Uniqueness of derivative of an entire function that shares pair values with derivative of its difference operator

Abstract
The main objective of this article is to examine the uniqueness of \(k\)th derivative of transcendental entire function and \(i\)th derivative of its difference operator that share pair values \((\alpha_{1}, \alpha_{2})\), 0 IM and zeros of \(k\)th derivative of transcendental entire function contained in \(i\)th derivative of its difference operator. Meanwhile all of which greatly generalizes the outcomes obtained by X. Huang [11] and L. Sheng [16].
2020 Mathematics Subject Classification:
30D35Keywords
entire function, difference operator, sharing pair values
Author Details
Renukadevi S. Dyavanal
Department of Mathematics
Karnatak University
Dharwad – 580003, India
e-mails: rsdyavanal@kud.ac.in, renukadyavanal@gmail.com
Shakuntala B. Kalakoti
Department of Mathematics
Karnatak University
Dharwad – 580003, India
e-mail: shakkukalakoti1@gmail.com
References
- T. B. Cao. Difference analogues of the second main theorem for meromorphic functions in several complex variables. Math Nachr. 287, 5–6 (2014), 530–545.
- S. Chen. On uniqueness of meromorphic functions and their difference operator with partially shared values. Comput. Methods Funct. Theory 18, 3 (2018), 529–536.
- Z.-X. Chen, H.-X. Yi. On sharing values of meromorphic functions and their differences. Results Math. 63, 1–2 (2013), 557–565.
- Y.-M. Chiang, S.-J. Feng. On the Nevanlinna characteristic of f(z + η) and difference equations in the complex plane. Ramanujan J. 16, 1 (2008), 105–129.
- R. S. Dyavanal. Uniqueness and value-sharing of differential polynomials of meromorphic functions. J. Math. Anal. and Appl. 372, 1 (2010), 252–261.
- R. S. Dyavanal, A. M. Hattikal. Uniqueness of difference-differential polynomials of entire functions sharing one value. Tamkang J. Math. 47, 2 (2016) 193–206.
- R. G. Halburd, R. J. Korhonen. Nevanlinna theory for the difference operator. Ann. Acad. Sci. Fenn. Math. 31, 2 (2006), 463–478.
- W. K. Hayman. Meromorphic Functions. Oxford Math. Monogr. Oxford, Clarendon Press, 1964.
- J. Heittokangas, R. Korhonen, I. Laine, J. Rieppo. Uniqueness of meromorphic functions sharing values with their shifts. Complex Var. Elliptic Equ. 56, 1–4 (2011), 81–92.
- X. Huang. Unicity on entire function concerning its differential-difference operators. Results Math. 76, 3 (2021), Paper No. 147, 17 pp.
- X. Huang, Uniqueness of entire functions sharing two pairs of values with its difference operator. arXiv: 2109.15183, 2021, https://arxiv.org/abs/2109.15183
- X. Huang, B. Deng, M. L. Fang. Entire functions that share two pairs of small functions. Open Math. 19, 1 (2021), 144–156.
- X. Huang, M. L. Fang. Unicity of entire functions concerning their shifts and derivatives. Comput. Methods Funct. Theory 21, 3 (2021), 523–532.
- I. Lahiri. Value distribution of certain differential polynomials, Int. J. Math. Math. Sci. 28, 2 (2001), 83–91.
- X. Qi, L. Yang. Uniqueness of meromorphic functions concerning their shifts and derivatives. Comput. Methods Funct. Theory 20, 1 (2020) 159–178.
- L. Sheng, D. Mei, B. Chen. Uniqueness of entire functions sharing two values with their difference operators. Adv. Difference Equ. (2017), Paper No. 390, 9 pp.
- C.-C. Yang, H.-X. Yi. Uniqueness Theory of Meromorphic Functions. Math. Appl., vol. 557. Dordrecht, Kluwer Academic Publishers Group, 2003.
- L. Yang. Value Distribution Theory. Berlin, Springer-Verlag; Beijing, Science Press Beijing, 1993.
- Q. C. Zhang. Uniqueness of meromorphic functions with their derivatives. Acta Math. Sinica (Chinese Ser.) 45, 5 (2002), 871–876.