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|f(z)|=1 if |z|=1, find f if holomorphic

Source: VJIMC 1997 2.2

October 7, 2021
complex analysiscalculus

Problem Statement

Let f:C→Cf:\mathbb C\to\mathbb C be a holomorphic function with the property that ∣f(z)∣=1|f(z)|=1 for all z∈Cz\in\mathbb C such that ∣z∣=1|z|=1. Prove that there exists a θ∈R\theta\in\mathbb R and a k∈{0,1,2,…}k\in\{0,1,2,\ldots\} such that f(z)=eiθzkf(z)=e^{i\theta}z^kfor all z∈Cz\in\mathbb C.
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