I am trying to show the following:
Denote with $|\cdot|_p$ the p-norm of a vector. Given $x_1, x_2 \in R^n$ so that all components of $x_1, x_2$ are strictly positive, $|x_1|_1=|x_2|_1=1$ and $|x_1|_2 \leq |x_2|_2$. Does this imply that $|x_1|_p \leq |x_2|_p$ for $p>2$?
Any help is much appreciated!
P-norms are non-increasing with p so I tried taking derivatives of the above to show that one side decreases faster, without success. This problem feels connected with Rényi entropy but I found nothing useful there. I found examples where this is true and no counter example.