Fast escaping points of entire functions: a new regularity condition

TL;DR

This paper introduces a new regularity condition in transcendental dynamics, showing that for certain entire functions, the newly defined set $Q_2(f)$ coincides with the fast escaping set $A(f)$, and explores their relationships.

Contribution

It defines $Q_2(f)$ based on a weaker condition and proves its equality with $A(f)$ for functions of finite order, positive lower order, and their finite compositions.

Findings

$Q_2(f)$ equals $A(f)$ for functions of finite order and positive lower order

Finite compositions of such functions also satisfy $Q_2(f)=A(f)$

Constructs a function where $Q_2(f) eq Q(f)=A(f)$

Abstract

Let $f$ be a transcendental entire function. The fast escaping set, $A(f)$, plays a key role in transcendental dynamics. The quite fast escaping set, $Q(f)$, defined by an apparently weaker condition is equal to $A(f)$ under certain conditions. Here we introduce $Q_2(f)$ defined by what appears to be an even weaker condition. Using a new regularity condition we show that functions of finite order and positive lower order satisfy $Q_2(f)=A(f)$. We also show that the finite composition of such functions satisfies $Q_2(f)=A(f)$. Finally, we construct a function for which $Q_2(f) \neq Q(f)= A(f)$.