Let $p(t)$ be any continuous increasing function on $[0,\pi/2]$ with $p(0)=0, p(\pi/2)=1$. What is the following infimum?
$$\inf_p \int_0^{\frac{\pi}{2}} \frac{\left( p'(t) \right)^2}{4p(t) \left(1 - p(t) \right)} \, {\rm d} t$$
It seems that we could not estimate
$$p(t) \left(1 - p(t) \right) \leq \left(\frac{1}{2}\right)^2$$
since the minimum could not be attained.
$$\int_0^{\pi/2} \frac{p'^2(t)}{4p(t)[1-p(t)]}dt=\int_0^1 [(\sqrt{p(t)})'^2+(\sqrt{1-p(t)})'^2]dt$$
But we could not use the Schwarz inequality to estimate from below, since if the equality occurs, $p$ is a linear funtion,