An Injectivity Condition

★ 9/10complex numbers Absolute Value Estimation

Problem

Find all complex numbers $\alpha$ such that for any distinct $z_1, z_2$ with $|z_1|, |z_2| < 1$ ,

\left(z_1 + \alpha\right)^2 + \alpha\overline{z_1} \ne \left(z_2 + \alpha\right)^2 + \alpha\overline{z_2}.

Answer

Solution

Difficulty	9/10
Topics	complex numbers, Absolute Value, Estimation

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