For $v \in \mathbb{R}^n$ denote by $||v||_p$ its p-norm. That is $(\sum_{i=1}^nv_i^p)^{\frac{1}{p}}$ where $v_i$ are the componenets of $v$.
I'm looking for a way to bound the following expression: $$M_v=\frac{||v||_2^2||v||_4^4}{||v||_3^6}$$
Using Holder's inequality, and the fact that $||v||_2 \leq ||v||_3$ One can show that $M_v \leq n^{\frac{1}{3}}$. I'm wondering if that's the best possible bound.
If, for example, $v_i \sim \frac{1}{\sqrt{i}}$ then it's easy to verify that $M_v$ scales like $\ln(n)$.
I'm unable to find an example for which $M_v$ is much larger than that. Intuitively, I would like to say that $M_v$ is bound by some logarithm, but I can't find a suitable proof or a counterexample.
Thanks a lot.