Summary: In this note, some operator inequalities for operator means and positive linear maps are investigated. The conclusion based on operator means is presented as follows: Let \(\Phi : B(\mathcal{H}) \to B(\mathcal{K})\) be a strictly positive unital linear map and \(h^{-1}_1 I_H \leq A \leq h_1 I_H\) and \(h_2^{-1} I_H \leq B \leq h_2 I_H\) for positive real numbers \(h_1, h_2 \geq 1\). Then for \(p>0\) and an arbitrary operator mean \(\sigma\), \[ (\Phi(A)\sigma\Phi(B))^p \leq \alpha_p \Phi^p(A\sigma^* B), \] where \(\alpha_p = \max \left\{ \left(\frac{\alpha^2(h_1 ,h_2)}{4}\right)^p, \frac{1}{16} \alpha^{2p}(h_1, h_2)\right\}\), \(\alpha(h_1, h_2) = (h_1 + h^{-1}_{1})\sigma(h_2 + h^{-1}_{2})\). Likewise, a \(p\)-th (\(p \geq 2\)) power of the Diaz-Metcalf type inequality is also established