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Refinements of Jensen's Inequality for Twice-Differentiable Convex Functions with Bounded Hessian

Sambhab Mishra

TL;DR

The paper addresses the need for precise refinements of Jensen's inequality for twice-differentiable convex functions with bounded Hessian. It develops an integral remainder framework, Grüss-type refinements, and a fourth-order moment expansion to tighten Jensen bounds and incorporate skewness and kurtosis. Key contributions include variance-based bounds with domain partitioning, Grüss-type remainder bounds, Green function representations, and a corrected Jensen inequality with higher moments; these are applied to Shannon entropy and Rayleigh fading capacity. The results yield tighter, practically useful bounds for information-theoretic and wireless-communication problems and are validated numerically.

Abstract

Jensen's inequality, attributed to Johan Jensen -- a Danish mathematician and engineer noted for his contributions to the theory of functions -- is a ubiquitous result in convex analysis, providing a fundamental lower bound for the expectation of a convex function. In this paper, we establish rigorous refinements of this inequality specifically for twice-differentiable functions with bounded Hessians. By utilizing Taylor expansions with integral remainders, we tried to bridge the gap between classical variance-based bounds and higher-precision estimates. We also discover explicit error terms governed by Gruss-type inequalities, allowing for the incorporation of skewness and kurtosis into the bound. Using these new theoretical tools, we improve upon existing estimates for the Shannon entropy of continuous distributions and the ergodic capacity of Rayleigh fading channels, demonstrating the practical efficacy of our refinements.

Refinements of Jensen's Inequality for Twice-Differentiable Convex Functions with Bounded Hessian

TL;DR

The paper addresses the need for precise refinements of Jensen's inequality for twice-differentiable convex functions with bounded Hessian. It develops an integral remainder framework, Grüss-type refinements, and a fourth-order moment expansion to tighten Jensen bounds and incorporate skewness and kurtosis. Key contributions include variance-based bounds with domain partitioning, Grüss-type remainder bounds, Green function representations, and a corrected Jensen inequality with higher moments; these are applied to Shannon entropy and Rayleigh fading capacity. The results yield tighter, practically useful bounds for information-theoretic and wireless-communication problems and are validated numerically.

Abstract

Jensen's inequality, attributed to Johan Jensen -- a Danish mathematician and engineer noted for his contributions to the theory of functions -- is a ubiquitous result in convex analysis, providing a fundamental lower bound for the expectation of a convex function. In this paper, we establish rigorous refinements of this inequality specifically for twice-differentiable functions with bounded Hessians. By utilizing Taylor expansions with integral remainders, we tried to bridge the gap between classical variance-based bounds and higher-precision estimates. We also discover explicit error terms governed by Gruss-type inequalities, allowing for the incorporation of skewness and kurtosis into the bound. Using these new theoretical tools, we improve upon existing estimates for the Shannon entropy of continuous distributions and the ergodic capacity of Rayleigh fading channels, demonstrating the practical efficacy of our refinements.
Paper Structure (30 sections, 10 theorems, 48 equations, 1 table)

This paper contains 30 sections, 10 theorems, 48 equations, 1 table.

Table of Contents

  1. Introduction
  2. The Need for Refinement
  3. Analytical Framework
  4. Mathematical Preliminaries and Definitions
  5. Convexity and Generalized Convexity
  6. Taylor's Theorem with Integral Remainder
  7. The Grüss and Chebysev Inequalities
  8. Variance-Based Refinements for Bounded Hessian
  9. The Standard Variance Bounds
  10. Refinement via Domain Partitioning
  11. Grüss-Type Refinements for Taylor Remainders
  12. Applying the Grüss Inequality to the Remainder
  13. Bounds via Green Functions
  14. Fourth-Order Moment Refinements
  15. The Fourth-Order Expansion Theorem
  16. ...and 15 more sections

Key Result

Theorem 2.1

Let $\phi: I \to \mathbb{R}$ be a function such that $\phi^{(n)}$ is absolutely continuous on $I$. Then for any $a, x \in I$: where the remainder term is given by:

Theorems & Definitions (15)

  • Definition 2.1: Convex Function
  • Definition 2.2: Strongly Convex Function
  • Definition 2.3: $(m, M)$-Convexity
  • Theorem 2.1: Taylor's Theorem
  • Definition 2.4: Chebysev Functional
  • Theorem 2.2: Grüss Inequality
  • Theorem 2.3: Pre-Grüss Inequality
  • Theorem 3.1
  • proof
  • Theorem 3.2: Partitioned Variance Bound
  • ...and 5 more