This inclusion is continuous

Question

Let $H$ be the real space of all absolute continous functions $F$ from $[0,1]$ to the reals such that $f(0)=0$. I'd like to prove that there is $K>0$ such that for all $f\in H$

$\int_0^1f(x)^2dx\leq K\int_0^1f'(x)^2dx$

Any hint is welcome.

Answer 1

Write

$$f(x) = \int_0^x f'(t)\,dt,$$

apply the Cauchy-Schwarz inequality to the inner integral, change the order of integration.

Without any tricky estimates, you can achieve $K \leqslant \frac12$.