Young-like inequality

Question

I'm having trouble with the following statement:

For positive variables $\{x_1,\dots,x_n\}\subset \mathbb{R_+}$ and $k\geq 1$ an integer, there exists $C=C(k)$ a numerical constant such that

\begin{align*} (\sum_n x_n)^k\leq C \sum_n x_n^k. \end{align*}

For $k=2$ this might be done by the Young inequality, what about the case $k\geq 3$?

Answer 1

I was able to solve this question using Jensen's inequation:

The above setting is equivalent to \begin{align*} n^k\frac{1}{n^k}(\sum_n x_n)^k=n^k(\frac{1}{n}\sum_n x_n)^k\leq n^k\frac{1}{n}\sum_n x_n^k \end{align*} where the inequation comes from Jensen's inequality and $n^{k-1}$ is our $C$.