Operator mean inequalities for sector matrices

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.