Regularity and Fast Escaping Points of Entire Functions

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 ) = 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 B.

Baker's conjecture for functions with real zeros

“…We remark that Beurling proved this estimate with the first constant on the right‐hand side equal to , but for simplicity we use the value here. We also remark that further applications of Lemma were given in .…”

Eremenko’s Conjecture for Functions with Real Zeros: The Role of the Minimum Modulus

“…In an earlier paper [23], we saw other families of functions of order less than 1/2 for which Q(f ) contains a spider's web. For many of these, Q(f ) = A(f ) (see [21]), but there are examples for which Q(f ) contains A(f ) strictly (see [22]).…”

Regularity and growth conditions for fast escaping points of entire functions

“…In this context, Rippon and Stallard introduced the quite fast escaping set Q(f ) in [17], which is defined as follows:…”

“…The function f in [17, Example 6.1] was found by first constructing a real, increasing, convex function φ with certain properties and then using a result of Clunie and Kövari from [7] (see [17,Lemma 6.3]) in order to obtain a transcendental entire function f such that ψ(t) = log M(e t ) ∼ φ(t). Let µ(t) = exp(t 1/2 ), t ≥ 0.…”