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Find all analytic function $f: \mathbb C \rightarrow \mathbb C$ such that $|f^`(z)|$ constant on curves of the form $Ref$ constant.

This is one of the past comp question. Seriously I do not know where to start. I do not even understand what the question is asking here. I have difficulty understanding what kind of curve has the form $Ref$ constant (example please). When I need to find entire function in other problem, I usually think of Liouville as a rescue but this time I don't think Louiville is going to save me. Any rigorous solution will be much appreciated.

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    One approach exploits the answers to the closely related question at http://math.stackexchange.com/questions/267514: consider the real part of the inverse of $f'$.2012-12-31

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