Consider a continuously differentiable function $f:[0,1]\to\mathbb{R}$ which is integrable and has expectation zero with respect to a measure $\mu$ on the space $[0,1]$, that is $\int_{[0,1]} f(x) d\mu = 0$.
Is there some way to express the $L_2$ norm of this function in terms of the $L_2$ norm of its derivative?
That is, for some $G: \mathbb{R}\to\mathbb{R}$ independent of $f$ is it possible to have$$\int_{[0,1]} f(x)^2 d\mu = G\Bigg(\int_{[0,1]} f'(x)^2 d\mu\Bigg)$$
Impose as many regularity conditions on the function $f$ as you like, i.e. I encourage answers considering nontrivial restricted spaces of functions.
Intuitively, it feels like a continuously differentiable function with mean zero can only get so far from zero as the size of its derivative allows, but I'm having trouble pinning down the exact relationship.