Results on the growth of solutions of complex linear differential equations with meromorphic coefficients

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].