  In this paper, we will give suitable conditions on differential polynomials
$Q(f)$ such that they take every finite non-zero value infinitely often, where
$f$ is a meromorphic function in complex plane. These results are related to
Problem 1.19 and Problem 1.20 in a book of Hayman and Lingham \cite{HL}. As
consequences, we give a new proof of the Hayman conjecture. Moreover, our
results allow differential polynomials $Q(f)$ to have some terms of any degree
of $f$ and also the hypothesis $n>k$ in \cite[Theorem 2]{BE} is replaced by
$n\ge 2$ in our result.
