Some Estimations for the Generalized Relative Operator Entropy

In this paper, we investigate a notion of the generalized relative operator entropy, which develops the theory of the relative operator entropy introduced by Fujii and Kamei, and a notion of the Csiszar operator f-divergence mapping. We estimate some upper and lower bounds of the generalized relative operator entropy and generalized operator Shannon entropy. In particular, we reach some new bounds for the relative operator entropy, the operator q-geometric mean, and the χ2\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\chi ^2$$\end{document}-divergence. Mainly, our results extend some known operator inequalities.