Is $(\sum_{i=1}^m (x _ {i})^{n})^{n+1} \le (\sum_{i=1}^m (x_{i})^{n+1})^{n}$ true for sufficiently large $n$

Question

I'm working on this problem about the relationship between $\ell_{p}$ norm and $\ell_{\infty}$. This leads me to evaluate the below inequality.

$$\left( \sum_{i=1}^m \left( x _ { i } \right) ^ { n } \right) ^ { n + 1 } \le \left( \sum_{i=1}^m \left( x _ { i } \right) ^ { n + 1 } \right) ^ { n }, \quad (x_1, \ldots,x_m) \in {(\mathbb R^+)}^m$$

I would like to ask whether there exists $N \in \mathbb N$ such that this inequality holds for all $n \ge N$.

Thank you for your help!

Answer 1

Your inequality is equivalent to $$\|x\|_{\ell_p^{n}}\leq \|x\|_{\ell_p^{n+1}}$$ which is not true. Actually, the reverse inequality is valid.

Answer 2

Take $x_i=1$ for all $i$ to see that it is false for any $m>1$ however large you take $N$ to be.