Open Journal of Mathematical Sciences
Vol. 7 (2023), Issue 1, pp. 248-268
ISSN: 2523-0212 (Online) 2616-4906 (Print)
DOI: 10.30538/oms2023.0209
Results on the growth of solutions of complex linear differential equations with meromorphic coefficients
Mansouria Saidani\(^{1,*}\), and Benharrat Belaïdi\(^1\)
\(^{1}\) Department of Mathematics, Laboratory of Pure and Applied Mathematics, University of Mostaganem (UMAB), B. P. 227 Mostaganem-(Algeria)
Copyright © 2023 Mansouria Saidani and Benharrat Belaïdi. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
Received: March 03, 2023 – Accepted: April 30, 2023 – Published: November 08, 2023
Abstract
The purpose of this paper is the study of the growth of solutions of higher order linear differential equations \(f^{\left( k\right) }+\left( A_{k-1,1}\left( z\right) e^{P_{k-1}\left(z\right) }+A_{k-1,2}\left( z\right) e^{Q_{k-1}\left( z\right) }\right)f^{\left( k-1\right) }+\cdots +\left( A_{0,1}\left( z\right) e^{P_{0}\left( z\right)
}+A_{0,2}\left( z\right) e^{Q_{0}\left( z\right) }\right) f=0\) and \(f^{\left( k\right) }+\left( A_{k-1,1}\left( z\right) e^{P_{k-1}\left(z\right) }+A_{k-1,2}\left( z\right) e^{Q_{k-1}\left( z\right) }\right)f^{\left( k-1\right) }+\cdots +\left( A_{0,1}\left( z\right) e^{P_{0}\left( z\right)}+A_{0,2}\left( z\right) e^{Q_{0}\left( z\right) }\right) f=F\left( z\right),\) where \(A_{j,i}\left( z\right) \left( \not\equiv 0\right) \left(j=0,…,k-1;i=1,2\right) ,\) \(F\left( z\right) \) are meromorphic functions of finite order and \(P_{j}\left( z\right) ,Q_{j}\left( z\right) \) \((j=0,1,…,k-1;i=1,2)\) are polynomials with degree \(n\geq 1\). Under some others conditions, we extend the previous results due to Hamani and Belaïdi [1].
}+A_{0,2}\left( z\right) e^{Q_{0}\left( z\right) }\right) f=0\) and \(f^{\left( k\right) }+\left( A_{k-1,1}\left( z\right) e^{P_{k-1}\left(z\right) }+A_{k-1,2}\left( z\right) e^{Q_{k-1}\left( z\right) }\right)f^{\left( k-1\right) }+\cdots +\left( A_{0,1}\left( z\right) e^{P_{0}\left( z\right)}+A_{0,2}\left( z\right) e^{Q_{0}\left( z\right) }\right) f=F\left( z\right),\) where \(A_{j,i}\left( z\right) \left( \not\equiv 0\right) \left(j=0,…,k-1;i=1,2\right) ,\) \(F\left( z\right) \) are meromorphic functions of finite order and \(P_{j}\left( z\right) ,Q_{j}\left( z\right) \) \((j=0,1,…,k-1;i=1,2)\) are polynomials with degree \(n\geq 1\). Under some others conditions, we extend the previous results due to Hamani and Belaïdi [1].
Keywords:
Order of growth; Hyper-order; Exponent of convergence of zero sequence; Differential equation; Meromorphic function