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

TL;DR

This paper investigates the growth properties of transcendental entire functions of order 1/2 minimal type, establishing conditions under which their minimum modulus iterates tend to infinity, and provides examples demonstrating the sharpness of these conditions.

Contribution

It proves that functions of order 1/2 minimal type with sufficiently regular maximum modulus growth exhibit the iterated minimum modulus property, and constructs examples to show the results are optimal.

Findings

The iterated minimum modulus property holds under certain growth conditions.

Regular growth of maximum modulus is crucial for the property.

Examples confirm the sharpness of the theoretical results.

Abstract

For a transcendental entire function $f$, the property that there exists $r>0$ such that $m^n(r)\to\infty$ as $n\to\infty$, 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 $m(r)$.