New Sharp Inequalities Involving Non-Relative, Relative and Cross Informational Functionals

Jul 16, 2026

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Several new, sharp information-theoretic inequalities are established by combining Stam-like and moment-entropy-like inequalities in a relative setting, together with a recent result that couples Rényi entropy, Rényi divergence, and Rényi cross-entropy for appropriate probability densities. A Stam-like inequality is proved relating Rényi entropy power, a scaling-invariant relative Fisher information, and Rényi cross-entropy. Additional sharp bounds are derived using only Fisher-like measures or only moment-like non-relative, relative, and cross functionals. Minimizers include Gaussian or stretched Gaussian densities and generalized Beta densities.

LanguageEnglish

CategoryResearch & Report

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Pages17

Reading Time43 min

Introduction

Information-theoretic inequalities and sharp bounds

Objectives and main inequalities

arXiv :2607 .08599v 1 [ cs .IT] 9 Jul 2026 New sharp inequalities involving non-relative, relative and cross informational functionals with some remarkable minimizers of generalized Gaussian and Beta types Razvan Gabriel Iagar ∗1 and David Puertas-Centeno†1,2 1 Departamento de Matem´atica Aplicada, Ciencia e Ingenier´ıa de los Materiales y Tecnolog´ıa Electr´onica, Universidad Rey Juan Carlos, 28933 M´ostoles (Madrid), Spain 2 Data, Complex Networks and Cybersecurity Research Institute, Universidad Rey Juan Carlos, 28028 (Madrid), Spain July 10, 2026 Abstract Several new and sharp informational inequalities are derived as a byproduct of Stamlike and moment-entropy-like inequalities in the relative framework and a recently established inequality mixing the R´enyi entropy, the R´enyi divergence and the R´enyi cross entropy of suitable probability density functions. More precisely, we obtain a Stam-like inequality connecting the R´enyi entropy power, the recently introduced scaling-invariant relative Fisher information and the R´enyi 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. ∗ e-mail: [razvan.iagar@urjc.es](razvan.iagar@urjc.es)[ ](razvan.iagar@urjc.es)†e-mail: [david.puertas@urjc.es](david.puertas@urjc.es) 1 Introduction The derivation of sharp inequalities has been a cornerstone in the development of scientific knowledge. Fundamental inequalities such as the Cauchy–Schwarz, Jensen, H¨older or Gagliardo–Nirenberg inequalities have played a pivotal role across numerous areas of science, with strong implications not only in the theoretical advances but also in the understanding of physical phenomena. As a representative example, in statistical mechanics, the prominent role of the Gaussian probability density function in stochastic processes is supported by the central limit theorem, which at its turn is closely connected to the Cram´er–Rao inequality as noted in [1] and to several entropy-related inequalities [2–4], whose optimizing distribution is precisely the Gaussian density. Beyond the range of validity of the central limit theorem, the so-called stretched Gaussian densities have proven to be highly effective in describing a broad variety of physical systems (see for example [5–8]) . Furthermore, the entire family of stretched Gaussian densities can be characterized as the family of minimizers of certain generalizations of the Cram´er–Rao inequality. Consequently, the derivation of sharp inequalities involving information-theoretic functionals has become an active and important area of research, as seen in [9–14] . Since the most employed classical informational functionals are (Shannon and R´enyi) entropies, absolute moments and Fisher information, the inequalities bounding these quantities have become essential in the theory. Some of the most well-known informational inequalities are the Stam inequality, introduced in [15], the moment-entropy inequality introduced in its first form in [16] and in modern form in the seminal work [17] and the aforementioned Cram´erRao inequality, which is actually an immediate consequence of the previous two inequalities but has been established before them in [18,19] . For a modern treatment of these inequalities and extensions of them, we refer the reader to [20–22] (see also references therein) . More recently, these three inequalities have been extended by the authors and their collaborators to new functionals [23,24] and to a mirrored domain of validity for the entropic parameters [25], with the aid of Sundm

FAQs

What types of informational inequalities are derived in the document?

The document derives new sharp informational inequalities, including a Stam-like inequality, a moment-like inequality, and a Fisher-like inequality, spanning non-relative, relative, and cross frameworks.

Which information functionals are combined in the Stam-like result?

The Stam-like inequality links Rényi entropy power, a scaling-invariant relative Fisher information, and Rényi cross-entropy.

What probability densities minimize the stated inequalities?

Minimizers for the Stam-like inequality are given in certain cases by Gaussian or stretched Gaussian densities, while the minimizers for the moment-like inequality are probability densities of the generalized Beta distribution.

Jul 16, 2026

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