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Geometric inequality with complex numbers

Source: CWMI 2015 Q7

August 20, 2015
inequalitiesgeometric inequalitycomplex numbers

Problem Statement

Let a∈(0,1)a\in (0,1), f(z)=z2−z+a,z∈Cf(z)=z^2-z+a, z\in \mathbb{C}. Prove the following statement holds: For any complex number z with ∣z∣≥1|z| \geq 1, there exists a complex number z0z_0 with ∣z0∣=1|z_0|=1, such that ∣f(z0)∣≤∣f(z)∣|f(z_0)| \leq |f(z)|.
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