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

Abstract

n this paper we discuss the growth of solutions of the higher order nonhomogeneous linear differential equation f (k)+ Ak− 1f (k− 1)+...+ A2f′′+(D1 (z)+ A1 (z) eaz) f′+(D0 (z)+ A0 (z) ebz) f= F (k⩾ 2), 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 {̺ (Aj)(j= 0, 1,..., k− 1), ̺ (Dj)(j= 0, 1)}< 1. We also investigate the relationship between small functions and the solutions of the above equation.