Different terms such as variability, inequality and dispersion, which occur in various engineering problems and scientific fields, in mathematics are most simply described by the concept of majorization, a powerful mathematical tool which allows one to see the existing connections between vectors that can be used. In majorization theory, majorization inequalities play an important role. In this paper, using known properties of superquadratic functions, extensions and improvements of majorization inequalities are obtained. Also their converse inequalities are presented. For superquadratic functions, which are not convex, results analog ones for convex functions are presented. For superquadratic functions which are convex, improvements are given. At the end, applications to \(\varphi \)-divergences are discussed. New estimates for the Rényi entropy are derivated.

在数学中的各种工程问题和科学领域中出现的不同术语，例如变异性，不等式和离散，最简单地用专业化概念来描述，专业化是一种功能强大的数学工具，可以让人们看到可以使用的向量之间的现有联系。 。在专业化理论中，专业化不平等起着重要的作用。在本文中，利用超二次函数的已知性质，获得了对不等式的扩展和改进。还介绍了它们的逆不等式。对于非凸的超二次函数，给出了凸函数的模拟结果。对于凸的超二次函数，给出了改进。最后，应用到\（\ varphi \）-分歧被讨论。推导了Rényi熵的新估计。