An estimate of euclidean norm via multinomial theorem

Question

I have a problem with the following inequality: suppose $k \in \mathbb{N},$ $x \in \mathbb{R}^{n}$ ($n \in \mathbb{N}$), is it true that $$ (x_1^2 + \ldots + x_n^2)^k \leq C_{k,n}(|x_1|^k + \ldots + |x_n|^k)^2 $$ with some constants $C_{k,n}$ depending on $k$ and $n$ only? I have the suspicion that multinomial theorem and the inequality $|x^{\alpha}| \leq |x|^{|\alpha|}$ valid for any $x \in \mathbb{R}^n$ and any multiindex $\alpha \in \mathbb{Z}_{+}^{n}$ might be somehow involved. For example, I tried \begin{eqnarray*} (x_1^2 + \ldots + x_n^2)^k & \leq & \sum_{|\alpha| = k} \frac{k!}{\alpha!} |x^{2\alpha}| \leq \sum_{|\alpha| = k}\frac{k!}{\alpha!} \left(|x|^{|\alpha|}\right)^2, \end{eqnarray*} but cannot see any conclusion. Is this the right way? Any clues? Thank you.

Answer 1

Well, it seems that the correct answer has very little to do with either the multinomial theorem or the basic multiindex inequality mentioned above. In fact it is a consequence of Hölder's inequality.

First, we note that for $k=1,2$ the inequality is pretty trivial (in case $k=2$ it is even equality with $C_{k,n}=1$ whereas in case $k=1$ it is in general inequality with $C_{k,n}=1$). Now for the case $k \geq 3,$ the proposition can be reduced to equivalence between euclidean norm and general $\|\cdot\|_k$ norm: $$ \|x\|_2 \leq C_{k,n} \|x\|_{k}, $$ where according to one of the answers to this post: Relations between p norms the $C_{k,n}$'s are given by the formula $C_{k,n} = n^{(1/2 - 1/k)}.$