Inequality for $p$ vector norm

Question

Is it true for $2 \leq q \leq p$ that $$ \|x\|_q\leq n^{\frac{p-q}{pq}} \|x\|_p $$ where $x$ is an $n$-dimensional vector. I only need the inequality for $n=2$, so that would suffice. I'm just curious if it's true for any $n$. Just from plugging it into a graphing calculator I believe it to be true, but how would I prove it?

Answer 1

$$ \lVert x\rVert_q^{p }=\left(\frac 1n\sum_{i=1}^n\lvert x_i\rvert^q\right)^{p/q}n^{p/q} $$ and using convexity of the map $t\mapsto t^{p/q}$, one gets $$ \lVert x\rVert_q^{p }\leqslant \frac 1n\sum_{i=1}^n\lvert x_i\rvert^p n^{p/q} $$ or in other words, $$ \lVert x\rVert_q^{p }\leqslant n^{p/q-1}\lVert x\rVert_p^p. $$