Norm inequalities related to operator monotone functions

Let A, B be positive semidefinite matrices and ||| · ||| any unitarily invariant norm on the space of matrices. We show |||f(A) + f(B)||| ≥ |||f(A + B)||| for any non-negative operator monotone function f(t) on [0, ∞), and |||g(A) + g(5)||| ≤ |||g(A + B)||| for non-negative increasing function g(t) on [0, ∞) with g(0) = 0 and g(∞) = ∞, whose inverse function is operator monotone.