I just want to hear from you if I am understanding it correctly. No proofs here, just yes or no.
It is well-know that, if $a_i,b_i$ are complex numbers, then $$ \left | \sum_{i}a_i\overline{b_i} \right |^2\leq \sum_{i}\left | a_i \right |^2\sum_{i}\left | b_i \right |^2 $$ Would this still hold, if we replace $\overline{b_i}$ with $b_i$? Or, using this inequality, could we then say that $$ \left | \sum_{i}a_ib_i \right |^2=\left | \sum_{i}a_i\overline{(\overline{b_i})} \right |^2\leq \sum_{i}\left | a_i \right |^2\sum_{i}\left | \overline{b_i} \right |^2=\sum_{i}\left | a_i \right |^2\sum_{i}\left | b_i \right |^2? $$