Functions of small growth with no unbounded Fatou components

TL;DR

This paper establishes new criteria for transcendental entire functions of order zero to lack unbounded Fatou components, extending previous results and providing examples and conditions related to the structure of their Fatou and escaping sets.

Contribution

It introduces a generalized $ ext{cos} \\pi ho$ theorem for order zero functions and new sufficient conditions for the absence of unbounded Fatou components, expanding understanding of their dynamics.

Findings

A strong estimate for the minimum modulus of order zero functions.

A generalized condition ensuring no unbounded Fatou components.

Connection between conditions for Fatou components and the connectedness of the escaping set.

Abstract

We prove a form of the $\cos \pi \rho$ theorem which gives strong estimates for the minimum modulus of a transcendental entire function of order zero. We also prove a generalisation of a result of Hinkkanen that gives a sufficient condition for a transcendental entire function to have no unbounded Fatou components. These two results enable us to show that there is a large class of entire functions of order zero which have no unbounded Fatou components. On the other hand we give examples which show that there are in fact functions of order zero which not only fail to satisfy Hinkkanen's condition but also fail to satisfy our more general condition. We also give a new regularity condition that is sufficient to ensure that a transcendental entire function of order less than 1/2 has no unbounded Fatou components. Finally, we observe that all the conditions given here which guarantee that a transcendental entire function has no unbounded Fatou components, also guarantee that the escaping set is connected, thus answering a question of Eremenko for such functions.