On Bloch's "Principle of topological continuity''

TL;DR

This paper explores the conditions under which certain properties of entire and meromorphic functions hold when relaxing the assumption of total ramification, proving a new result related to Bloch's principle for functions of order less than one.

Contribution

It extends Bloch's principle by showing that entire functions of order less than one must have a simple island over at least one of two disjoint Jordan domains, under weaker conditions.

Findings

Entire functions of order less than 1 have a simple island over at least one of two disjoint Jordan domains.

Relaxing total ramification to multiple islands still preserves some classical results.

The paper provides a new perspective on Bloch's principle in complex analysis.

Abstract

We discuss to what extent certain results about totally ramified values of entire and meromorphic functions remain valid if one relaxes the hypothesis that some value is totally ramified by assuming only that all islands over some Jordan domain are multiple. In particular, we prove a result suggested by Bloch which says that an entire function of order less than $1$ has a simple island over at least one of two given Jordan domains with disjoint closures.