Let $f$ be a continuously differentiable complex function on $\mathbf{R}$ s.t. the functions $x \mapsto xf(x)$ and $f'$ are in $L_{\mathbf{C}}^2(\mathbf{R},\lambda)$.
Show that $$ x|f(x)|^2dx \le 4\left(\int_x^\infty t^2|f(t)|^2dt\right)^{\frac{1}{2}}\left(\int_x^\infty |f'(t)|^2dt\right)^{\frac{1}{2}} $$
and use that to prove Heisenberg uncertainty inequality:
$$\int_{-\infty}^\infty|f(x)|^2 \le 2\left(\int_{-\infty}^\infty x^2|f(x)|^2dx\right)^{\frac{1}{2}}\left(\int_{-\infty}^\infty |f'(x)|^2\right)^{\frac{1}{2}}dx$$
Well I'm reading through complex measures on Hilbert spaces and I'm not able to solve this. Any help appreciated.