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On an open problem of value sharing and differential equations

Part of: Entire and meromorphic functions, and related topics

Published online by Cambridge University Press:  16 April 2025

Molla Basir Ahamed Open the ORCID record for Molla Basir Ahamed [Opens in a new window]  and
Vasudevarao Allu Open the ORCID record for Vasudevarao Allu [Opens in a new window]
Molla Basir Ahamed
Affiliation:
Department of Mathematics, School of Basic Science, Indian Institute of Technology Bhubaneswar, Bhubaneswar, Odisha, India
Vasudevarao Allu*
Affiliation:
Department of Mathematics, School of Basic Science, Indian Institute of Technology Bhubaneswar, Bhubaneswar, Odisha, India
*
Corresponding author: Vasudevarao Allu; email: [email protected]
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Abstract

Let f be a non-constant meromorphic function. We define its linear differential polynomial $ L_k[f] $ by

\begin{equation*}L_k[f]=\displaystyle b_{-1}+\sum_{j=0}^{k}b_jf^{(j)}, \text{where}\; b_j (j=0, 1, 2, \ldots, k) \; \text{are constants with}\; b_k\neq 0.\end{equation*}
In this paper, we solve an open problem posed by Li [J. Math. Anal. Appl. 310 (2005) 412-423] in connection with the problem of sharing a set by entire functions f and their linear differential polynomials $ L_k[f] $. Furthermore, we study the Fermat-type functional equations of the form $ f^n+g^n=1 $ to find the meromorphic solutions (f, g) which enable us to answer the question of Li completely. This settles the long-standing open problem of Li.

Keywords

meromorphic functiondifferential polynomialsolutionsshared setFermat-type functional equation

MSC classification

Primary: 30D35: Distribution of values, Nevanlinna theory
Secondary: 30D30: Meromorphic functions, general theory
Type
Research Article
Information
Proceedings of the Edinburgh Mathematical Society , First View , pp. 1 - 27
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Copyright
© The Author(s), 2025. Published by Cambridge University Press on Behalf of The Edinburgh Mathematical Society.

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Footnotes

*

Present address: Department of Mathematics, Jadavpur University, Kolkata, India.

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