This is an inequality used in a proof which I do not know how to prove.
$$\left(\sum_{k = 2^j +1}^{2^{j+1}} \frac{\sin(k\pi t)}{k}G_k\right)^2 \leq \left|\sum_{k = 2^j +1}^{2^{j+1}} \frac{e^{ik\pi t}}{k}G_k\right|^2 =\sum_{k = 2^j +1}^{2^{j+1}} \sum_{l = 2^j +1}^{2^{j+1}} \frac{e^{ik\pi t} e^{-il\pi t}}{kl} G_k G_l,$$ where $G_k \sim N(0,1).$
Update: There is a further inequality following from the previous one. It seems that here we use the fact that $e^{im\pi t}$ is a unit complex number. Once again, however, I cannot see why we can compare a complex number with a real number.
$$\sum_{l = 2^j+1}^{2^{j+1}-m} \frac{e^{im\pi t}}{l(l+m)} G_lG_{l+m} \leq \left|\sum_{l = 2^j+1}^{2^{j+1}-m} \frac{G_lG_{l+m}}{l(l+m)} \right|.$$
Thank you for your help.