Norm inequalities related to operator monotone functions

Abstract: Let $A$ be a positive definite operator on a Hilbert space $H$, and $|||.|||$ be a unitarily invariant norm on $B(H)$. We show that if $f$ is an operator monotone function on $(0,\infty)$ and $n\in \mathbb{N}$, then $|||D^n f(A)|||\leq|f^{(n)}(A)|$ and $|f^{(n)}(\cdot)|$ is a quasi-convex function on the set of all positive definite operators in $B(H)$. We establish some estimates of the right hand side of some Hermite-Hadamard type inequalities in which differentiable functions are involved, and norms of the maps induced by them on the set of self adjoint operators are convex, quasi-convex or $s$-convex. As applications, we obtain some of bounds for $|||f(B)-f(A)|||$ in term of $|||B-A|||$. For instance, Let $f,g$ be two operator monotone functions on $(0,\infty)$. Then, for every unitarily invariant norm $|||.|||$ and every positive definite operators $A,B$, \begin{align*} &\left|\left|\left|f(A)g(A)-f(B)g(B)\right|\right|\right|\notag\ &\leq|||B-A|||\Big[\max\left{|f'(A)|,|f'(B)|\right}\times\max\left{|g(A)|,|g(B)|\right}\notag\ &+\max\left{|f(A)|,|f(B)|\right}\times \max\left{|g'(A)|,|g'(B)|\right}\Big]. \end{align*}