An inequality with complex numbers.

Given $n$ complex numbers $z_1,\ldots,z_n$, is it true that $$|z_j|\sum_{k=1}^n|z_k|\leq\sum_{k=1}^n|z_k|^2$$ for $j\in\{1,\ldots,n\}$ ? Thank u for any help!

• It is usually not true for every $j$, but there is always at least one for which it is true. Namely that where $|z_j|$ is minimal. – quid May 23 '16 at 18:00

No. It is only true (i.e. it only needs to be true) if $j$ is the index of the minimum among all the $|z_k|$'s.
Example: $z_1 = 1$, $z_2 = 2$ and $j=2$.