Generalized Schur Functions as Multivalent Functions

The multivalency approach to generalized Nevanlinna functions established in Wietsma (Indag Math 29:997–1008, 2018) is here extended to the related class of generalized Schur functions giving thereby rise to new characterizations for this class of functions as well as a straightforward function-theoretical proof of its factorization. In particular, this multivalency approach explains how the well-known factorizations of the two mentioned classes of functions differ from each other. Indeed, by this approach a new factorization of generalized Schur functions is obtained which is more directly connected to the factorization of generalized Nevanlinna functions. These results demonstrate that multivalency is a valuable concept for the complete understanding of the mentioned classes of functions.