Let $A$ and $B$ be two positive operators with $0 < m \leqslant A, B \leqslant M$ 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 $$\Phi^{p}(A \sigma_{1} B) \leqslant \alpha^{p} \Phi^{p}(A \sigma_{2} B),$$ 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. In addition, for $p\geqslant 4$ $$\Phi^{p}(A \sigma_{1} B) \leqslant \dfrac{1}{16}\left (\dfrac{ K(M^{2}+m^{2})}{mM}\right )^{p} \Phi^{p}(A \sigma_{2} B).$$

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