Inequality for $p>1$ with nonnegative real numbers Let $1<p<\infty$ and $x=(x_1,x_2,\ldots,x_N)\in\mathbb{R}^N$, $N\geq 2$. Then does the following inequality holds:
$$
(|x_1|^{2(p-1)}+\ldots+|x_N|^{2(p-1)})^\frac{1}{2}\leq(|x_1|^p+\ldots+|x_N|^p)^\frac{p-1}{p}.
$$
I can only see it holds for $p=2$. Can anyone please help to prove or disprove it for $p\neq 2$. Thanks.

In $p$-Norm notation, we have $\|x\|_{2(p-1)}^{p-1}\leq \|x\|_p^{p-1}$. This is equivalent to $\|x\|_{2(p-1)}\leq \|x\|_p$, which is true, because $2(p-1)\geq p$ for $p\geq 2$.
This is true by the monotonicity of the $p$-norms.
We prove $\|x\|_p\leq \|x\|_r$ for $r<p$. W.l.o.g. suppose that $\|x\|_p=1$. Then $\|x\|_p^p=1$ and $|x_i|\le 1$ for all $i$. Since $p>r$, we obtain $|x_i|^p \le |x_i|^r$ for all $i$ and we get
$$1=\|x\|_p^p = \sum_{i} |x_i|^p \le \sum_{i} |x_i|^r = \|x\|_r^r.$$
The proof of the monociticy is not my own proof. You can find it on several places in the internet.
