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

2021/02/03 by Nicks, Daniel A., Rippon, Philip J., Stallard, Gwyneth M. · 1 citation
#Complex Variables (math.CV) #Dynamical Systems (math.DS) #FOS: Mathematics

paper · doi:10.48550/arxiv.2102.02158

Abstract

For a transcendental entire function f, the property that there exists r>0 such that mⁿ(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).

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• The escaping set in transcendental dynamics

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