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Let $z_1,z_2,\ldots,z_n$ be complex numbers. Is it true that $\displaystyle\sum_{1\le i,j\le n} |z_i+z_j| \ge \displaystyle\sum_{1\le i,j\le n} |z_i-z_j|$?

I know the inequality holds for reals and probably complex numbers... Unfortunately I do not have a proof of either scenario.

This seems like it would fall to some kind of induction and triangle inequalities but I have tried without too much success.

A proof of the inequality for $z_i\in \mathbb R$ or $z_i\in \mathbb C$ would be appreciated.

Note: The sigma includes when $i<j,i=j, i>j$ so please keep this in mind.

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  • $\begingroup$ Note that $\displaystyle\sum_{i,j}$ includes when $i=j$ so it still holds :) $\endgroup$ – James Li Feb 15 '16 at 16:58
  • $\begingroup$ I need more caffeine. Deleting my embarrassing comments... $\endgroup$ – copper.hat Feb 15 '16 at 17:08

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