I find an interesting theorem,but have no idea to prove it.
$f(x) \in C^2[0,1]$ and $f(0)=f(1)=0$ , $f(x) \not = 0 \ \ , x\in (0,1) $
Prove that if $\displaystyle\int_{0}^{1} \left|\frac {f^{''}(x)}{f(x)}\right| dx$ exists, $$\int_{0}^{1} \left|\frac {f^{''}(x)}{f(x)}\right| dx \geq 4$$
It is a problem for fun. If anyone knows how to do, plz tell me, not just vote " It's useless".
Edit: I submit a proof in the same question. And there are more two excellent answers.