A note on Hayman's problem

Abstract: 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.