Regularity and fast escaping points of entire functions

TL;DR

This paper explores the relationships between regularity conditions, the fast escaping set, and the quite fast escaping set of transcendental entire functions, establishing new equivalences and examples in complex dynamics.

Contribution

It proves a weak regularity condition is necessary and sufficient for the equality of Q(f) and A(f), and shows log-regularity holds for many functions including those in class B.

Findings

Q(f)=A(f) if and only if a weak regularity condition holds

Examples where Q(f) differs from A(f) are provided

Log-regularity applies to a broad class of functions, including class B

Abstract

Let $f$ be a transcendental entire function. The fast escaping set $A(f)$, various regularity conditions on the growth of the maximum modulus of $f$, and also, more recently, the quite fast escaping set $Q(f)$ have all been used to make progress on fundamental questions concerning the iteration of $f$. In this paper we establish new relationships between these three concepts. We prove that a certain weak regularity condition is necessary and sufficient for $Q(f)=A(f)$ and give examples of functions for which $Q(f)\neq A(f)$. We also apply a result of Beurling that relates the size of the minimum modulus of $f$ to the growth of its maximum modulus in order to establish that a stronger regularity condition called log-regularity holds for a large class of functions, in particular for functions in the Eremenko-Lyubich class ${\mathcal B}$.