Some matrix mean inequalities with Kantorovich constant

Let $0 \begin{eqnarray*} \Phi^2(A\sigma B) & \le & K^2(h)\Phi^2(A\tau B),\\ \Phi^2(A\sigma B) & \le & K^2(h)\left(\Phi(A)\tau \Phi(B)\right)^2,\\ (\Phi(A)\sigma\Phi(B))^2 & \le & K^2(h)\Phi^2(A\tau B),\\ (\Phi(A)\sigma\Phi(B))^2 & \le & K^2(h) (\Phi(A)\tau \Phi(B))^2, \end{eqnarray*} where $K(h)=\dfrac{(h+1)^2}{4h}$ and $h=\dfrac{M}{m}$ is the Kantorovich constant