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continous morphism from (C,+) to (C,+)

Source: Romanian Nationals RMO 2005 - grade 12, problem 1

March 31, 2005
superior algebrasuperior algebra solved

Problem Statement

Prove that the group morphisms f:(C,+)→(C,+)f: (\mathbb{C},+)\to(\mathbb{C},+) for which there exists a positive λ\lambda such that ∣f(z)∣≤λ∣z∣|f(z)| \leq \lambda |z| for all z∈Cz\in\mathbb{C}, have the form f(z)=αz+βz‾ f(z) = \alpha z + \beta \overline{z} for some complex α\alpha, β\beta. Cristinel Mortici
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