For any real number $p > 0$, is it true that
$$ |x_1 + \cdots x_n|^p \leq |x_1|^p + \cdots |x_n|^p ?$$
I believe it is not since $|x + y|^2 \leq |x|^2 + |y|^2$ is not true, but if there are any similar results that include a constant, those are what I'd be looking for. I have found a few similar questions citing Jensen's inequality but I need something general for any positive $p$, not just $p \geq 1$ or $0 < p < 1$. Thanks in advance.
