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THE DIFFERENCE ANALOGUE OF THE TUMURA–HAYMAN–CLUNIE THEOREM

Part of: Difference equations Entire and meromorphic functions, and related topics Difference and functional equations

Published online by Cambridge University Press:  06 November 2023

MINGLIANG FANG Open the ORCID record for MINGLIANG FANG [Opens in a new window] ,
HUI LI Open the ORCID record for HUI LI [Opens in a new window]  and
XIAO YAO Open the ORCID record for XIAO YAO [Opens in a new window]
MINGLIANG FANG
Affiliation:
School of Sciences, Hangzhou Dianzi University, Hangzhou 310012, PR China e-mail: mlfang@hdu.edu.cn
HUI LI*
Affiliation:
School of Science, China University of Mining and Technology-Beijing, Beijing 100083, PR China
XIAO YAO
Affiliation:
School of Mathematical Sciences and LPMC, Nankai University, Tianjin 300071, PR China e-mail: yaoxiao@nankai.edu.cn
*
e-mail: lihui2021@amss.ac.cn
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Abstract

We prove a difference analogue of the celebrated Tumura–Hayman–Clunie theorem. Let f be a transcendental entire function, let c be a nonzero constant and let n be a positive integer. If f and $\Delta _c^n f$ omit zero in the whole complex plane, then either $f(z)=\exp (h_1(z)+C_1 z)$, where $h_1$ is an entire function of period c and $\exp (C_1 c)\neq 1$, or $f(z)=\exp (h_2(z)+C_2 z)$, where $h_2$ is an entire function of period $2c$ and $C_2$ satisfies

$$ \begin{align*} \bigg(\frac{1+\exp(C_2c)}{1-\exp(C_2 c)}\bigg)^{2n}=1. \end{align*} $$

Keywords

entire functionsforward differencesvalue distribution

MSC classification

Primary: 30D35: Distribution of values, Nevanlinna theory
Secondary: 39A10: Difference equations, additive

Information

Type
Research Article
Information
Bulletin of the Australian Mathematical Society , Volume 110 , Issue 1 , August 2024 , pp. 145 - 154
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Copyright
© The Author(s), 2023. Published by Cambridge University Press on behalf of Australian Mathematical Publishing Association Inc.

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Footnotes

This work was partially supported by the National Natural Science Foundation of China (Grant Nos. 12171127, 12071047, 12301096, 11901311, 12371074) and National Key Technologies R&D Program of China (2020YFA0713300).

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