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On meromorphic solutions of functional-differential equations

Part of: Functional-differential and differential-difference equations Entire and meromorphic functions, and related topics

Published online by Cambridge University Press:  10 February 2022

Feng Lü
Feng Lü*
Affiliation:
College of Science, China University of Petroleum, Qingdao, Shandong266580, P.R. China (lvfeng18@gmail.com)
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Abstract

We consider meromorphic solutions of functional-differential equations

\[ f^{(k)}(z)=a(f^{n}\circ g)(z)+bf(z)+c, \]
where $n,\,~k$ are two positive integers. Firstly, using an elementary method, we describe the forms of $f$ and $g$ when $f$ is rational and $a(\neq 0)$, $b$, $c$ are constants. In addition, by employing Nevanlinna theory, we show that $g$ must be linear when $f$ is transcendental and $a(\neq 0)$, $b$, $c$ are polynomials in $\mathbb {C}$.

Keywords

functional-differential equationmeromorphic solutioncompositiongrowth

MSC classification

Primary: 30D30: Meromorphic functions, general theory 30D35: Distribution of values, Nevanlinna theory
Secondary: 34K06: Linear functional-differential equations
Type
Research Article
Information
Proceedings of the Edinburgh Mathematical Society , Volume 65 , Issue 1 , February 2022 , pp. 263 - 278
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Copyright
Copyright © The Author(s), 2022. Published by Cambridge University Press on Behalf of The Edinburgh Mathematical Society

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