On the growth of entire and meromorphic functions of infinite order
type: Item
instance of: scholarly article
title: On the growth of entire and meromorphic functions of infinite order
zbMATH Open document ID: 0722.30016
DOI: 10.1007/BF02789202
author: Chong Ji Dai
author: David Drasin
author: Bao Qin Li
published in: Journal d'Analyse Mathématique
publication date: 1990-01-01T00:00:00Z
review text: Let \(\phi\) (x) be a positive increasing function satisfying \(\int^{\infty}dx/\phi (x)<\infty\). It is shown that if f is a meromorphic function, then there exists a set E of logarithmic density 1 such that \[ \lim_{r\to \infty,r\in E}\frac{\log M(r)}{T(r)\phi (\log T(r))\log (\phi (\log T(r)))}=0. \] Here M(r) and T(r) denote the maximum modulus and Nevanlinna characteristic of f. If f is entire, then \[ \lim_{r\to \infty,r\in E}\frac{\log M(r)}{T(r)\phi (\log T(r))}=0. \] The proof depends on the Poisson-Jensen formula, a growth lemma for real functions, and - in the case that f is meromorphic - Cartan's lemma. Next it is shown that the condition that \(\int^{\infty}dx/\phi (x)<\infty\) cannot be weakened. To construct examples which show this, the authors first use conformal mappings of strips to construct a function which is harmonic in the plane except on certain curves where it is subharmonic or superharmonic. Then they approximate this function by log\(| f|\) where f is meromorphic and has zeros (respectively poles) on these curves. The estimation of the asymptotic behavior of the conformal mapping of the strip seems to require some regularity hypothesis on \(\phi\) (x) which is not stated in the paper, but will be given in a forthcoming correction (same journal, to appear).
reviewed by: Walter Bergweiler
Mathematics Subject Classification ID: 30D35
Mathematics Subject Classification ID: 30D15
Mathematics Subject Classification ID: 30D20
Mathematics Subject Classification ID: 30D30
zbMATH DE Number: 4190097
zbMATH Keywords: infinite order
zbMATH Keywords: maximum modulus
zbMATH Keywords: Nevanlinna characteristic
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 4190097