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

$f\prime \prime +A_1\left(z\right)e^{az}f\prime +A_0\left(z\right)e^{bz}f=F$,

where $a$, $b$ are complex constants and $A_j\left(z\right)\not\equiv 0$ $(j=0,1)$, and $F\not\equiv 0$ are entire functions such that $max\left\{\rho \right(A_j\left)\right(j=0,1),\rho (F\left)\right\}<1$. We also investigate the relationship between small functions and differential polynomials $g_f\left(z\right)=d_{2}f\prime \prime +d_{1}f\prime +d_{0}f$, where $d_0\left(z\right),d_1\left(z\right),d_2\left(z\right)$ are entire functions that are not all equal to zero with $\rho \left(d_j\right)<1$ $(j=0,1,2)$ generated by solutions of the above equation.