Regularity and growth conditions for fast escaping points of entire functions

TL;DR

This paper introduces a family of sets $Q_m(f)$ for transcendental entire functions and establishes conditions under which these sets coincide with the fast escaping set $A(f)$, especially for functions of finite order.

Contribution

It generalizes existing sets related to escaping points and provides new regularity and growth conditions ensuring their equality with $A(f)$.

Findings

All functions of finite order and positive lower order satisfy $Q_m(f)=A(f)$ for any $m$.

A regularity condition implies $Q_m(f)=A(f)$, linking growth and regularity to escaping set equality.

The new conditions relate to recent criteria for $Q_2(f)=A(f)$, unifying previous results.

Abstract

Let $f$ be a transcendental entire function. The quite fast escaping set, $Q(f)$, and the set $Q_2(f),$ which was defined recently, are equal to the fast escaping set, $A(f),$ under certain conditions. In this paper we generalise these sets by introducing a family of sets $Q_m(f)$, $m \in \mathbb{N}.$ We also give one regularity and one growth condition which imply that $Q_m(f)$ is equal to $A(f)$ and we show that all functions of finite order and positive lower order satisfy $Q_m(f)=A(f)$ for any $m$. Finally, we relate the new regularity condition to a sufficient condition for $Q_2(f)=A(f)$ introduced in recent work.