Hostname: page-component-f554764f5-246sw Total loading time: 0 Render date: 2025-04-20T07:02:58.488Z Has data issue: false hasContentIssue false
  • English
  • Français

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]
Get access Rights & Permissions [Opens in a new window]

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
Check for updates
Copyright
© The Author(s), 2025. Published by Cambridge University Press on Behalf of The Edinburgh Mathematical Society.

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

Article purchase

Temporarily unavailable

Footnotes

*

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

References

Clunie, J., On integral and meromorphic functions, J. London Math. Soc. 37(1) (1962), 17–27.CrossRefGoogle Scholar
Fang, C. Y. and Fang, M. L., Uniqueness of meromorphic functions and differential polynomials, Comput. Math. Appl. 44(5–6) (2002), 607–617).CrossRefGoogle Scholar
Frei, M., Sur l’ordre des solutions entieres d’une equation differentiable lineare, C. R. Acad. Sci. Paris 236 (1953), 38–40.Google Scholar
Gross, F., On the equation $f^n+g^n=h^n$, Amer. Math. Monthy 73(10) (1966), 1093–1096.CrossRefGoogle Scholar
Gundersen, G. G., Finite order solutions of second order linear differential equations, Trans. Amer. Math. Soc. 305 (1988), 415–429.CrossRefGoogle Scholar
Halburd, R. and Korhonen, R., Value distribution and linear operators, Proc. Edinburg Math. Soc. 57(2) (2014), 493–504.CrossRefGoogle Scholar
Hayman, W. K., Meromorphic functions, (Oxford University Press, Oxford, 1964).Google Scholar
He, Y. Z. and Xiao, X. Z., Algebroid functions and ordinary differential equations, (Science Press, Beijing, 1988) (in Chinese).Google Scholar
Heittokangas, J., Korhonen, R. J. and Rättyä, J., Linear differential equations with coefficients in weighted Bergman and Hardy spaces, Trans. Amer. Math. Soc. 360(2): (2008), 1035–1056.CrossRefGoogle Scholar
Lü, F. and Yi, H. X., On the uniqueness problems of meromorphic functions and their linear differential polynomials, J. Math. Anal. Appl. 362(2): (2010), 301–312.CrossRefGoogle Scholar
Lahiri, I. and Das, S., Brück conjecture and linear differential polynomials, Comput. Method Funct. Theory 18 (2018), 125–142.CrossRefGoogle Scholar
Lahiri, I. and Mukherjee, R., Uniqueness of entire functions sharing a value with linear differential polynomial, Bull. Aust. Math. Soc. 85(2) (2012), 295–306.CrossRefGoogle Scholar
Li, P., Value sharing and differential equations, J. Math. Anal. Appl. 310(2) (2005), 412–423.CrossRefGoogle Scholar
Li, P., Meromorphic functions sharing three values im with its linear differential polynomials, Complex Var. Elliptic Equ, 41(3) (2007), 221–229.Google Scholar
Li, P. and Yang, C. C., Value sharing of an entire function and its derivatives I, J. Math. Soc. Japan 51(4) (1999), 781–799.CrossRefGoogle Scholar
Li, P. and Yang, C. C., When an entire function and its linear differential polynomial share two values, Illinois J. Math. 44(2): (2000), 349–362.CrossRefGoogle Scholar
Li, X. M. and Yi, H. X., Meromorphic functions sharing three values with their linear differential polynomials in some angular domains, Results Math, 68 (2015), 441–453.CrossRefGoogle Scholar
Li, Y., Sharing set and normal families of entire functions, Results Math. 63 (2013), 543–556.CrossRefGoogle Scholar
Mao, Z. Q., Uniqueness theorems on entire functions and their linear differential polynomials, Results Math, 55 (2009), 447–456.Google Scholar
Mues, E. and Steinmetz, N., Meromorphe Funktionen die unit ihrer Ableitung Werte teilen, Manuscripta Math. 29 (1979), 195–206.CrossRefGoogle Scholar
Nevanlinna, R., Le th$\acute{e}$or$\grave{e}$me de Picard–Borel et la th$\acute{e}$orie des fonctions m$\acute{e}$romorphes. In: Collections de monographies sur la th $\acute{e}$orie des fonctions, VolumeVII, , (Gauthier-Villars, Paris, 1929).Google Scholar
Rubel, L. A. and Yang, C. C., Values shared by an entire function and its derivative, Complex Analysis (Proc. Conf. Univ. Kentucky, Lexington, Ky. 1976), Lecture Notes in Mathematics, Volume599, , (Springer, Berlin, 1977).Google Scholar
Saleeby, R. S., On complex analytic solutions of certain trinomial functional and partial differential equations, Aequat. Math. 85 (2013), 553–562.CrossRefGoogle Scholar
Wang, W. and Li, P., Unicity of entire functions and their linear differential polynomials, Complex Var. Elliptic Equ. 49(11): (2004), 821–832.Google Scholar
Yang, C. C. and Yi, H. X., Uniqueness theory of meromorphic functions, (Kluwer Academic Publishers, Science Press, 2003).CrossRefGoogle Scholar
Yang, L., Value distribution theory, (Springer-Verlag, Berlin, 1993).Google Scholar