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

TL;DR

This paper presents refined Jensen's inequalities for twice-differentiable convex functions with bounded Hessians, providing higher-precision bounds that incorporate skewness and kurtosis, with applications to entropy and channel capacity.

Contribution

It introduces new bounds for Jensen's inequality using Taylor expansions and error terms, enhancing accuracy over classical variance-based estimates.

Findings

Improved bounds for Shannon entropy of continuous distributions.

Enhanced estimates for ergodic capacity of Rayleigh fading channels.

Explicit error terms involving skewness and kurtosis.

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.