Mean inequalities for sector matrices involving positive linear maps

Abstract

Let ${\mathcal {S}}_{\alpha } (0 \le \alpha <\frac{\pi }{2} )$ stand for the set of all complex sector matrices and $\sigma _1, \sigma _2$ be two operator means satisfying $\sigma _1 \le \sigma _2.$ Except some other assertions, it is also shown that for $A, B \in {\mathcal {S}}_{\alpha },$

$$\begin{aligned} \Re (A\sigma _1 B)\le \sec ^2\alpha \ \Re (A\sigma _2 B) \end{aligned}$$

and

$$\begin{aligned} \Re (A\sigma _2 B)^{-1}\le \sec ^2\alpha \ \Re (A\sigma _1 B)^{-1}. \end{aligned}$$

In addition, if $\sigma _i^{*} \le \sigma _i,$ for $i=1, 2$ and $\Phi$ is a unital positive linear map, then

$$\begin{aligned} \Phi \Re (A \sigma _1 B)^{-1}\le \sec ^2\alpha \ \Re \big (\Phi (A^{-1})\sigma _2\Phi (B^{-1})\big ). \end{aligned}$$