Composition of entire functions with finite iterated order
type: Item
instance of: scholarly article
title: Composition of entire functions with finite iterated order
zbMATH Open document ID: 1169.30014
author: Jin Tu
author: Zongxuan Chen
author: Xiu-Min Zheng
published in: Journal of Mathematical Analysis and Applications
publication date: 2009-04-29T00:00:00Z
review text: Let \(f\) and \(g\) be two entire functions. The authors investigate the growth of the composite function \(f(g)\) with iterated order. For sufficiently large \(r\), define \(\log_1r=\log r\) and \(\log_{n+1}r=\log(\log_n r)\), \(n\in\mathbb{N}\). The iterated order and the iterated lower order are defined by \(\rho_n(f)=\lim\sup_{r\to \infty}\log_{n+1}M(r,f)/\log r\) and \(\mu_n(f)=\lim\inf_{r\to\infty}\log_{n+1} M(r,f)/\log r\), respectively. The finiteness degree \(i(f)\) of the order is defined by \(i(f)=0\) if \(f\) is a polynomial, \(i(f)=\min\{n\in\mathbb{N}:\rho_n(f)< \infty\}\) if \(f\) is transcendental with \(\rho_n(f)< \infty\) for some \(n\in\mathbb{N}\), \(i(f)=\infty\) if \(\rho_n(f)=\infty\) for all \(\in \mathbb{N}\). See e.g., [\textit{L. Kinnunen}, Southeast Asian Bull. Math. 22, No. 4, 385--405 (1998; Zbl 0934.34076)]. One of the main results is the follwing: If \(f\) and \(g\) are of finite iterated order with \(i(f)=p\), \(i(g)=q\) and \(\mu_p(f)>0\), then \(i(f(g))=p+q\) and \(\rho_{p+g} (f(g))=\rho_q(g)\). The authors give an application to the factorization problem of the solutions of linear differential equations.
reviewed by: Katsuya Ishizaki
Mathematics Subject Classification ID: 30D35
Mathematics Subject Classification ID: 30D20
Mathematics Subject Classification ID: 34M10
zbMATH DE Number: 5549428
zbMATH Keywords: finite iterated order
zbMATH Keywords: composition of entire functions
zbMATH Keywords: differential equation
MaRDI profile type: Publication
full work available at URL: j.jmaa.2008.12.007
OpenAlex ID: W2088395838
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DOI: 10.1016/J.JMAA.2008.12.007
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description: scientific article; zbMATH DE number 5549428