2022/12/02 by Jiaxing Huang, Huang, Jiaxing, Yuefei Wang +1
Mathematics · #Complex Variables (math.CV) #FOS: Mathematics #Meromorphic and Entire Functions
paper · pdf · doi:10.48550/arxiv.2212.01063
openalex publication_date 2022/12/02 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
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 (fn)'-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.