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.
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