The growth of entire and harmonic functions along asymptotic paths
type: Item
instance of: scholarly article
title: The growth of entire and harmonic functions along asymptotic paths
zbMATH Open document ID: 0569.30021
DOI: 10.1007/BF02567395
author: John Rossi
author: Allen Weitsman
published in: Commentarii Mathematici Helvetici
publication date: 1985-01-01T00:00:00Z
full work available at URL: https://eudml.org/doc/139996
review text: The authors continue work of \textit{J. Lewis} and the authors, Ark. Mat. 22, 109-119 (1984; Zbl 0547.31003) and the first author, J. Lond. Math. Soc., II. Ser. 30, 73-78 (1984; Zbl 0554.31001). Theorem 1. let f be an entire function such that for some \(K>0\), at least one of the level curves \(| f| =K\) tends to \(\infty\). Then there exists a path \(\Gamma\) from 0 to \(\infty\) such that \[ (*)\quad \log | f(z)| >| z|^{-o(1)} \] (z\(\to \infty\); \(z\in \Gamma)\), and \[ (**)\quad \int_{\Gamma}(\log | f|)^{-2-\lambda}| dz| <\infty \quad for\quad all\quad \lambda >0. \] Corollary 1. If u is a nonconstant harmonic function in \({\mathbb{C}}\) then there exists a curve \(\Gamma\) form 0 to \(\infty\) such that (*) and (**) hold with log \(| f|\) replaced by u. Theorem 2. Let f be an entire function of order \(\rho\leq \infty\) such that for some \(K>0\) the set \(\{| f| >K\}\) contains at least two components. Then there exists a path \(\Gamma\) from 0 to \(\infty\) such that log \(| f(z)| >| z|^{-(\rho /(2\rho -1))-o(1)}\) as \(z\to \infty\), \(z\in \Gamma\) and \[ \int_{\Gamma}(\log | f|)^{-((2\rho -1)/\rho)+\lambda}| dz| <\infty \quad for\quad all\quad \lambda >0. \] For some refinements of these results and further work of this type see \textit{J. M. Wu}, Length of paths for subharmonic functions (to appear).
Mathematics Subject Classification ID: 30C85
zbMATH DE Number: 3908787
reviewed by: David Drasin
MaRDI profile type: Publication
description: scientific article; zbMATH DE number 3908787