Mean type of functions of bounded characteristic and Martin functions in Denjoy domains

TL;DR

This paper extends classical factorization and mean type concepts from simply connected domains to infinitely connected Denjoy domains, highlighting the role of Martin functions and the complexities introduced by multiple-valued functions.

Contribution

It introduces a new approach to measure mean type in Denjoy domains using Martin functions, accounting for multiple-valued functions and differences from universal coverings.

Findings

Factorization in Denjoy domains involves multiple-valued functions.

Martin functions can measure mean type in complex domains.

Mean types differ between original and lifted functions in universal coverings.

Abstract

Functions of bounded characteristic in simply connected domains have a classical factorization to Blaschke, outer and singular inner parts. The latter has a singular measure on the boundary assigned to it. The exponential speed of change of a function when approaching a point of a boundary (mean type) corresponds to a point mass at this point. In this paper we consider the analogous relation for functions in arbitrary infinitely connected (Denjoy) domains. The factorization result holds of course with one important addition: all functions involved become multiple valued even though the initial function was single valued. The mean type now can be measured by using the Martin function of the domain. But this result does not follow from the lifting to the universal covering of the domain because of the simple (but interesting) reason that the mean types of the original and the lifted functions can be completely different.