On hyper-order of solutions of higher order linear differential equations with meromorphic coefficients

In this paper, we investigate the growth of meromorphic solutions of the differential equations $$f^{(k)}+A_{k-1}(z)f^{(k-1)}+\cdots+A_{1}(z)f'+A_{0}(z)f=0 $$ and $$f^{(k)}+A_{k-1}(z)f^{(k-1)}+\cdots+A_{1}(z)f'+A_{0}(z)f=F(z), $$ where $A_{0}(z)\not\equiv0, A_{1}(z), \ldots, A_{k-1}(z)$ and $F(z)\not \equiv0$ are meromorphic functions. A precise estimation of the hyper-order of meromorphic solutions of the above equations is given provided that there exists one dominant coefficient, which improves and extends previous results given by Belaïdi, Chen, etc.