Meromorphic functions sharing small functions with their difference polynomial in several variables

Year 2023, Volume: 52 Issue: 1, 73 - 90, 15.02.2023

Minzhuo Wang , Zhixue Liu

https://doi.org/10.15672/hujms.1090815

Abstract

The aim of this paper is to deal with the uniqueness problem on meromorphic functions in $ \mathbb{C}^{m} $ sharing small functions with their difference polynomial, and the results obtained can be seen as some extensions of previous results from one complex variable to several complex variables.

Keywords

Meromorphic function, uniqueness, difference operators, several variables

References

• [1] C. Berenstein, D.C. Chang and B.Q. Li, A note on wronskians and linear dependence of entire functions, Complex var. elliptic equ. 24 (1-2), 131-144, 1994.
• [2] Z.X. Chen, Complex differences and difference equations, Science press, Beijing, 2013.
• [3] Z.X. Chen and H.X. Yi, On sharing values of meromorphic functions and their differences, Results Math. 63 (1), 557-565, 2013.
• [4] Y.M. Chiang and S.J. Feng, On the Nevanlinna characteristic of $f(z+\eta)$ and difference equations in the complex plane, Ramanujan J. 16 (1), 105-129, 2008.
• [5] N. Cui and Z.X. Chen, The conjecture on unity of meromorphic functions concerning their differences, J. difference equ. appl. 22 (10), 1452-1471, 2016.
• [6] H. Fujimoto, On meromorphic maps into the complex projective space, J. math. soc. Japan, 26 (2), 272-288, 1974.
• [7] Z.S. Gao, R. Korhonen, J.L. Zhang and Y.Y. Zhang, Uniqueness of meromorphic functions sharing values with their nth order exact differences, Anal. Math. 45 (2), 321-334, 2019.
• [8] G.G. Gundersen, Meromorphic functions that share two finite values with their derivative, Pacific J. Math. 105 (2), 299-309, 1983.
• [9] 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 (2), 477-487, 2006.
• [10] R.G. Halburd and R.J. Korhonen, Nevanlinna theory for the difference operator, Ann. acad. sci. fenn. Math. 31, 463-478, 2006.
• [11] J. Heittokangas, R. Korhonen, I. Laine and J. Rieppo, Uniqueness of meromorphic functions sharing values with their shifts, Complex var. elliptic equ. 56 (1-4), 81-92, 2011.
• [12] P.C. Hu, P. Li and C.C. Yang, Unicity of meromorphic mappings, Springer, New York, 2013.
• [13] R. Korhonen, A difference picard theorem for meromorphic functions of several variables, Comput. methods funct. theory, 12v(1), 343-361, 2012.
• [14] I. Laine, Nevanlinna theory and complex differential equations, Berlin, Walter de Gruyter, 1993.
• [15] P. Li, Unicity of meromorphic functions and their derivatives, J. math. anal. appl. 285 (2), 651-665, 2003.
• [16] Y.H. Li and J.Y. Qiao, The uniqueness of meromorphic functions concerning small functions, Sci. China ser. A-math., 43 (6), 581-590, 2000.
• [17] Z.X. Liu and Q.C. Zhang, Difference uniqueness theorems on meromorphic functions in several variables, Turkish J. Math. 42 (5), 2481-2505, 2018.
• [18] F. Lü and W. Lü, Meromorphic functions sharing three values with their difference operators, Comput. methods funct. theory, 17 (3), 395-403, 2017.
• [19] E. Mues and N. Steinmetz, Meromorphe Funktionen, die mit ihrer Ableitung zwei Werte teilen, Results Math. 6 (1), 48-55, 1983.
• [20] R. Nevanlinna, Einige Eindeutigkeitss¨atze in der Theorie der meromorphen Funktionen, Acta Math. 48 (3), 367-391, 1926.
• [21] M. Ru, Nevanlinna theory and its relation to diophantine approximation, World Scientific Publishing Co., Singapore, 2001.
• [22] L.A. Rubel and C.C. Yang, Values shared by an entire function and its derivative, Complex anal. synerg. 599, 101-103, 1976.
• [23] P. Russmann, Uber Differentialpolynome, die mit ibres erzeugenden Funktion zwei werte teilen, Dissertation, Technischen Universitit Berlin, 1993.
• [24] B.V. Shabat, Distribution of values of holomorphic mappings, Translations of Mathematical Monographs, Amer. math. soc. 1985.
• [25] W.N. Wu and T.B. Cao, Uniqueness theorems of meromorphic functions and their differences in several complex variables, Comput. methods funct. theory, 22v(2), 379- 399, 2021.
• [26] C.C. Yang and H.X. Yi, Uniqueness theory of meromorphic functions, Dordrecht, the Netherlands: Kluwer Academic, 2003.