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

Abstract

We show that for many families of transcendental entire functions f the property that m n (r) → ∞ as n → ∞, for some r > 0, where m(r) = min{|f (z)| : |z| = r}, implies that the escaping set I(f ) of f has the structure of a spider's web. In particular, in this situation I(f ) is connected, so Eremenko's conjecture holds. We also give new examples of families of functions for which this iterated minimum modulus condition holds and new families for which it does not hold.

Iterating the Minimum Modulus: Functions of Order Half, Minimal Type

“…respectively, and f is said to be real if f (z) = f (z), for z ∈ C. In view of Theorem 1.1, it is natural to ask which transcendental entire functions satisfy property (1.2); in particular, which amongst those that satisfy the other hypotheses of this theorem. In [8] we proved several results in relation to this question, including the following.…”

“…More recently, it has been observed that property (1.2), when it is true, can also play an important role in relation to this conjecture; see [8][9][10]. For example, in [8] we obtained the following result, which gives a family of transcendental entire functions for which Eremenko's conjecture holds in a particularly strong way. Theorem 1.1 Let f be a real transcendental entire function of finite order with only real zeros for which there exists r > 0 such that m n (r ) → ∞ as n → ∞.…”

“…It has been known for some time that the sequence M n (r ) is of importance in relation to work on Eremenko's conjecture, since it plays a key role in the definition of a subset of I ( f ) called the fast escaping set, all of whose components are unbounded; see, for example [12]. More recently, it has been observed that property (1.2), when it is true, can also play an important role in relation to this conjecture; see [8][9][10]. For example, in [8] we obtained the following result, which gives a family of transcendental entire functions for which Eremenko's conjecture holds in a particularly strong way.…”

Iterating the minimum modulus: functions of order half, minimal type

“…More recently, it has been observed that property (1.2), when it is true, can also play an important role in relation to this conjecture; see [9], [10] and [8]. For example, in [8] we obtained the following result, which gives a family of transcendental entire functions for which Eremenko's conjecture holds in a particularly strong way.…”

“…It has been known for some time that the sequence M n (r) is of importance in relation to work on Eremenko's conjecture, since it plays a key role in the definition of a subset of I(f ) called the fast escaping set, all of whose components are unbounded; see, for example, [12]. More recently, it has been observed that property (1.2), when it is true, can also play an important role in relation to this conjecture; see [9], [10] and [8]. For example, in [8] we obtained the following result, which gives a family of transcendental entire functions for which Eremenko's conjecture holds in a particularly strong way.…”