Horváth László
Department of Mathematics, University of Pannonia, Egyetem u. 10., 8200 Veszprém, Hungary.
Entropy (Basel). 2023 Jun 19;25(6):954. doi: 10.3390/e25060954.
In this paper, we present a general framework that provides a comprehensive and uniform treatment of integral majorization inequalities for convex functions and finite signed measures. Along with new results, we present unified and simple proofs of classical statements. To apply our results, we deal with Hermite-Hadamard-Fejér-type inequalities and their refinements. We present a general method to refine both sides of Hermite-Hadamard-Fejér-type inequalities. The results of many papers on the refinement of the Hermite-Hadamard inequality, whose proofs are based on different ideas, can be treated in a uniform way by this method. Finally, we establish a necessary and sufficient condition for when a fundamental inequality of -divergences can be refined by another -divergence.
在本文中，我们提出了一个通用框架，该框架对凸函数和有限符号测度的积分优超不等式提供了全面且统一的处理。除了新的结果外，我们还给出了经典命题的统一且简单的证明。为了应用我们的结果，我们处理了埃尔米特 - 哈达玛 - 费耶尔型不等式及其改进形式。我们提出了一种通用方法来改进埃尔米特 - 哈达玛 - 费耶尔型不等式的两边。许多基于不同思路证明的关于埃尔米特 - 哈达玛不等式改进的论文结果，都可以用这种方法统一处理。最后，我们建立了一个关于何时一个(\alpha -)散度的基本不等式可以被另一个(\alpha -)散度改进的充要条件。
