Question 22
Consider $v, w$ two distinct non-zero complex numbers. Prove that
$$
|zw + \bar{w}| \leq |zv + \bar{v}|,
$$
for any $z \in \mathbb{C}$, $|z| = 1$, if and only if there exists $k \in [-1, 1]$ such that $w = kv$.
$$
|zw + \bar{w}| \leq |zv + \bar{v}|,
$$
for any $z \in \mathbb{C}$, $|z| = 1$, if and only if there exists $k \in [-1, 1]$ such that $w = kv$.