Let $A,B\\\\in \\\\mathbb{B}(\\\\mathscr{H})$ be such that $0<b_{1}I \\\\leq A \\\\leq a_{1}I$ and $0<b_{2}I \\\\leq B \\\\leq a_{2}I$ for some scalars $0<b_{i}< a_{i},\\\\;\\\\; i=1,2$ and $\\\\Phi:\\\\mathbb{B}(\\\\mathscr{H})\\\\rightarrow\\\\mathbb{B}(\\\\mathscr{K})$ be a positive linear map. We show that for any operator mean $\\\\sigma$ with the representing function $f$, the double inequality $$ \\\\omega^{1-\\\\alpha}(\\\\Phi(A)\\\\#_{\\\\alpha}\\\\Phi(B))\\\\le (\\\\omega\\\\Phi(A))\\\\nabla_{\\\\alpha}\\\\Phi(B)\\\\leq \\\\frac{\\\\alpha}{\\\\mu}\\\\Phi(A\\\\sigma B) $$ holds, where $\\\\mu=\\\\frac{a_{1}b_{1}(f(b_{2}a_{1}^{-1})-f(a_{2}b_{1}^{-1}))}{b_{1}b_{2}-a_{1}a_{2}},~$ $\\\\nu=\\\\frac{a_{1}a_{2}f(b_{2}a_{1}^{-1})-b_{1}b_{2}f(a_{2}b_{1}^{-1})}{a_{1}a_{2}-b_{1}b_{2}},~$ $\\\\omega=\\\\frac{\\\\alpha \\\\nu}{(1-\\\\alpha)\\\\mu}$ and $\\\\#_{\\\\alpha}$ ($\\\\nabla_{\\\\alpha}$, resp.) is the weighted geometric (arithmetic, resp.) mean for $\\\\alpha \\\\in (0,1)$. As applications we present several generalized operator inequalities including Diaz--Metcalf and reverse Ando type inequalities. We also give some related inequalities involving Hadamard product and operator means

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