# Some extensions of the Young and Heinz inequalities for Matrices

## TL;DR

This paper extends classical Young and Heinz inequalities to matrices using unitarily invariant norms, providing new bounds and inequalities for positive semidefinite matrices and arbitrary matrices.

## Contribution

It introduces novel matrix inequalities extending Young and Heinz inequalities for unitarily invariant norms, including bounds involving positive semidefinite matrices.

## Key findings

- New inequalities for matrices involving unitarily invariant norms.
- Extensions of Young and Heinz inequalities to matrix settings.
- Specific bounds for positive semidefinite matrices and arbitrary matrices.

## Abstract

In this paper, we present some extensions of the Young and Heinz inequalities for the Hilbert-Schmidt norm as well as any unitarily invariant norm. Furthermore, we give some inequalities dealing with matrices. More precisely, for two positive semidefinite matrices $A$ and $B$ we show that \begin{align*} \Big\|A^{\nu}XB^{1-\nu}+A^{1-\nu}XB^{\nu}\Big\|_{2}^{2}\leq\Big\|AX+XB\Big\|_{2}^{2}- 2r\Big\|AX-XB\Big\|_{2}^{2}-r_{0}\left(\Big\|A^{\frac{1}{2}}XB^{\frac{1}{2}}-AX\Big\|_{2}^{2}+ \Big\|A^{\frac{1}{2}}XB^{\frac{1}{2}}-XB\Big\|_{2}^{2}\right), \end{align*} where $X$ is an arbitrary $n\times n$ matrix, $0<\nu\leq\frac{1}{2}$, $r=\min\{\nu, 1-\nu\}$ and $r_{0}=\min\{2r, 1-2r\}$.