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ON THE PERIODICITY OF TRANSCENDENTAL ENTIRE FUNCTIONS

Part of: Entire and meromorphic functions, and related topics

Published online by Cambridge University Press:  13 February 2020

XINLING LIU Open the ORCID record for XINLING LIU [Opens in a new window]  and
RISTO KORHONEN Open the ORCID record for RISTO KORHONEN [Opens in a new window]
XINLING LIU
Affiliation:
Department of Mathematics,Nanchang University, Nanchang, Jiangxi, 330031, PR China Department of Physics and Mathematics,University of Eastern Finland, PO Box 111, 80101, Joensuu, Finland email liuxinling@ncu.edu.cn
RISTO KORHONEN*
Affiliation:
Department of Physics and Mathematics,University of Eastern Finland, PO Box 111, 80101, Joensuu, Finland email risto.korhonen@uef.fi
*
✉risto.korhonen@uef.fi
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Abstract

According to a conjecture by Yang, if $f(z)f^{(k)}(z)$ is a periodic function, where $f(z)$ is a transcendental entire function and $k$ is a positive integer, then $f(z)$ is also a periodic function. We propose related questions, which can be viewed as difference or differential-difference versions of Yang’s conjecture. We consider the periodicity of a transcendental entire function $f(z)$ when differential, difference or differential-difference polynomials in $f(z)$ are periodic. For instance, we show that if $f(z)^{n}f(z+\unicode[STIX]{x1D702})$ is a periodic function with period $c$, then $f(z)$ is also a periodic function with period $(n+1)c$, where $f(z)$ is a transcendental entire function of hyper-order $\unicode[STIX]{x1D70C}_{2}(f)<1$ and $n\geq 2$ is an integer.

Keywords

entire functionsperiodicitygrowth

MSC classification

Primary: 30D35: Distribution of values, Nevanlinna theory
Type
Research Article
Information
Bulletin of the Australian Mathematical Society , Volume 101 , Issue 3 , June 2020 , pp. 453 - 465
Copyright
© 2020 Australian Mathematical Publishing Association Inc.

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Footnotes

The first author was partially supported by the NSFC (No. 11661052) and EDUFI Fellowship No. 18-11020. The second author was partially supported by the Academy of Finland Grant No. 286877.

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