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.

翻译：Jensen不等式源于丹麦数学家兼工程师约翰·Jensen在函数论领域的重要贡献，是凸分析中普遍存在的结果，为凸函数的期望提供了基本下界。本文针对具有有界Hessian矩阵的二阶可微函数，建立了该不等式的严格改进形式。通过采用带积分余项的泰勒展开，我们试图弥合基于经典方差的界与更高精度估计之间的差距。同时，我们发现了由Gruss型不等式控制的显式误差项，使得偏度和峰度能够被纳入界中。利用这些新的理论工具，我们改进了连续分布香农熵与瑞利衰落信道遍历容量的现有估计，从而证明了所提改进方法的实际有效性。