Some operator inequalities for operator means

Let $A$ and $B$ be two positive operators with $0 m \leqslant A, B \leqslant M$ for positive real numbers $ M, m$ and $\sigma$ and $\sigma^*$ be two adjoint operator means and $\Phi$ is a positive unital linear map. If $\sigma\leqslant \sigma_1,\sigma_2\leqslant \sigma^*$ then $$\Phi^{p}(A \sigma_{1} B) \leqslant \alpha^{p} \Phi^{p}(A \sigma_{2} B),$$ where $$ \alpha= \max \left \lbrace \dfrac{(M+m)^{2}}{4mM}, \dfrac{(M+m)^{2}}{4^{\frac{2}{p}}mM} \right \rbrace.$$ In addition, for $p\geqslant 4$ $$\Phi^{p}(A \sigma_{1} B) \leqslant \dfrac{1}{16}\left (\dfrac{ K(h)(M^{2}+m^{2})}{mM}\right )^{p} \Phi^{p}(A \sigma_{2} B),$$ where $ K(h) $ is the kantorovich constant with $ h=\dfrac{M}{m}. $