On the valence structure of analytic functions (Q1191904)

The main result of the paper is: There exist functions \(f\) and \(g\) analytic on the closed unit disk \(\overline\mathbb{D}\) such that for every \(w\in\mathbb{C}\) \[ \#[f^{-1}(w)\cap\mathbb{D}]=\#[g^{-1}(w)\cap\mathbb{D}] \] but \(f\circ\psi\neq g\) for any homeomorphism \(\psi\) of the unit circle \(\mathbb{T}\). This nice theorem gives a negative answer to a question posed by K. Stephenson.