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Inequality iff w=kv

Source: Romanian MO 2010 Grade 10

August 6, 2012
inequalitiesalgebra unsolvedalgebra

Problem Statement

Consider v,wv,w two distinct non-zero complex numbers. Prove that ∣zw+wˉ∣≤∣zv+vˉ∣,|zw+\bar{w}|\le |zv+\bar{v}|, for any z∈C,∣z∣=1z\in\mathbb{C},|z|=1, if and only if there exists k∈[−1,1]k\in [-1,1] such that w=kvw=kv.
Dan Marinescu
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