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

Abstract

For a transcendental entire function f , the property that there exists r > 0 such that m n (r) → ∞ as n → ∞, where m(r) = min{|f (z)| : |z| = r}, is related to conjectures of Eremenko and of Baker, for both of which order 1/2 minimal type is a significant rate of growth. We show that this property holds for functions of order 1/2 minimal type if the maximum modulus of f has sufficiently regular growth and we give examples to show the sharpness of our results by using a recent generalisation of Kjellberg's method of constructing entire functions of small growth, which allows rather precise control of m(r).

On Subharmonic and Entire Functions of Small Order: After Kjellberg

“…In this paper we use subharmonic functions to give a very general method for constructing examples of functions of given small order, including order 0, which allows precise control over the size and shape of the set where the minimum modulus of the function is relatively large. The original motivation for this work was related to further progress on Eremenko's conjecture, which we report on in forthcoming work [22], but our new results here are of independent interest. and we say that a function has small order if it has order less than 1/2.…”

“…In forthcoming work [22] we will use the results in this paper to construct entire functions of order 1/2, minimal type, with dynamically interesting properties related to their minimum modulus. Our next theorem shows that to construct such examples the key will be to choose a set E with a sufficiently large complement in the negative real axis.…”