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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!

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  • $\begingroup$ 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. $\endgroup$ – quid May 23 '16 at 18:00
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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 $j=2$.

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