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Inequalities for Pairs of Measure Spaces and Applications

P. D. Johnson, R. N. Mohapatra, Shankhadeep Mondal

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 Hölder'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.

Inequalities for Pairs of Measure Spaces and Applications

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 Hölder'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.
Paper Structure (7 sections, 14 theorems, 57 equations)

This paper contains 7 sections, 14 theorems, 57 equations.

Table of Contents

  1. Introduction and background
  2. Main Inequality on Product Measure Spaces
  3. Quantitative and Variational Refinements
  4. Variational Extremality
  5. Power, Entropy, and Mean-Type Consequences
  6. Robustness Under Erasures
  7. A family of Special Cases

Key Result

Theorem 1.1

Suppose $M = [m_{ij}]$ is a $p \times q$ matrix of real numbers with constant column sum $c$ and row sums $d_1, \dots, d_p$. Suppose that $\varphi$ is a function such that $f(x) = x \varphi (x)$ is convex on an interval containing $d_1, \dots, d_p$. Let $\bar{d} = \frac{1}{p} \sum _{i=1}^p d_i (= \f If $f$ is strictly convex then equality holds if and only if $d_1 = \cdots = d_p$.

Theorems & Definitions (26)

  • Theorem 1.1: JP
  • Theorem 1.2: PJ
  • Theorem 2.1
  • Remark 2.2
  • proof : Proof of Theorem \ref{['t3']}
  • Theorem 3.1
  • proof
  • Proposition 3.2
  • proof
  • Proposition 3.3
  • ...and 16 more