The structure of the semigroup of proper holomorphic mappings of a planar domain to the unit disc (Q934543)

The authors show that there are two Ahlfors maps \(f_1\) and \(f_2\) associated with the domain such that any such mapping is given by a fixed linear fractional transformation mapping of the right half plane to the unit disc composed with \(cR+ iC\), where \(R\) is a rational function of the \((2n+2)\) functions \(f_1(z)\), \(f_2(z)\) and \(f_1(b_1),f)2(b_1),\dots, f_1(b_n)\), \(f_2(b_n)\) and \(c\) and \(C\) are arbitrary real constants subject to the condition \(c> 0\). The proof of the above statement modifies a proof by H. Grunsky of Bieberbach's result. The authors also show how all proper holomorphic mappings to the unit disc are generated via the rational function \(R\).