On further refinements for Young inequalities

TL;DR

This paper presents sharp refinements and reverses of the operator Young's inequality, along with an improved additive inequality and estimates for specific operator differences, advancing the theoretical understanding of operator inequalities.

Contribution

It introduces new sharp multiplicative refinements and reverses for operator Young's inequality, and improves a known additive inequality, extending the scope of operator inequality analysis.

Findings

Sharp multiplicative refinements of operator Young's inequality

Reverses of the operator Young's inequality

Improved additive inequality related to Tominaga's inequality

Abstract

In this paper, sharp results on operator Young's inequality are obtained. We first obtain sharp multiplicative refinements and reverses for the operator Young's inequality. Secondly, we give an additive result, which improves a well-known inequality due to Tominaga. We also provide some estimates for $A{{\sharp}_{v}}B-A{{\nabla }_{v}}B$ in which $v\notin \left[ 0,1 \right]$.