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

Daniel A. Nicks; Philip J. Rippon; Gwyneth M. Stallard

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arXiv · 2102.02158

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

Daniel A. Nicks·Philip J. Rippon·Gwyneth M. Stallard

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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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Daniel A. Nicks, Philip J. Rippon, Gwyneth M. Stallard. 2021-02-03. Iterating the minimum modulus: functions of order half, minimal type. https://arxiv.org/abs/2102.02158

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