Subadditivity of eigenvalue sums

Let $f(t)$ be a nonnegative concave function on $0 \leq t <\infty$ with $f(0)=0$, and let $X, Y$ be $n\times n$ matrices. Then it is known that $\Vert f(\vert X+Y\vert)\Vert_1\leq \Vert f(\vert X\vert)\Vert_1 +\Vert f(\vert Y\vert)\Vert_1$, where $\Vert \cdot \Vert_1$ is the trace norm. We extend this result to all unitarily invariant norms and prove some inequalities of eigenvalue sums.