Hi I am trying to understand a step of a proof, why is it true that
$$\left [ \int_{-\infty}^{\infty}|t{f(t)}{f'^*(t)}|\,dt\right]^2\ge\left [ \int_{-\infty}^{\infty}\frac{t}{2}[{f'(t)}f^*(t)+{f'^*(t)}f(t)]\,dt\right]^2?$$
EDIT: $f\in\mathbb{L}^2(\mathbb{R})$, $f^*$ is just the complex conjugate of $f$
With your edit, it's obvious because $|f(t)^{\star}f'(t)|=|f(t)f'(t)^{\star}|$ and $$ \begin{align} \left|\int_{-\infty}^{\infty}\frac{t}{2}\left[f(t)f'(t)^{\star}+f(t)^{\star}f'(t)\right]dt\right| & \le \int_{-\infty}^{\infty}\frac{|t|}{2}(|f(t)f'(t)^{\star}|+|f(t)^{\star}f'(t)|)dt \\ & =\int_{-\infty}^{\infty}|tf(t)f'(t)^{\star}|dt. \end{align} $$