Matrix inequalities between \(f(A)\sigma f(B)\) and \(A\sigma B\)

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