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 Problems 1.19 and 1.20 in the book of research problems by Hayman and Lingham [ Research Problems in Function Theory, Springer, 2019 ]. 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 [ Bull. Hong Kong Math. Soc. 1 (1997), p. 97–101].