Blaschke-type conditions for analytic functions in the unit disk: inverse problems and local analogs

TL;DR

This paper investigates conditions under which analytic functions in the unit disk with prescribed singular and zero sets exist, extending previous work and exploring local analogs of Blaschke conditions for finite order functions.

Contribution

It advances the understanding of inverse problems for finite order analytic functions with singular points on the unit circle, introducing local analogs of Blaschke conditions.

Findings

Established criteria for the existence of functions with given singular and zero sets.

Extended Blaschke-type conditions to local settings for finite order functions.

Provided new insights into inverse problems in complex analysis.

Abstract

We continue the study of analytic functions in the unit disk of finite order with arbitrary set of singular points on the unit circle, introduced in \cite{FG}. The main focus here is made upon the inverse problem: the existence of a function from this class with a given singular set and zero set subject to certain Blaschke-type condition. We also discuss the local analog of the main result from \cite{FG} similar to the standard local Blaschke condition for analytic and bounded functions in the unit disk.