The question is as in the above.
In all literature, I only find that on $[0,1]$ with the Lebesgue measure, $\lVert \cdot \rVert_q \leq \lVert \cdot \rVert_p$ for $p>q$.
(I deleted the last question because it was poorly worded...) but in the last question, what I wanted to know is the possibility of reversing this inequality.
Namely, for a given pair $p>q$ and all $f \in L^q[0,1]$ with $\lVert f \rVert_q \geq 1$, is it possible to have some very large but "fixed" $n$ such that \begin{equation} \lVert f \rVert_p^{1/n} \leq \lVert f \rVert_q \end{equation}