TITLE: $\int_0^1\frac{f^2(t)}{t(1-t)}dt \leq \frac{1}{2}\int_0^1 f'(t)^2 dt$
QUESTION [9 upvotes]: Let $f\in C^1([0,1],\mathbb R)$ such that $f(0)=f(1)=0$
Prove that $\displaystyle \int_0^1\frac{f^2(t)}{t(1-t)}dt \leq \frac{1}{2}\int_0^1 f'(t)^2 dt$

First thing that bothers me is that the LHS is an improper integral.
Note that $\displaystyle \int_0^1 f'(t)dt =0$.
I tried integration by parts, but I  can't get the RHS.

REPLY [6 votes]: This is more like a comment than an answer but we have:
$$\displaystyle \int_0^1 \left(\frac{1}{2} f'(t)^2 - \frac{f^2(t)}{t(1-t)}\right) dt = \frac{1}{2}\int_0^1 \left(f'(t) - \frac{1-2t}{t(1-t)}f(t) \right)^2\,dt \ge 0$$
Q.E.D.