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 $
