More results on weighted means

TL;DR

This paper refines Young's inequality, explores bounds for the logarithmic mean, and extends these results to operator inequalities involving weighted means and entropies, advancing the theoretical understanding of inequalities among means.

Contribution

It introduces generalized scalar and operator inequalities for weighted means, including extensions involving Tsallis operator relative entropy, building upon and broadening previous results.

Findings

Refined Young inequality generalizing Zou--Jiang

Upper bounds for the logarithmic mean using weighted means

Operator inequalities involving weighted relative entropy

Abstract

We give a refined Young inequality which generalizes the inequality by Zou--Jiang. We also show the upper bound for the logarithmic mean by the use of the weighted geometric mean and the weighted arithmetic mean. Furthermore, we show some inequalities among the weighted means. Based on the obtained essential scalar inequalities, we give some operator inequalities. In particular, we give a generalization of the result by Zou--Jiang, that is, we show the operator inequalities with the operator relative entropy with the weighted parameter. Finally, we give the further generalized inequalities by the Tsallis operator relative entropy.