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# Least Value of a Sum of Complex Numbers

If $$z_1 , z_2 , z_3 , z_4 \in \mathbb{C}$$ satisfy $$z_1 + z_2 + z_3 + z_4 = 0$$ and $$|z_1|^2 + |z_2|^2 + |z_3|^2 + |z_4|^2 = 1$$ then the least value of $$|z_1 – z_2 |^2 + |z_1 – z_4|^2 + |z_2 – z_3|^2 + |z_3 – z_4|^2$$ is 2

True

Discussion:

$$|z_1 – z_2|^2 = (z_1 – z_2)(\bar{z_1} – \bar{z_2}) = |z_1|^2 + |z_2|^2 – (z_1 \bar {z_2} + \bar {z_1} {z_2} )$$

Similarly we compute the others to get the total sum as

$$2 ( |z_1|^2 + |z_2|^2 + |z_3|^2 + |z_4|^2 )$$ –  $$(z_1 + z_3) (\bar{z_2} + \bar {z_4} )$$

– $$( \bar {z_1} + \bar {z_3} ) (z_2 + z_4 )$$

Since $$z_1 + z_3 = – (z_2 + z_4)$$ thus $$\bar {z_2 } + \bar {z_4} = – ( \bar {z_1} + \bar {z_3} )$$ the above expression reduces to

$$2 ( |z_1|^2 + |z_2|^2 + |z_3|^2 + |z_4|^2 ) + 2 |z_1 + z_3|^2 \ge 2$$

November 14, 2013