Relation between differential polynomials and small functions

Abstract

In this article, we discuss the growth of solutions of the second-order nonhomogeneous linear differential equation f + A1(z)eazf + A0(z)ebzf = F, where a, b are complex constants and Aj (z) ≡ 0 (j = 0, 1), and F ≡ 0 are entire functions such that max{ρ(Aj ) (j = 0, 1), ρ(F)} < 1. We also investigate the relationship between small functions and differential polynomials gf (z) = d2f + d1f + d0f, where d0(z), d1(z), d2(z) are entire functions that are not all equal to zero with ρ(dj ) < 1 (j = 0, 1, 2) generated by solutions of the above equation.