Norm inequalities related to operator monotone functions

Abstract.

Let A,B be positive semidefinite matrices and $|||\cdot|||$ any unitarily invariant norm on the space of matrices. We show $|||f(A) + f(B)||| \geq |||f(A + B)|||$ for any non-negative operator monotone function f(t) on $[0,\infty)$, and $|||g(A) + g(B)||| \leq |||g(A + B)|||$ for non-negative increasing function g(t) on $[0,\infty)$ with g(0) = 0 and $g(\infty) = \infty$, whose inverse function is operator monotone.