Refined Heinz Mean Operator Inequality

TL;DR

This paper refines Heinz mean operator inequalities for positive operators, providing a sequence of tighter bounds and applications to unitarily invariant norms, advancing the theoretical understanding of operator inequalities.

Contribution

It introduces refined bounds for Heinz mean operator inequalities, improving existing results and extending their applicability to unitarily invariant norms.

Findings

Established a sequence of increasingly tight inequalities for positive operators.

Provided applications to unitarily invariant norms.

Enhanced the theoretical framework of Heinz mean inequalities.

Abstract

It is shown that if $A,B\in \mathbb{B}\left( \mathcal{H} \right)$ be positive operators, then \begin{equation*} \begin{aligned} A\#B&\le \frac{1}{1-2\mu }{A^{\frac{1}{2}}}{{F}_{\mu }}\left( {A^{-\frac{1}{2}}}B{A^{-\frac{1}{2}}} \right){A^{\frac{1}{2}}}\\ & \le \frac{1}{2}\left[ A\#B+{{H}_{\mu }}\left( A,B \right) \right]\\ & \le \frac{1}{2}\left[ \frac{1}{1-2\mu }{A^{\frac{1}{2}}} {F_{\mu }}\left( {A^{-\frac{1}{2}}}B{A^{-\frac{1}{2}}} \right){A^{\frac{1}{2}}}+{H_{\mu }}\left( A,B \right) \right]\\ & \le \cdots \le \frac{1}{{{2}^{n}}}A\#B+\frac{{{2}^n}-1}{{2^n}}{H_\mu }\left( A,B \right)\\ & \le \frac{1}{{{2}^{n}}\left( 1-2\mu \right)}{A^{\frac{1}{2}}}{F_{\mu }}\left( {A^{-\frac{1}{2}}}B{A^{-\frac{1}{2}}} \right){A^{\frac{1}{2}}}+\frac{{{2}^{n}}-1}{{{2}^{n}}}{H_\mu }\left( A,B \right)\\ & \le \frac{1}{{2^{n+1}}}A\#B+\frac{{{2}^{n+1}}-1}{{2^{n+1}}}{H_\mu }\left( A,B \right)\\ & \le \cdots \le {H_\mu }\left( A,B \right). \end{aligned} \end{equation*} for each $\mu \in \left[ 0,1 \right]\backslash \left\{ \frac{1}{2} \right\}$. As an application, we present several inequalities for unitarily invariant norms. Our results are refinements of some existing inequalities.