Question 4
Let $\alpha_1,\alpha_2,\alpha_3$ be distinct nonzero complex numbers whose points in the complex plane are noncollinear, and suppose $\alpha_1+\alpha_2+\alpha_3=0$. Prove that
$$
\sum_{i=1}^{3}\frac{|\alpha_{i+1}-\alpha_{i+2}|}{\sqrt{|\alpha_i|}}
\left(\frac1{\sqrt{|\alpha_{i+1}|}}+\frac1{\sqrt{|\alpha_{i+2}|}}-\frac2{\sqrt{|\alpha_i|}}\right)\le0.\tag{*}
$$
Here $\alpha_4=\alpha_1$ and $\alpha_5=\alpha_2$. Determine the necessary and sufficient condition for equality.
$$
\sum_{i=1}^{3}\frac{|\alpha_{i+1}-\alpha_{i+2}|}{\sqrt{|\alpha_i|}}
\left(\frac1{\sqrt{|\alpha_{i+1}|}}+\frac1{\sqrt{|\alpha_{i+2}|}}-\frac2{\sqrt{|\alpha_i|}}\right)\le0.\tag{*}
$$
Here $\alpha_4=\alpha_1$ and $\alpha_5=\alpha_2$. Determine the necessary and sufficient condition for equality.