Refined Heinz Mean Operator Inequality

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.