On the growth of solutions of some higher order linear differential equations

In this paper we discuss the growth of solutions of the higher order nonhomogeneous linear differential equation \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${f^{(k)}} + {A_{k - 1}}{f^{(k - 1)}} + ... + {A_2}f" + ({D_1}(z) + {A_1}(z){e^{az}})f' + ({D_0}(z) + {A_0}(z){e^{bz}})f = F(k \ge 2),$$\end{document} where a, b are complex constants that satisfy ab(a − b) ≠ 0 and Aj(z) (j = 0, 1, …, k − 1), Dj (z) (j = 0, 1), F(z) are entire functions with max{ϱ(Aj ) (j = 0, 1, …, k − 1), ϱ(Dj) (j = 0, 1) < 1. We also investigate the relationship between small functions and the solutions of the above equation.