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.