Let $A$ and $ B$ be $n\\\\\\\\\\\\\\\\times n$ positive definite complex matrices, let $\\\\\\\\\\\\\\\\sigma$ be a matrix mean, and let $f : [0,\\\\\\\\\\\\\\\\infty)\\\\\\\\\\\\\\\\to [0,\\\\\\\\\\\\\\\\infty)$ be a differentiable convex function with $f(0)=0$. We prove that $$f^{\\\\\\\\\\\\\\\\prime}(0)(A \\\\\\\\\\\\\\\\sigma B)\\\\\\\\\\\\\\\\leq \\\\\\\\\\\\\\\\frac{f(m)}{m}(A\\\\\\\\\\\\\\\\sigma B)\\\\\\\\\\\\\\\\leq f(A)\\\\\\\\\\\\\\\\sigma f(B)\\\\\\\\\\\\\\\\leq \\\\\\\\\\\\\\\\frac{f(M)}{M}(A\\\\\\\\\\\\\\\\sigma B)\\\\\\\\\\\\\\\\leq f^{\\\\\\\\\\\\\\\\prime}(M)(A\\\\\\\\\\\\\\\\sigma B),$$ where $m$ represents the smallest eigenvalues of $A$ and $B$ and $M$ represents the largest eigenvalues of $A$ and $B$. If $f$ is differentiable and concave, then the reverse inequalities hold. We use our result to improve some known subadditivity inequalities involving unitarily invariant norms under certain mild conditions. In particular, if $f(x)/x$ is increasing, then $$|||f(A)+f(B)|||\\\\\\\\\\\\\\\\leq\\\\\\\\\\\\\\\\frac{f(M)}{M} |||A+B|||\\\\\\\\\\\\\\\\leq |||f(A+B)|||$$ holds for all $A$ and $B$ with $M\\\\\\\\\\\\\\\\leq A+B$. Furthermore, we apply our results to explore some related inequalities. As an application, we present a generalization of Minkowski\\\\\\\\\\\\\\\'s determinant inequality.

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