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.