Matrix Inequalities between $f(A)\sigma f(B)$ and $A\sigma B$

Let $A$ and $B$ be $n\times n$ positive definite complex matrices, let $\sigma$ be a matrix mean, and let $f:[0,\mathrm{\infty })\to [0,\mathrm{\infty })$ be a differentiable convex function with $f(0)=0$. We prove that $$f^{\mathrm{\prime }}(0)(A\sigma B)\le \frac{f(m)}{m}(A\sigma B)\le f(A)\sigma f(B)\le \frac{f(M)}{M}(A\sigma B)\le f^{\mathrm{\prime }}(M)(A\sigma B),$$ 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 $f(x)\mathrm{/}x$ is increasing, then $$\mathrm{\mid }\mathrm{\mid }\mathrm{\mid }f(A)+f(B)\mathrm{\mid }\mathrm{\mid }\mathrm{\mid }\le \frac{f(M)}{M}\mathrm{\mid }\mathrm{\mid }\mathrm{\mid }A+B\mathrm{\mid }\mathrm{\mid }\mathrm{\mid }\le \mathrm{\mid }\mathrm{\mid }\mathrm{\mid }f(A+B)\mathrm{\mid }\mathrm{\mid }\mathrm{\mid }$$ holds for all $A$ and $B$ with $M\le A+B$. Furthermore, we apply our results to explore some related inequalities. As an application, we present a generalization of Minkowski’s determinant inequality.equality.