A matrix subadditivity inequality for symmetric norms

Let $f(t)$ be a non-negative concave function on $[0,\infty)$. We prove that

$$\Vert f(\vert A+B\vert) \Vert \le \Vert f(\vert A\vert)+f(\vert B\vert) \Vert$$

for all normal $n$-by-$n$ matrices $A$, $B$ and all symmetric norms. This result has several applications. For instance, for a Hermitian ${\mathbb{A}}=[A_{i, j}]$ partitioned in blocks of the same size,

$$\left\Vert f(\vert{\mathbb{A}}\vert) \right\Vert \le \left\Vert \sum f(\vert A_{i, j}\vert) \right\Vert.$$

We also prove, in a similar way, that given $Z$ expansive and $A$ normal of the same size,

$$\Vert f(\vert Z^*AZ\vert) \Vert \le \Vert Z^*f(\vert A\vert)Z \Vert.$$