RO-NMO 2010 Problem 102

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

Dan Marinescu