Let $f, g$ be $\mathbb C$-valued functions defined on $\mathbb R$ and $f, g \in L^2$. To prove the inequality in this title, I proceed as follows but got a weaker bound. Recall that $\mathrm{Re}\ a \leq |a|$ and $\mathrm{Im}\ a \leq |a|, \forall a \in \mathbb C$.
$$\left|(f, g)\right| = \left|\int f\bar g \right| = \left|\int \mathrm{Re}\ f\bar g + i\int\mathrm{Im}\ f\bar g\right| = \sqrt{\left(\int \mathrm{Re}\ f\bar g\right)^2 + \left(\int\mathrm{Im}\ f\bar g\right)^2} \leq \sqrt{\left(\int \left|f\bar g\right|\right)^2 + \left(\int\left |f\bar g\right| \right)^2} = \sqrt{2}\int\left |f\bar g\right|.$$
How do I get the desired bound, please? Thank you!