RO-NMO 2005 Problem 121

Prove that the group morphisms $f: (\mathbb{C},+)\to(\mathbb{C},+)$ for which there exists a positive $\lambda$ such that $|f(z)| \leq \lambda |z|$ for all $z\in\mathbb{C}$ , have the form $$f(z) = \alpha z + \beta \overline{z}$$ for some complex $\alpha$ , $\beta$ .

Cristinel Mortici