On the growth of meromorphic functions of infinite order
type: Item
instance of: scholarly article
title: On the growth of meromorphic functions of infinite order
zbMATH Open document ID: 0828.30013
DOI: 10.1007/BF03008414
author: Heinrich Bock
author: Walter Bergweiler
published in: Journal d'Analyse Mathématique
publication date: 1995-12-18T00:00:00Z
review text: For a meromorphic function \(f\) let \(T(r,f)\) denote its Nevanlinna (or Ahlfors-Shimizu) characteristic and let \(M(r,f)\) denote its maximum modulus. In this paper, it is proved that if \(\gamma\) is an increasing and differentiable function with \(T(r,f)\leq \gamma(r)\) for large \(r\), then \[ \liminf_{r\to \infty} {{\log M(r,f)} \over {r\gamma' (r)}} \leq \pi \] and that if \(\psi\) is a positive and continuously differentiable function such that \(\psi(x) /x\) is nondecreasing, \(\psi' (x)\leq \sqrt {\psi (x)}\), and \(\int_{x_0}^\infty dx/ \psi(x)< \infty\), then \[ \liminf_{r\to \infty} {{\log M(r,f)} \over {T(r,f) \psi (\log T(r,f))}} =0. \] The proofs of these results are based on the method of Petrenko which was modified by \textit{W. H. J. Fuchs} [Topics in Nevanlinna theory, Washington D.C., Proc. NRL Conf. Classical function theory, 1-32 (1970; Zbl 0294.30021)]. The proof in Fuchs' paper uses Pólya-peaks which exist in general only for meromorphic functions of finite lower order. For the proofs of the results in this paper the Pólya-peaks are replaced are replaced by a suitable other sequence of \(r\)-values which can be considered as Pólya-peaks of infinite order. \textit{C. J. Dai}, \textit{D. Drasin} and \textit{B. Q. Li} [J. Anal. Math. 55, 217-228 (1990; Zbl 0722.30016); Correction: J. Anal. Math. 57, 299-300 (1991; Zbl 0767.30027)] have shown that \[ \lim_{r\to \infty} {{\log M(r,f)} \over {T(r,f) \varphi (\log T(r,f)) \log \varphi (\log T(r,f))}} =0 \] on a set of logarithmic density 1, where \(\varphi\) is an increasing, positive function with \(\int_{x_0}^\infty {{dx} \over {\varphi (x)}} <\infty\). For a meromorphic function \(f\) and a complex number \(a\) let \(b(a,f):= \liminf_{r\to \infty} {{\log M(r, 1/(f- a))} \over {rT' (r,f)}}\), \(b(\infty, f):= \liminf_{r\to \infty} {{\log M(r,f)} \over {rT' (r,f)}}\). The first result of this paper shows that \(b(\infty, f)\leq \pi\) for functions of infinite order. Recently, A. Eremenko has prove…
Mathematics Subject Classification ID: 30D30
Mathematics Subject Classification ID: 30D35
zbMATH DE Number: 718525
zbMATH Keywords: growth properties
zbMATH Keywords: maximum modulus
zbMATH Keywords: Nevanlinna characteristic
reviewed by: G. Jank
MaRDI profile type: Publication
cites work: Maximum Modulus, Characteristic, and Area on the Sphere
cites work: Q3961090
cites work: On the growth of entire and meromorphic functions of infinite order
cites work: Q4046348
cites work: Q5593933
cites work: Q5723337
cites work: Q3995409
cites work: Growth of entire functions
cites work: Q5597066
cites work: GROWTH OF MEROMORPHIC FUNCTIONS OF FINITE LOWER ORDER
Recommended article: Q4696486
Recommended article: Q4512596
Recommended article: Q4518676
Recommended article: Q4241166
Recommended article: Q4133898
Recommended article: Growth of meromorphic functions
Recommended article: On the growth of entire and meromorphic functions of infinite order
Recommended article: Q4009568
Recommended article: Q4283812
Recommended article: Anwenclungen einer methode von goldberg und fuchs in der theorie der ganzen und meromorphen funktionen
description: scientific article; zbMATH DE number 718525