropy. We lastly explore applications of our Rényi-entropic isoperimetric inequalities in entropy power inequalities. We establish a sharp Rényi entropy power inequality under the assumption that one of two independent random vectors is Gaussian.

In this paper, we generalize the log-Sobolev inequalities to Rényi–Sobolev inequalities by replacing the entropy with the two-parameter entropy, which is a generalized version of entropy and closely related to Rényi divergences. We derive the sharp nonlinear dimension-free version of this kind of inequalities. Interestingly, the resultant inequalities show a transition phenomenon depending on the parameters. We then connect Rényi–Sobolev inequalities to contractive and data-processing inequalities, concentration inequalities, and spectral graph theory. Our proofs in this paper are based on the information-theoretic characterization of the Rényi–Sobolev inequalities, as well as the method of types.