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A note on Hayman’s problem

Part of: Entire and meromorphic functions, and related topics

Published online by Cambridge University Press:  18 December 2024

Jiaxing Huang  and
Yuefei Wang Open the ORCID record for Yuefei Wang [Opens in a new window]
Jiaxing Huang
Affiliation:
School of Mathematical Sciences, Shenzhen University, Guangdong, 518060, P. R. China
Yuefei Wang*
Affiliation:
Institute of Mathematics, Academy of Mathematics and Systems Sciences, Chinese Academy of Sciences, Beijing 100190, China
*
e-mail: wangyf@math.ac.cn
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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].

Keywords

Hayman’s problemdifferential polynomialsmeromorphic functionszero distributions

MSC classification

Primary: 30D35: Distribution of values, Nevanlinna theory
Secondary: 30D45: Bloch functions, normal functions, normal families 30D30: Meromorphic functions, general theory

Information

Type
Article
Information
Canadian Mathematical Bulletin , Volume 68 , Issue 2 , June 2025 , pp. 620 - 627
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Copyright
© The Author(s), 2024. Published by Cambridge University Press on behalf of Canadian Mathematical Society

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Footnotes

The authors were supported by NSF of China (No.12231013, 12201420) and NCAMS.

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