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 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 >= 2), where a, b are complex constants that satisfy ab(a - b) not equal 0 and A(j)(z) (j = 0, 1,..., k - 1), D-j(z) (j = 0, 1), F(z) are entire functions with max{rho(A(j)) (j = 0, 1, ..., k - 1), rho(D-j) (j = 0, 1)} < 1. We also investigate the relationship between small functions and the solutions of the above equation.