Advanced refinements of Young and Heinz inequalities

TL;DR

This paper presents new multi-term refinements of Young inequalities for real numbers and operators, along with Heinz-type inequalities involving the Hilbert-Schmidt norm, advancing the theoretical understanding of these inequalities.

Contribution

The paper introduces novel multi-term refinements of Young inequalities and establishes new Heinz-type inequalities with Hilbert-Schmidt norm, improving existing results in operator inequalities.

Findings

Refined Young inequalities for operators with multiple terms.

New Heinz-type inequalities involving Hilbert-Schmidt norm.

Improved bounds for positive operators in the context of Young inequalities.

Abstract

In this article, we prove several multi-term refinements of Young type inequalities for both real numbers and operators improving several known results. Among other results, we prove \begin{eqnarray*} A\#_{\nu}B&+&\sum_{j=1}^{N}s_{j}(\nu)\left(A\#_{\alpha_j(\nu)}B+A\#_{2^{1-j}+\alpha_j(\nu)}B-2A\#_{2^{-j}+\alpha_j(\nu)}B\right)\leq A\nabla_{\nu}B, \end{eqnarray*} for the positive operators $A$ and $B$, where $0\leq \nu\leq 1, N\in\mathbb{N}$ and $\alpha_j(\nu)$ is a certain function. Moreover, some new Heinz type inequalities involving the Hilbert-Schmidt norm are established.