Suppose complex function $f$ in the Schwartz Space, its definition see http://en.wikipedia.org/wiki/Schwartz_space how can we argue that $$\int_{-\infty}^{\infty}f '\bar{f}'+x^2 f\bar{f}dx\geq \int_{-\infty}^{\infty}|f|^2 dx ~?$$ PS: $\bar{f}$ is conjugate to $f$, $f'=\frac{df}{dx}.$
I try to integrate by parts, but I get nothing.
Write $$|f'(x)+xf(x)|^2=|f'(x)|^2+x^2|f(x)|^2+xf(x)\overline{f'(x)}+x\overline{f(x)}f'(x),$$ then integrate by parts in the integral $\int_{-\infty}^{+\infty}xf(x)\overline{f'(x)}dx$.