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Complex inequalities

Source: 2021 Taiwan TST Round 2 Independent Study 1-A

July 22, 2021
complex numbersinequalitiesTaiwan

Problem Statement

Prove that if non-zero complex numbers α1,α2,α3\alpha_1,\alpha_2,\alpha_3 are distinct and noncollinear on the plane, and satisfy α1+α2+α3=0\alpha_1+\alpha_2+\alpha_3=0, then there holds ∑i=13(∣αi+1−αi+2∣∣αi∣(1∣αi+1∣+1∣αi+2∣−2∣αi∣))≤0......(∗)\sum_{i=1}^{3}\left(\frac{|\alpha_{i+1}-\alpha_{i+2}|}{\sqrt{|\alpha_i|}}\left(\frac{1}{\sqrt{|\alpha_{i+1}|}}+\frac{1}{\sqrt{|\alpha_{i+2}|}}-\frac{2}{\sqrt{|\alpha_{i}|}}\right)\right)\leq 0......(*) where α4=α1,α5=α2\alpha_4=\alpha_1, \alpha_5=\alpha_2. Verify further the sufficient and necessary condition for the equality holding in (∗)(*).
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