Taking the triangle inequality for norm of a matrix to some power

Question

I'm doing a derivation and was wondering whether the known triangle inequality for the norm of a sum of two matrices can be generalized when the norm is to some power $n$, i.e.

\begin{equation} ||A + B||^n \leq \left(||A|| + ||B|| \right)^n. \end{equation}

In particular, I'm considering the $\infty$-norm. I know this holds for when $n=1$ but am not sure whether it can be generalized to other $n > 1$.

Answer 1

Notice that $x^n$ is an increasing function over the nonnegative region.

Since we know $\|A+B \| \le \|A\|+\|B\|$ and they are nonnegative, the inequality holds.