The Role of Nevanlinna and Carleson Functions in Analytic Maps

Abstract

We demonstrate that the maximal Nevanlinna counting function and the Carleson function for analytic self-maps of the unit disk are comparable, up to constant factors. This equivalence establishes a fundamental connection between these two classical concepts in complex analysis, providing insights into composition operators and their compactness criteria. The results unify previous findings and highlight the role of Carleson measures and Nevanlinna counting functions in the study of analytic maps