Inequalities for Pairs of Measure Spaces and Applications

TL;DR

This paper introduces a generalized Jensen-type inequality for pairs of measure spaces, extending classical inequalities like Hölder's and Minkowski, with applications to various mathematical and probabilistic models.

Contribution

It establishes a broad, unifying inequality framework that encompasses and extends classical measure inequalities, with sharp equality conditions and multiple applications.

Findings

Unified Jensen-type inequality for measure spaces

Extensions of classical inequalities with sharp equality conditions

Applications to convolution operators and probabilistic models

Abstract

We study a family of inequalities on pairs of measure spaces involving functions defined on product domains. Our main result establishes a Jensen-type inequality under a general product-measure framework, extending classical inequalities such as H\"older's and Minkowski's as special cases. The inequality admits sharp characterizations of equality and yields quantitative, variational, and probabilistic refinements under additional convexity assumptions. Several corollaries illustrate power-mean, entropy-type, and erasure-robust inequalities, as well as applications to convolution-type operators and weighted discrete models.