  In this paper, we investigate zeros of difference polynomials of the form
$f(z)^nH(z, f)-s(z)$, where $f(z)$ is a meromorphic function, $H(z, f)$ is a
difference polynomial of $f(z)$ and $s(z)$ is a small function. We first obtain
some inequalities for the relationship of the zero counting function of
$f(z)^nH(z, f)-s(z)$ and the characteristic function and pole counting function
of $f(z)$. Based on these inequalities, we establish some difference analogues
of a classical result of Hayman for meromorphic functions. Some special cases
are also investigated. These results improve previous findings.
