Growth and oscillation related to a second order linear differential equation

Abstract

This paper is devoted to studying the growth and the oscillation of
solutions of the second order non-homogeneous linear differential equation
\begin{equation*}
f^{\prime \prime }+A_{1}\left( z\right) e^{P\left( z\right) }f^{\prime
}+A_{0}\left( z\right) e^{Q\left( z\right) }f=F,
\end{equation*}%
where $P\left( z\right)$, $Q\left( z\right) $ are nonconstant
polynomials such that $\deg P=\deg Q=n$ and $A_{j}\left( z\right) $\linebreak
$\left( \not\equiv 0\right) $ $(j=0,1),$ $F\left( z\right) $ are
entire functions with $\max \{\rho \left( A_{j}\right) :j=0,1\}<n$. We also
investigate the relationship between small functions and differential
polynomials $g_{f}\left( z\right) \linebreak =d_{2}f^{\prime \prime }+d_{1}f^{\prime
}+d_{0}f$, where $d_{0}\left( z\right) ,d_{1}\left( z\right) ,d_{2}\left(
z\right) $ are entire functions such that at least one of $d_{0},d_{1},d_{2}$%
\ does not vanish identically with $\rho \left( d_{j}\right) <n$%
$\left( j=0,1,2\right) $ generated by solutions of the above
equation.