TITLE: Holomorphic functions with equal inverse images of unit circle
QUESTION [5 upvotes]: Let $f,g:\mathbb{C} \to \mathbb{C}$ be holomorphic and have the property $f^{-1}(S)=g^{-1}(S)$ where S is the unit circle centered at 0. What can be said about $f$ and $g$.

REPLY [5 votes]: An exhaustive study of this question is contained in the paper
MR0825840
Stephenson, Kenneth
Analytic functions sharing level curves and tracts. 
Ann. of Math. (2) 123 (1986), no. 1, 107–144 (freely available online).