Hyper-order estimates for transcendental meromorphic solutions of linear complex differential equations

  • Mansouria Saidani Laboratory of Pure and Applied Mathematics, University of Mostaganem (UMAB), B. P. 227 Mostaganem-(Algeria
  • Benharrat Belaidi

Abstract

In this paper, we give some results about of the growth of transcendental meromorphic solutions of linear differential equations%
\begin{equation*}
f^{\left( k\right) }+\sum_{j=0}^{k-1}\left( D_{j}\left( z\right)
+A_{j}\left( z\right) e^{P_{j}\left( z\right) }+B_{j}\left( z\right)
e^{Q_{j}\left( z\right) }\right) f^{\left( j\right) }=F,
\end{equation*}%
where $A_{j}\left( z\right) \left( \not\equiv 0\right) ,$ $B_{j}\left( z\right) \left( \not\equiv 0\right) $ and $D_{j}\left( j=0,1,...,k-1\right) $ are meromorphic functions with finite order,\textit{\ }$P_{j}\left( z\right)
=a_{j,n}z^{n}+\cdots +a_{j,0}$\textit{\ }and\textit{\ }$Q_{j}\left( z\right) =b_{j,n}z^{n}+\cdots +b_{j,0}$\textit{\ }$\left( j=0,...,k-1\right) $\ are nonconstant polynomials, $a_{j,0},...,a_{j,n},b_{j,0},...,b_{j,n}$ ($%
j=0,1,...,k-1$)\textit{\ }are complex numbers. Under some conditions, we prove that the hyper-order of every transcendental meromorphic solution of the above equation and of its corresponding homogeneous equation coincides with $n$.

Published
2024-05-20