Hensley's problem for complex and non-Archimedean meromorphic functions
type: Item
instance of: scholarly article
title: Hensley's problem for complex and non-Archimedean meromorphic functions
zbMATH Open document ID: 1246.30076
author: Ta Thi Hoai An
author: Julie Tzu-Yueh Wang
published in: Journal of Mathematical Analysis and Applications
publication date: 2011-06-20T00:00:00Z
review text: The authors consider Hensley's \(n\)-th power problem for complex meromorphic functions and for non-Archimedean meromorphic functions. In the complex case they use classical Nevanlinna theory, and in the non-Archimedean case they use \(p\)-adic Nevanlinna theory. One of the results in the complex case is the following: Suppose that \(f\), \(g\) and \(x_1,\dots, x_M\) are meromorphic functions and at least one of the \(x_i\) is not constant. Let \(n\) be an integer no less than \(2\) and \((c_1, \dots, c_M)\) be a sequence of distinct constants in \(\mathbb C\). Suppose that \[ x_i^n =(f+c_i)^n-g \quad \text{for }1\leq i\leq M. \] Then \(g=0\) if (i) \(M> \frac{7n^4-n^3-4n+3}{(n-1)^2}\), or (ii) \(f\) and \(g\) are entire functions and \(M>\frac{4n^4-n^3-4n+3}{n-1}\). When \(n\geq 9\), the size of \(M\) can be much smaller. The authors also obtain the corresponding result in the non-Archimedean case.
reviewed by: Katsuya Ishizaki
Mathematics Subject Classification ID: 30G06
Mathematics Subject Classification ID: 11J97
Mathematics Subject Classification ID: 12Y05
Mathematics Subject Classification ID: 30D30
zbMATH DE Number: 5909847
zbMATH Keywords: Büchi's problem
zbMATH Keywords: Hensley's problem
zbMATH Keywords: meromorphic function
zbMATH Keywords: Nevanlinna theory
zbMATH Keywords: value distribution theory
zbMATH Keywords: non-Archimedean meromorphic functions
MaRDI profile type: Publication
full work available at URL: j.jmaa.2011.03.025
OpenAlex ID: W1973558958
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DOI: 10.1016/J.JMAA.2011.03.025
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description: scientific article; zbMATH DE number 5909847