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

Abstract

Let $A,B\in \mathbb{B}(\mathcal{H})$ be such that $0<b_{1}I\le A\le a_{1}I$ and $0<b_{2}I\le B\le a_{2}I$ for some scalars $0<b_i<a_i,i=1,2$ and $\Phi :\mathbb{B}(\mathcal{H})\to \mathbb{B}(\mathcal{K})$ be a positive linear map. We show that for any operator mean $\sigma$ with the representing function $f$, the double inequality $$ ω^{1-α}(Φ(A)#_αΦ(B))\le (ωΦ(A))\nabla_αΦ(B)\leq \fracαμΦ(AσB) $$ 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 $#_α$ (${\nabla }_{\alpha }$, 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.

12 pages, to appear in Linear Algebra Appl. (LAA)