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

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 $Φ:\mathbb{B}(\mathscr{H})\rightarrow\mathbb{B}(\mathscr{K})$ be a positive linear map. We show that for any operator mean $σ$ 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 $μ=\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}}, $ $ν=\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}}, $ $ω=\frac{αν}{(1-α)μ}$ and $#_α$ ($\nabla_α$, resp.) is the weighted geometric (arithmetic, resp.) mean for $α\in (0,1)$. 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.