Upper bound for the sum $\left|\sum_{i=1}^N x_i\right|^p$

Question

Since it is a simple looking question, this might be asked before (however I was not able to find it).

I am looking for a bound in the form, $$\left|\sum_{i=1}^N x_i\right|^p \leq C(p,N) \sum_{i=1}^N |x_i|^p$$ for $x_i\in\mathbb{R}$, with an explicit constant $C(p,N)$ (at least with its dependence to $N$). I would be very happy if I could learn the order of $C(p,N)$ in terms of $N$ (or $N,p$ together).

There is a related question in math.SE for $N = 2$ case, where $C(p,N) = 2^{p-1}$. In general, I am interested in this constant for general $N$.

Thank you very much.

Answer 1

https://en.wikipedia.org/wiki/Generalized_mean $$$$if $p\ge 1$ then $$ \\\left|\dfrac{\sum_{i=1}^N x_i}{N}\right|^p\le\dfrac{\left|\sum_{i=1}^N |x_i|^p\right|}{N} \\C(p, N)=N^{p-1} $$