Abstract
We investigate the uniqueness problems of meromorphic functions and their difference operators by using a new method. It is proved that if a non-constant meromorphic function f shares a non-zero constant and ∞ counting multiplicities with its difference operators Δcf(z) and \(\Delta_{c}^{2}f(z)\), then \(\Delta_{c}f(z)\equiv\Delta_{c}^{2}f(z)\). In particular, we give a difference analogue of a result of Jank–Mues–Volkmann. Our method has two distinct features: (i) It converts the relations between functions into the corresponding vectors. This makes it possible to deal with the uniqueness problem by linear algebra and combinatorics. (ii) It circumvents the obstacle of the difference logarithmic derivative lemma for meromorphic functions of infinite order, since this method does not depend on the growth of the functions. Furthermore, the idea in this paper can also be applied to the case for several variables.
Similar content being viewed by others
References
Chen, B. Q., Chen, Z. X., Li, S.: Uniqueness theorems on entire functions and their difference operators and shifts. Abstr. Appl. Anal., ID 906893 (2012)
Chen, B. Q., Li, S.: Uniqueness theorems on entire functions that share small functions with their difference operators. Adv. Diff. Equ., 311, 1–11 (2014)
Chiang, Y. M., Feng, S. J.: On the Nevanlinna characteristic of f(z + η) and difference equations in the complex plane. Ramanujan J., 16, 105–129 (2008)
Chiang, Y. M., Feng, S. J.: On the growth of logarithmic differences, difference quotients and logarithmic derivatives of meromorphic functions. Trans. Amer. Math. Soc., 361, 3767–3791 (2009)
Cui, N., Chen, Z. X.: Entire functions sharing one small function CM with their shifts and difference operators. Acta. Math. Sci., 37, 786–798 (2017)
El Farissi, A., Latreuch, Z., Asiri, A.: On the uniqueness theory of entire functions and their difference operators. Complex Anal. Oper. Theory, 10, 1317–1327 (2016)
El Farissi, A., Latreuch, Z., Beladi, B., et al.: Entire functions that share a small function with their difference operators. Electron. J. Differ. Equ., 32, 1–13 (2016)
Gundersen, G. G.: Meromorphic functions that share two finite values with their derivative. Pacific J. Math., 105, 299–309 (1983)
Halburd, R. G., Korhonen, R. J.: Difference analogue of the lemma on the logarithmic derivative with applications to difference equations. J. Math. Anal. Appl., 314, 477–487 (2006)
Halburd, R. G., Korhonen, R. J.: Nevanlinna theory for the difference operator. Ann. Acad. Sci. Fenn. Math., 31, 463–478 (2006)
Halburd, R. G., Korhonen, R. J., Tohge, K.: Holomorphic curves with shift-invariant hyperplane preimages. Trans. Amer. Math. Soc., 366, 4267–4298 (2014)
Hayman, W. K.: Meromorphic Functions, Clarendon Press, Oxford, 1964
Jank, G., Mues, E., Volkmann, L.: Meromorphe Funktionen, die mit ihrer ersten und zweiten Ableitung einen endlichen Wert teilen. Complex Variables Theory Appl., 6, 51–71 (1986)
Mues, E., Steinmetz, N.: Meromorphe Funktionen, die mit ihrer Ableitung zwei Werte teilen. Results Math., 6, 48–55 (1983)
Rubel, L. A., Yang, C. C.: Values shared by an entire function and its derivative. In: Complex Analysis, Lecture Notes in Math., Vol. 599, Springer-Verlag, Berlin, 101–103 (1977)
Yang, C. C., Yi, H. X.: Uniqueness Theory of Meromorphic Functions, Kluwer Academic Publishers, Dordrecht, 2003; Chinese original: Science Press, Beijing, 1995
Yang, L.: Value Distribution Theory, Springer-Verlag, Berlin, 1993
Yang, L. Z.: Entire functions that share finite values with their derivatives. Bull. Austral. Math. Soc., 41, 337–342 (1990)
Acknowledgements
The authors would like to thank the referees for their valuable suggestions and comments. The authors appreciate the help of Dr. Wenqiang Shen and Yiming Zhang during the preparation of this manuscript.
Author information
Authors and Affiliations
Corresponding author
Ethics declarations
Conflict of Interest The authors declare no conflict of interest.
Additional information
Supported by National Natural Science Foundation of China (Grant Nos. 12071047, 12171127, 11901311) and National Key Technologies R&D Program of China (Grant No. 2020YFA0713300)
Rights and permissions
About this article
Cite this article
Li, H., Fang, M.L. & Yao, X. Uniqueness on Difference Operators of Meromorphic Functions of Infinite Order. Acta. Math. Sin.-English Ser. 40, 511–527 (2024). https://doi.org/10.1007/s10114-023-2300-x
Received:
Revised:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s10114-023-2300-x