Abstract
In this paper, we investigate uniqueness of finite-order transcendental meromorphic solutions of the following two equations:
and
where \(R(z,f)\) is an irreducible rational function in \(f(z)\), \(a(z)\), \(a_{m}\) and \(b_{n}\) are small functions of \(f(z)\). Such solutions \(f(z)\) are uniquely determined by their poles and the zeros of \(f(z)-e_{j}\) (counting multiplicities) for two complex numbers \(e_{1}\neq e_{2}\).
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REFERENCES
Yi.-M. Chiang and Sh.-J. Feng, ‘‘On the Nevanlinna characteristic of \(f(z+\eta)\) and differences equations in the complex plane,’’ Ramanujan J. 16, 105–129 (2008). https://doi.org/10.1007/s11139-007-9101-1
Yi. Du and J. Zhang, ‘‘Painlevé III and V types delay differential equations,’’ Comput. Methods Funct. Theory (2022). https://doi.org/10.1007/s40315-022-00442-8
W. K. Hayman, Meromorphic Functions, Oxford Mathematical Monographs (Clarendon Press, Oxford, 1964).
R. G. Halburd and R. J. Korhonen, ‘‘Difference analogue of the lemma on the logarithmic derivative with applications to difference equations,’’ J. Math. Anal. Appl. 314, 477–487 (2006). https://doi.org/10.1016/j.jmaa.2005.04.010
R. G. Halburd and R. J. Korhonen, ‘‘Nevanlinna theory for the difference operator,’’ Ann. Acad. Sci. Fenn. Math. 31, 463–478 (2006).
R. G. Halburd, R. J. Korhonen, and K. Tohge, ‘‘Holomorphic curves with shift-invariant hyperplane preimages,’’ Trans. Am. Math. Soc. 336, 4267–4298 (2014). https://doi.org/10.1090/S0002-9947-2014-05949-7
R. G. Halburd and R. J. Korhonen, ‘‘Growth of meromorphic solutions of delay differential equations,’’ Proc. Am. Math. Soc. 145, 2513–2526 (2017). https://doi.org/10.1090/proc/13559
P.-Ch. Hu and Q.-Ya. Wang, ‘‘Growth on meromorphic solutions of non-linear delay differential equations,’’ Bull. Belg. Math. Soc. Simon Stevin 26, 131–147 (2019). https://doi.org/10.36045/bbms/1553047233
I. Laine, Nevanlinna Theory and Complex Differential Equations, De Gruyter Studies in Mathematics, Vol. 15 (De Gruyter, Berlin, 1993). https://doi.org/10.1515/9783110863147
R. Nevanlinna, Le théorème de Picard–Borel et la théorie des fonctions méromorphes (1929).
X. Qi, N. Li, and L. Yang, ‘‘Uniqueness of meromorphic functions concerning their differences and solutions of difference Painlevé equations,’’ Comput. Methods Funct. Theory 18, 567–582 (2018). https://doi.org/10.1007/s40315-018-0241-7
Ch.-Ch. Yang and H.-X. Yi, Uniqueness Theory of Meromorphic Functions, Mathematics and Its Applications, Vol. 557 (Springer, Dordrecht, 2003; Science Press, Beijing, 1995).
H. X. Yi, ‘‘Meromorphic functions with two deficient values,’’ Acta Math. Sin. 30, 588–597 (1987).
ACKNOWLEDGMENTS
The authors would like to thank the referee for his or her valuable suggestions to the present paper.
Funding
This research was supported by the National Natural Science Foundation of China, project no. 11201014. This research was also supported by the youth talent program of Beijing no. 29201443.
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Du, Y., Zhang, J. On Uniqueness of Meromorphic Solutions to Delay Differential Equation. J. Contemp. Mathemat. Anal. 58, 142–151 (2023). https://doi.org/10.3103/S1068362323030044
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DOI: https://doi.org/10.3103/S1068362323030044