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

Abstract

Abstract We consider the class of real transcendental entire functions $f$ of finite order with only real zeros and show that if the iterated minimum modulus tends to $\infty $, then the escaping set $I(\,f)$ of $f$ has the structure of a spider’s web, in which case Eremenko’s conjecture holds. This minimum modulus condition is much weaker than that used in previous work on Eremenko’s conjecture. For functions in this class, we analyse the possible behaviours of the iterated minimum modulus in relation to the order of the function $f$.