The de Bruijn identity states that Fisher information is the half of the derivative of Shannon differential entropy along heat flow. In the same spirit, in this paper we introduce a generalized version of Fisher information, named as the R\'enyi--Fisher information, which is the half of the derivative of R\'enyi information along heat flow. Based on this R\'enyi--Fisher information, we establish 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. Lastly, we use these generalized Cram\'er--Rao inequalities to determine the signs of derivatives of entropy along heat flow, strengthening existing results on the complete monotonicity of entropy.

翻译：de Bruijn恒等式指出，Fisher信息是Shannon微分熵沿热流导数的一半。基于同样的思想，本文引入了一种广义的Fisher信息，称为Rényi–Fisher信息，它被定义为Rényi信息沿热流导数的一半。基于此Rényi–Fisher信息，我们建立了尖锐的Rényi熵等周不等式，从而将经典的熵等周不等式推广到Rényi框架。利用这些等周不等式，我们将经典的Cramér–Rao不等式从Fisher信息推广到Rényi–Fisher信息。最后，我们运用这些广义的Cramér–Rao不等式来确定熵沿热流导数的符号，从而强化了关于熵完全单调性的现有结果。