On a certain generalization of the Carathéeodory-Julia-Wolff theorem

Abstract

Given an analytic self-mapping $s$ of the open unit disk $\mathbb D$ and given a Blaschke product $b$ of degree $k$, we present necessary and sufficient conditions for $s-b$ to have exactly $k$ zeros inside $\mathbb D$. As a corollary, we obtain a Carathéeodory-Julia-Wolff type theorem for meromorphic functions of the form $s/b$.