Permutability of entire functions satisfying certain differential equations
type: Item
instance of: scholarly article
title: Permutability of entire functions satisfying certain differential equations
zbMATH Open document ID: 0638.30024
author: Zhengzhong Zhou
author: Jian-Hua Zheng
published in: Tôhoku Mathematical Journal. Second Series
publication date: 1988-01-01T00:00:00Z
review text: Two entire functions f(z) and g(z) are said to be permutable if they satisfy the relation \[ f(g(z))=g(f(z)) \] in the z-plane. In this paper, we have investigated the permutability of entire functions combining the differential equations, i.e. under what condition, two entire functions are permutable, and obtained the result that under the conditions \(0<\lambda (f)\leq \rho (f)<+\infty\) and \(\rho (g)<+\infty\) (\(\lambda\) (f) and \(\rho\) (f) denote the lower order and order of f(z)), if f(z) satisfies a linear differential equation with coefficients of polynomials, then so does g(z). Then with the help of the above result, we completely determined the entire functions permutable with the entire functions like the two forms: \[ f(z)=Q+He\quad P\quad and\quad f(z)=\sin P, \] where Q, H (\(\not\equiv 0)\) and P (\(\not\equiv\) constant) are all polynomials. The former form is a generalization of \textit{T. Kobayashi}'s theorem 1 [see Kodai Math. J. 3, 8-25 (1980; Zbl 0433.30023)].
Mathematics Subject Classification ID: 30D05
Mathematics Subject Classification ID: 30D20
Mathematics Subject Classification ID: 34A30
zbMATH DE Number: 4040249
MaRDI profile type: Publication
full work available at URL: https://doi.org/10.2748/tmj/1178227977
OpenAlex ID: W1968348202
cites work: Sums of deficiencies of meromorphic functions
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DOI: 10.2748/TMJ/1178227977
description: scientific article; zbMATH DE number 4040249