On holomorphic functions with cluster sets of finite linear measure

TL;DR

This paper proves that for holomorphic functions on the unit disc with cluster sets of finite linear measure and finitely many components in their complement, the derivative belongs to the Hardy space H^1.

Contribution

It establishes a new connection between the geometric properties of cluster sets and the Hardy space membership of derivatives of holomorphic functions.

Findings

Cluster set C(f) has finite linear measure.

Complement of C(f) has finitely many components.

Derivative of f belongs to Hardy space H^1.

Abstract

We prove that if f is a holomorphic function on the open unit disc in C whose cluster set C(f) has finite linear measure and is such that the complement of C(f) has finitely many components, then the derivative of f belongs to the Hardy space H^1.