Operator mean inequalities for sector matrices

Abstract: In this note, some inequalities involving operator means of sectorial matrices are proved which are generalizations and refinements of previous known results. Among them, let $A$ and $B$ be two accretive matrices with $A,B\in\mathcal{S}{\theta}$, $0 < mI \leqslant A, B \leqslant MI$ for positive real numbers $ M, m, \, \sigma$ be an operator mean and $\sigma^{*}$ be the adjoint mean of $ \sigma.$ If $\sigma^*\leqslant \sigma_1,\sigma_2\leqslant \sigma$ and $\Phi$ is a positive unital linear map, then $$\Phi^{p}\Re(A \sigma{1} B) \leqslant \sec^{2p}\theta\alpha^{p} \Phi^{p}\Re(A \sigma_{2} B),$$ where $$ \alpha= \max \left \lbrace K, 4^{1-\frac{2}{p}}K \right \rbrace,$$ and $ K= \frac{(M+m)^2}{4mM}$ is the Kantorovich constant.