A problem on the bounds of Lp-norms

Question

Let $L>0$ and $\Omega$ be the set of all integrable functions from $[0,L]$ to $[0,+\infty]$. Also, Let $f\in \Omega$ such that $\left \| f \right \|_{1}=1$. Find the tightest possible bounds for:
\begin{align}\left \| f \right \|_{2}\times \left \| \frac{1}{f} \right \|_{2}\end{align}

Answer 1

Hint: The information about $\|f\|_1$ is a red herring.

Answer 2

Here are a couple of hints, hopefully you can work it through yourself.

Hölder's inequality with the obvious indices is all you need for the lower bound - you can show tightness using constant functions.

For the upper bound, you can try setting it up with Hölder's inequality; but you can't get the required indices to stay $\ge 1$. You should try looking at some families of simple examples to see if such a bound is possible - in particular note that we can make $\Vert f \Vert$ very large while keeping $\Vert f^{-1} \Vert$ fairly steady if the function is highly peaked in a small region.