Some new inequalities of Karamata type are established with a convex function in this paper. The methods of our proof allow us to obtain an extended version of the reverse of Jensen inequality given by Pe{\v} caric and Micic. Applying the obtained results, we give reverses for information inequality (Shannon inequality) in different types, namely ratio type and difference type, under some conditions. Also, we provide interesting inequalities for von Neumann entropy and quantum Tsallis entropy which is a parametric extension of von Neumann entropy. The inequality for von Neumann entropy recovers the non-negativity and gives a refinement for the weaker version of Fannes's inequality for only special cases. Finally, we estimate bounds for the Tsallis relative operator entropy.

中文翻译：

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一些新的 Karamata 型不等式及其对某些熵的应用

本文用凸函数建立了一些新的 Karamata 型不等式。我们的证明方法允许我们获得由 Pe{\v} caric 和 Micic 给出的 Jensen 不等式的逆的扩展版本。应用得到的结果，我们给出了在一定条件下不同类型的信息不等式（香农不等式）的反演，即比率型和差异型。此外，我们为冯诺依曼熵和量子 Tsallis 熵提供了有趣的不等式，这是冯诺依曼熵的参数扩展。冯诺依曼熵的不等式恢复了非负性，并仅在特殊情况下对范内不等式的较弱版本进行了改进。最后，我们估计了 Tsallis 相对算子熵的界限。