On the radial distribution of Julia sets of entire solutions of \(f^{(n)} + A(z)f = 0\)
type: Item
instance of: scholarly article
title: On the radial distribution of Julia sets of entire solutions of \(f^{(n)} + A(z)f = 0\)
zbMATH Open document ID: 1238.37012
author: Jun Wang
author: Zhigang Huang
published in: Journal of Mathematical Analysis and Applications
publication date: 2012-01-04T00:00:00Z
review text: For a transcendental entire (or meromorphic) function \(f\) denote by \(J(f)\) the Julia set of \(f\). The set \(\{ z\in\mathbb{C}: \arg z=\theta\}\), \(\theta\in[0,2\pi)\), is said to be a radial distribution of \(f\) if \(\Omega(\theta-\varepsilon,\theta+\varepsilon)\cap J(f)\) is unbounded for every \(\varepsilon>0\), where, for \(0\leq\alpha <\beta<2\pi\), the set \(\Omega(\alpha,\beta)\) denotes the open sector bounded by the rays \(\arg z=\alpha\) and \(\arg z=\beta\). Denote by \(\Delta(f)\) the set of all such angles for \(f\). The study of the linear measure \(\operatorname{mes} \Delta(f)\) of \(\Delta(f)\) has a long history. \textit{I. N. Baker} [J. Lond. Math. Soc. 40, 499--502 (1965; Zbl 0136.37804)] constructed an entire function with infinite lower order for which the Julia set is contained in a horizontal strip. On the other hand, it is proved in [\textit{J. Qiao}, Acta Math. Sin. 37, No. 5, 702--708 (1994; Zbl 0814.30019)] that for \(f\) transcendental entire with finite lower order \(\mu(f)\), \(\operatorname{mes}\Delta(f) = 2\pi\) if \(\mu(f)\) is less than \(1/2\) and \(\operatorname{mes}\Delta(f) \geq \pi/\mu(f)\) otherwise. Hence it is interesting to study \(\operatorname{mes}\Delta(f)\) for functions with infinite lower order. In the article under review, the authors restrict to the solutions of the differential equation \[ f^{(n)}(z) + A(z) f(z) = 0, \] where \(n\geq 2\) and \(A\) is transcendental entire. It is not hard to see that the nontrivial solutions of this equation are transcendental entire functions of infinite lower order. As the main result, it is proved that if \(E\) is a finite product of solutions to the above equation, then \[ \operatorname{mes} \Delta(E)\geq \min\left\{2\pi, \frac{\pi}{\sigma(A)}\right\}, \] where \(\sigma\) denotes the usual order of growth. If \(\operatorname{mes}\Delta(E)=2\pi\), this result implies that \(E\) has no unbounded Fatou components.
Mathematics Subject Classification ID: 37F10
Mathematics Subject Classification ID: 37F75
Mathematics Subject Classification ID: 30D30
Mathematics Subject Classification ID: 34M05
zbMATH DE Number: 5994366
zbMATH Keywords: radial distribution
zbMATH Keywords: entire function
zbMATH Keywords: Julia set
zbMATH Keywords: differential equation
reviewed by: Helena Mihaljević-Brandt
MaRDI profile type: Publication
full work available at URL: j.jmaa.2011.10.016
OpenAlex ID: W1204977504
Wikidata QID: Q122663862
cites work: Unbounded domains of normality
cites work: Sets of Non-Normality in Iteration Theory
cites work: The domains of normality of an entire function
cites work: Q3929977
cites work: ON A THEOREM OF BESICOVITCH
cites work: Iteration of meromorphic functions
cites work: Meromorphic Functions with Three Radially Distributed Values
cites work: On the Growth of Meromorphic Functions With Several Deficient Values
cites work: Q3522266
cites work: Q5723337
cites work: Growth and Values of Functions Regular in an Angle
cites work: An estimate of the modulus of the logarithmic derivative of a function which is meromorphic in an angular region, and its application
cites work: Q4320228
cites work: Radial distributions of Julia sets of meromorphic functions
cites work: Q4284396
cites work: Bounded domains of the Fatou set of an entire function
cites work: On the Location of Zeros of Solutions of $f'' + A(z)f = 0$ where $A(z)$ is Entire.
cites work: Q4827955
cites work: Some properties of Fatou and Julia sets of transcendental meromorphic functions
cites work: Unbounded domains of normality of entire functions of small growth
DOI: 10.1016/J.JMAA.2011.10.016
description: scientific article; zbMATH DE number 5994366