I have tried to write a proof for the following inequality, but I couldn't constitute a rigorous one yet.
Suppose $f$ is absolutely continuous on $[−a,a]$ for all $a \in \mathbb R$ so that $f′$ exists a.e. on $\mathbb R$. Then $$\int_{- \infty}^{\infty} |f(x)|^2 dx \leq 2 \Big ( \int_{- \infty}^{\infty} |xf(x)|^2 dx \Big )^{1/2} \Big ( \int_{- \infty}^{\infty} |f'(x)|^2 dx \Big )^{1/2}$$
Any help will be appreciated.