New sharp inequalities involving non-relative, relative and cross informational functionals with some remarkable minimizers of generalized Gaussian and Beta types
2026-07-09 • Information Theory
Information Theory
AI summaryⓘ
The authors present several new sharp mathematical inequalities related to measures of uncertainty and information in probability distributions. They extend known results by connecting concepts like Rényi entropy, Fisher information, and cross entropy in a more generalized framework. Some of these inequalities identify special distributions, such as Gaussian or Beta distributions, as optimal cases. Their work helps better understand how different informational measures interact and relate.
Rényi entropyFisher informationStam inequalityCross entropyRényi divergenceGaussian distributionBeta distributionInformation theoryMoment inequalities
Authors
Razvan Gabriel Iagar, David Puertas-Centeno
Abstract
Several new and sharp informational inequalities are derived as a byproduct of Stam-like and moment-entropy-like inequalities in the relative framework and a recently established inequality mixing the Rényi entropy, the Rényi divergence and the Rényi cross entropy of suitable probability density functions. More precisely, we obtain a Stam-like inequality connecting the Rényi entropy power, the recently introduced scaling-invariant relative Fisher information and the Rényi cross entropy. Furthermore, we derive an inequality involving only Fisher-like informational measures and another inequality involving only moment-like functionals of non-relative, relative and cross types, respectively. All the inequalities are sharp. The minimizers of the Stam-like inequality are, in certain cases, pairs of Gaussian or stretched Gaussian probability densities; in contrast, each minimizer of the moment-like inequality is the probability density of the generalized Beta distribution.