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

Nicks, Daniel; Rippon, Philip and Stallard, Gwyneth (2021). Iterating the minimum modulus: functions of order half, minimal type. Computational Methods and Function Theory (Early Access).

DOI: https://doi.org/10.1007/s40315-021-00400-w

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 $m(r)$.

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