Uniqueness of certain polynomials of meromorphic functions sharing a set with finite weight

Abstract
We investigated the Brück type conjecture and generalized the existing result by extending them up to a difference-differential polynomial \(\mathcal{P}[\xi]\) sharing a small function with a certain differential polynomial \(\mathcal{L}[\xi]\) of a meromorphic function. The class of all meromorphic solutions of the differential equation \(\mathcal{P}[\xi] \equiv \mathcal{L}[\xi]\) has been explored. Our result will generalize and extend the result due to A. Banerjee and B. Chakraborty [3]. For the generalization of our main result, some relevant questions have finally been posed for further study in this direction.
2020 Mathematics Subject Classification:
30D35Keywords
uniqueness, meromorphic functions, difference-differential polynomial, differential polynomial, weighted sharing, set sharing
Author Details
Harina P. Waghamore
Department of Mathematics
Jnanabharathi Campus
Bangalore University
Bengaluru-560056, India
e-mails: harina@bub.ernet.in, harinapw@gmail.com
B. N. Naveenkumar
Department of Mathematics
Jnanabharathi Campus
Bangalore University
Bengaluru-560056, India
emails: bnk_maths@bub.ernet.in, nknateshmaths1@gmail.com
References
- T. C. Alzahary. Meromorphic functions with weighted sharing of one set. Kyungpook Math. J. 47, 1 (2007), 57–68.
- A. Banerjee, M. B. Ahamed. Meromorphic Function Sharinga Small Function with its Differential Polynomial. Acta Univ. Palack. Olomuc. Fac. Rerum Natur. Math. 54, 1 (2015), 33–45.
- A. Banerjee, B. Chakraborty. Uniqueness of the power of a meromorphic functions with its differential polynomial sharing a set. Math. Morav. 20, 2 (2016), 1–14.
- A. Banerjee, B. Chakraborty. On the uniqueness of power of a meromorphic function sharing a set with its k-th derivative. J. Indian Math. Soc. (N.S.) 85, 1–2 (2018), 1–15.
- J. Chang, M. Fang, L. Zalcman. Entire functions that share a set with their derivatives. Arch. Math. (Basel) 89, 6 (2007), 561–569.
- J. Chang, L. Zalcman. Meromorphic functions that share a set with their derivatives. J. Math. Anal. Appl. 338, 2 (2008), 1020–1028.
- M. Fang, L. Zalcman. Normal families and uniqueness theorems for entire functions. J. Math. Anal. Appl. 280, 2 (2003), 73–283.
- W. K. Hayman. Meromorphic functions. Oxford Math. Monogr. Oxford, Clarendon Press, 1964.
- I. Lahiri. Weighted sharing and uniqueness of meromorphic functions. Nagoya Math. J. 161 (2001), 193–206.
- I. Lahiri, A. Sarkar. Uniqueness of a meromorphic function and its derivative. JIPAM J. Inequal. Pure Appl. Math. 5, 1 (2004), Article 20, 9 pp.
- I. Laine. Nevanlinna theory and complex differential equations. De Gruyter Stud. Math., vol. 15. Berlin, Walter de Gruyter & Co., 1993
- W. C. Lin, H.-X. Yi. Uniqueness theorems for meromorphic functions that share three sets. Complex Var. Theory Appl. 48, 4 (2003), 315–327.
- Y. Liu and J. P. Wang, F. H. Liu. Some results on value distribution of the difference operator. Bull. Iranian Math. Soc. 41, 3 (2015), 603–611.
- F. Lü. A note on meromorphic functions that share a set with their derivatives. Arch. Math. (Basel) 96, 4 (2011), 369–377.
- A. Z. Mohon’ko. The Nevanlinna characteristics of certain meromorphic functions. Teor. Funkciǐ Funkcional. Anal. i Prilozen. No. 14, (1971), 83–87 (in Russian).
- E. Mues, N. Steinmetz. Meromorphe Funktionen, die mit ihrer Ableitung Werte teilen. Manuscripta Math. 29, 2–4 (1979), 195–206.
- P. N. Raj, H. P. Waghamore. Results on uniqueness of a polynomial and difference differential polynomial. Adv. Stud. Euro-Tbil. Math. J. 16, 2 (2023), 79–96.
- L. A. Rubel, C. C. Yang. Values shared by an entire function and its derivative. In: Complex analysis (Eds J. D. Buckholtz, T. J. Suffridge) 101–103. Lecture Notes in Math., vol. 599. Berlin, Heidelberg, Springer, 1977, https://doi.org/10.1007/BFb0096830.
- C. C. Yang. H.-X. Yi. Uniqueness theory of meromorphic functions. Math. Appl., vol. 557. Dordrecht, Kluwer Academic Publishers Group, 2003.
- H.-X. Yi. Uniqueness theorems for meromorphic functions, II. Indian J. Pure Appl. Math. 28, 4 (1997), 509–519.
- H.-X. Yi. Uniqueness theorems for meromorphic functions whose nth derivatives share the same 1-points. Complex Variables Theory Appl. 34, 4 (1997), 421–436.
- H.-X. Yi. Uniqueness of meromorphic functions and a question of C. C. Yang. Complex Variables Theory Appl. 14, 1–4 (1990), 169–176.
- Q. C. Zhang. Meromorphic functions sharing three values. Indian J. Pure Appl. Math. 30, 7 (1999), 667–682.