  In this note, it is shown that the differential polynomial of the form
$Q(f)^{(k)}-p$ has infinitely many zeros, and particularly $Q(f)^{(k)}$ has
infinitely many fixed points for any positive integer $k$, where $f$ is a
transcendental meromorphic function, $p$ is a nonzero polynomial and $Q$ is a
polynomial with coefficients in the field of small functions of $f$. The
results are traced back to Problem 1.19 and Problem 1.20 in the book of
research problems by Hayman and Lingham. As a consequence, we give an
affirmative answer to an extended problem on the zero distribution of
$(f^n)'-p$, proposed by Chiang and considered by Bergweiler. Moreover, our
methods provide a unified way to study the problem of the zero distributions of
partial differential polynomials of meromorphic functions in one and several
complex variables with small meromorphic coefficients.
