Entire functions of small order of growth
type: Item
instance of: scholarly article
title: Entire functions of small order of growth
zbMATH Open document ID: 1236.30026
DOI: 10.1007/BF03321804
author: Katsuya Ishizaki
author: Niro Yanagihara
published in: Computational Methods and Function Theory
publication date: 2011-10-28T00:00:00Z
review text: The authors study the growth of composite entire functions. They prove that if \(f\) is a transcendental entire function and \(F\) is an entire function satisfying \[ \log M(r,F)=K(\log r)^p(1+o(1)),\tag{\(*\)} \] then, \[ \log M(r,F(f))=K\left(\log M(r,f)\right)^p(1+o(1)). \] Then, the authors apply this to the functional equation \[ f(sz)=F(f(z)), s\in \mathbb{C}, |s|>1, \] where \(F\) satisfies the equation (\(*\)) and prove that if \(f\) is a solution of the above functional equation, then \[ \log\log M(r,f)=A(r)r^{\rho}, \rho=\frac{\log p}{\log|s|}, \] where \(K_1<A(r)<K_2\) for some constants \(K_1\) and \(K_2\).
Mathematics Subject Classification ID: 30D05
Mathematics Subject Classification ID: 30D35
Mathematics Subject Classification ID: 37F10
zbMATH DE Number: 5965042
zbMATH Keywords: entire functions
zbMATH Keywords: Valiron-Mohon'ko's theorem
zbMATH Keywords: maximum modulus
zbMATH Keywords: Wiman-Valiron theory
zbMATH Keywords: composite function
zbMATH Keywords: functional equations
reviewed by: Liangwen Liao
MaRDI profile type: Publication
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full work available at URL: https://doi.org/10.1007/bf03321804
OpenAlex ID: W2087554996
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description: scientific article; zbMATH DE number 5965042