Subnormal solutions of second order linear differential equations with periodic coefficients
type: Item
instance of: scholarly article
title: Subnormal solutions of second order linear differential equations with periodic coefficients
zbMATH Open document ID: 0804.34009
DOI: 10.1007/BF03323411
author: Gary G. Gundersen
author: Enid M. Steinbart
published in: Results in Mathematics
publication date: 1994-07-12T00:00:00Z
review text: A very complete analysis is conducted for the second order linear differential equation \[ f''+ P(e^ z)f'+ Q(e^ z)f= R_ 1(e^ z)+ R_ 2(e^{-z}),\leqno{(*)} \] where \(P(z)\), \(Q(z)\), \(R_ 1(z)\), and \(R_ 2(z)\) are polynomials in \(z\), such that \(P(z)\) and \(Q(z)\) are not both constant. We say \(f\) is a subnormal solution of equation \((*)\) iff \(f\) satisfies \((*)\) and moreover satisfies the \(R\). Nevanlinna's condition, \[ \varlimsup_{r\to\infty} {\log T(r,f)\over r}= 0. \] It is well-known that the subnormal solution to the homogeneous problem, namely, \(R_ 1(e^ z)= 0\) and \(R_ 2(e^{-z})= 0\) is given by the function \(f(z)= e^{cz}(a_ 0+ a_ 1 e^ z+\cdots+ a_ m e^{mz})\), where \(m\geq 0\) is an integer and \(c,a_ 0,a_ 1,\dots,a_ m\) are constants with \(a_ 0\neq 0\) and \(a_ m\neq 0\). Four remarkable theorems are then stated with very excellent detailed proofs included in latter sections. They relate the degree of the polynomials, \(P\), \(Q\), \(R_ 1\) and \(R_ 2\) and this effect is then transposed to the solutions of the problem. For example if \(\deg P=\deg Q\geq 1\) then the subnormal solution \(f\) to equation \((*)\) must have one of the following two forms: \(f(z)= S_ 1(e^ z)+ S_ 2(e^{-z})+ e^{bz} S_ 3(e^ z)\), where \(S_ 1(z)\), \(S_ 2(z)\), and \(S_ 3(z)\) are polynomials in \(z\), and \(b\) is constant; or \[ f(z)= e^{nz}\left\{cz \sum^ m_{k=0} a_ k e^{- kz}+ S_ 1(e^ z)+ S_ 2(e^{-z})\right\}, \] where \(m\geq 0\) and \(n\) are integers, \(c,a_ 0,a_ 1,\dots,a_ m\) are constants with \(c\neq 0\), \(a_ 0\neq 0\) and \(a_ m\neq 0\). Again \(S_ 1(z)\) and \(S_ 2(z)\) are polynomials in \(z\). All proofs of the theorems are given in significant detail and several examples illustrating the results are also included.
Mathematics Subject Classification ID: 34A30
Mathematics Subject Classification ID: 34M99
zbMATH DE Number: 604854
zbMATH Keywords: meromorphic functions
zbMATH Keywords: Nevanlinna characteristic function
zbMATH Keywords: Landau notation
zbMATH Keywords: second order linear differential equation
zbMATH Keywords: subnormal solution
reviewed by: John Schmeelk
MaRDI profile type: Publication
cites work: Q5658395
cites work: Three results in the value-distribution theory of solutions of linear differential equations
cites work: On the explicit determination of certain solutions of periodic differential equations
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cites work: Estimates for the Logarithmic Derivative of a Meromorphic Function, Plus Similar Estimates
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full work available at URL: https://doi.org/10.1007/bf03323411
OpenAlex ID: W1988314575
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description: scientific article; zbMATH DE number 604854