A General Double Inequality Related to Operator Means and Positive   Linear Maps
  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. 
R. Kaur, M. Singh, J. S. Aujla and M. S. Moslehian
math.FA math.OA