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Let’s discuss a problem based on Least Value of a Sum of Complex Numbers. Try to solve it yourself before reading the solution.

Problem: 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

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