A General Double Inequality Related to Operator Means and Positive Linear Maps

TL;DR

This paper establishes a new double inequality involving operator means and positive linear maps, generalizing several known inequalities and providing new bounds in operator theory.

Contribution

It introduces a generalized double inequality for operator means under positive linear maps, extending existing inequalities like Diaz–Metcalf and reverse Ando inequalities.

Findings

Derived a new double inequality for operator means and positive linear maps.

Extended classical inequalities to more general operator mean settings.

Provided applications involving Hadamard products and operator inequalities.

Abstract

Let $A,B\in \mathbb{B}(\mathscr{H})$ be such that $0<b_{1}I \leq A \leq a_{1}I$ and $0<b_{2}I \leq B \leq a_{2}I$ for some scalars $0<b_{i}< a_{i},\;\; i=1,2$ and $\Phi:\mathbb{B}(\mathscr{H})\rightarrow\mathbb{B}(\mathscr{K})$ be a positive linear map. We show that for any operator mean $\sigma$ with the representing function $f$, the double inequality $$ \omega^{1-\alpha}(\Phi(A)#_{\alpha}\Phi(B))\le (\omega\Phi(A))\nabla_{\alpha}\Phi(B)\leq \frac{\alpha}{\mu}\Phi(A\sigma B) $$ holds, where $\mu=\frac{a_{1}b_{1}(f(b_{2}a_{1}^{-1})-f(a_{2}b_{1}^{-1}))}{b_{1}b_{2}-a_{1}a_{2}}, $ $\nu=\frac{a_{1}a_{2}f(b_{2}a_{1}^{-1})-b_{1}b_{2}f(a_{2}b_{1}^{-1})}{a_{1}a_{2}-b_{1}b_{2}}, $ $\omega=\frac{\alpha \nu}{(1-\alpha)\mu}$ and $#_{\alpha}$ ($\nabla_{\alpha}$, resp.) is the weighted geometric (arithmetic, resp.) mean for $\alpha \in (0,1)$. As applications, we present several generalized operator inequalities including Diaz--Metcalf and reverse Ando type inequalities. We also give some related inequalities involving Hadamard product and operator means.