On Unicity of Non-linear Differential Polynomial of Meromorphic Function With Its Shift and \lowercase{q}-difference

Abstract

This article is insisted to studying the problems on sharing value for the non-linear differential polynomial with its shift and q-difference. The results in this paper improve and generalize the recent results due to S. C. Kumar, S. Rajeshwari, T. Bhuvaneshwari [J. Math. Comput. Sci., 12(2022), 1-16.]

Keywords

• Uniqueness
• Meromorphic function
• Weighted sharing

References

1. [1] A. Banerjee, Uniqueness of meromorphic functions that share two sets, Southeast Asian Bull. Math. Vol:31, (2007), 7-17.
2. [2] D. C. Barnett, R. G. Halburd, R. J. Korhonen and W. Morgan, Nevanlinna theory for the q-difference operator and meromorphic solutions of q-difference equations, Proc. Roy. Soc. Edinburgh Sect. A Vol:137, (2007), 457-474.
3. [3] R. Bruck, On entire functions which share one value CM with their first derivative, Results Math. Vol:30, (1996), 21-24.
4. [4] Z. X. Chen and K. H. Shon, On conjecture of R. Bruck concerning the entire function sharing one value CM with its derivatives, Taiwanese J. Math. Vol:8, (2004), 235-244.
5. [5] G. G. Gundersen and L. Z. Yang, Entire functions that share one value with one or two of their derivatives, J. Math. Anal. Appl. Vol:223, (1998), 88-95.
6. [6] W. K. Hayman, Meromorphic Functions, The Clarendon Press, Oxford, 1964.
7. [7] J. Heittokangas, R. J. Korhonen, R. Laine, I. Rieppo and J. L. Zhang, Value sharing results for shifts of meromorphic functions and sufficient conditions for periodicity, J. Math. Anal. Appl. Vol:355, (2009), 352-363.
8. [8] S. C. Kumar, S. Rajeshwari and T. Bhuvaneshwari, Results on unicity of meromorphic function with its shift and q-difference, J. Math. Comput. Sci., Vol:12, (2022), 1-16.

9. [9] I. Laine, Nevanlinna Theory and Complex Differential Equations, Walter de Gruyter, Berlin/Newyork, 1993.
10. [10] I. Lahiri, Weighted value sharing and uniqueness of meromorphic functions, Complex Var. Theory Appl., Vol:46, (2001), 241-253.
11. [11] I. Lahiri, value distribution of certain differential polynomials, Int. J. Math. Math. Sci., Vol:28, (2001), 83-91.
12. [12] S. Li and Z. S. Gao, Entire functions sharing one or two finite values CM with their shifts or difference operators, Arch. Math. Vol:97, (2011), 475-483.
13. [13] C. Meng and G. Liu, On unicity of meromorphic functions concerning the shifts and derivatives, Journal of Mathematical inequalities, Vol:14, No.4, (2020), 1095-1112.
14. [14] X. G. Qi, N. Li and L. Z. Yang, Uniqueness of meromorphic functions concerning their differences and solutions of difference painleve equations, Comput. Methods Funct. Theory. Vol:18, (2018), 567-582.
15. [15] L. A. Rubel and C. C. Yang, Values shared by an entire function and its derivatives, In Complex Analysis, Kentucky 1976 (Proc. Conf), Lecture Notes in Mathematics, Springer-Verlag, Berlin, Vol:599, (1977), 101-103.
16. [16] H. P. Waghamore and S. H. Naveenkumar, Results on Uniqueness of meromorphic functions of differential polynomials. Malaya Journal of Matematik, Vol:6, No. 1, (2018), 14-20.
17. [17] C. C. Yang, On deficiencies of differential polynomials, Math. Z. Vol:125, (1972), 107-112.
18. [18] C. C. Yang and H. X. Yi., Uniqueness Theory of Meromorphic functions, Kluwer, Dordrecht, 2003.
19. [19] J. L. Zhang,Meromorphic functions sharing a small function with their differential polynomials, Kyungpook Math. J. Vol:50, No.3, (2010), 345-355.
20. [20] J. L. Zhang and R. J. Korhonen, On the Nevanlinna characteristic of f (qz) and its applications, J. Math. Anal. Appl. Vol:369, (2010), 537-544.
21. [21] Q. C. Zhang, Meromorphic function that share one small function with its derivative, J. Inequal. Pure Appl. Math. Vol:6, No.4, (2005), Art. 116.