Unique range sets for powers of meromorphic functions II

S. Mallick; Md T. Anwar

doi:10.30970/ms.53.2.134-146

S. Mallick Department of Mathematics Cooch Behar Panchanan Barma University West Bengal, India
Md T. Anwar Department of Mathematics Cooch Behar Panchanan Barma University West Bengal, India

DOI: https://doi.org/10.30970/ms.53.2.134-146

Keywords: Meromorphic function, entire function, uniqueness, unique range set, reduced unique range set.

Abstract

In this paper, we deal the problems of unique range sets in the light of powers of meromorphic functions and hence answer an open problem posed by Mallick-Sarkar in [18]. We also show the sharpness of our results exhibiting two counter examples. Moreover, our results improve the result of Mallick-Sarkar [18].

References

A. Banerjee, Uniqueness of meromorphic functions sharing two sets with finite weight II, Tamkang J. Math., 41 (2010), No4, 379–392.

A. Banerjee, Fujimoto’s theorem — a further study, J. Contemp. Math. Anal., 51 (2016), No4, 199–207.

A. Banerjee, I. Lahiri, A uniqueness polynomial generating a unique range set and vise versa, Comput. Methods Funct. Theory, 12 (2012), No2, 527–539.

A. Banerjee, S. Mallick, On the characterisations of a new class of strong uniqueness polynomials generating unique range sets, Comput. Methods Funct. Theory, 17 (2017), 19–45.

S. Bartels, Meromorphic functions sharing a set with 17 elements ignoring multiplicities, Complex Var. Elliptic Equ., 39 (1998), 85–92.

G. Frank, M. Reinders, A unique range set for meromorphic functions with 11 elements, Complex Var. Elliptic Equ., 37 (1998), No1, 185–193.

H. Fujimoto, On uniqueness of meromorphic functions sharing finite sets, Amer. J. Math., 122 (2000), 1175–1203.

F. Gross, Factorization of meromorphic functions and some open problems, Proc. Conf. Univ. Kentucky, Leixngton, Kentucky (1976); Lecture Notes in Math., 599 (1977), 51-69, Springer (Berlin).

F. Gross, C.C. Yang, On preimage and range sets of meromorphic functions, Proc. Japan Acad. Ser. A Math. Sci., 58 (1982), No1, 17–20.

W.K. Hayman, Meromorphic functions, The Clarendon Press, Oxford, 1964.

H.H. Khoai, V.H. An, N.X. Lai, Strong uniqueness polynomials of degree 6 and unique range sets for powers of meromorphic functions, Internat. J. Math., 29 (2018), No5, 1850037-1–1850037-19.

I. Lahiri, Value distribution of certain differential polynomials, Int. J. Math. Math. Sci., 28 (2001), No2, 83–91.

I. Lahiri, Weighted sharing and uniqueness of meromorphic functions, Nagoya Math. J., 161 (2001), 193–206.

I. Lahiri, Weighted value sharing and uniqueness of meromorphic functions, Complex Var. Elliptic Equ., 46 (2001), 241–253.

S. Mallick, D. Sarkar, Unique range sets for powers of meromorphic functions, Mat. Stud., 50 (2018), No2, 143–157.

C.C. Yang, H.X. Yi, Uniqueness theory of meromorphic functions, Kluwer Academic Publishers, 2003.

H.X. Yi, A question of Gross and the uniqueness of entire functions, Nagoya Math. J., 138 (1995), 169–177.

H.X. Yi, Meromorphic functions that share one or two values II, Kodai Math. J., 22 (1999), 264–272.

H.X. Yi, W.R. Lü, Meromorphic functions that share two sets II, Acta Math. Sci. Ser. B Engl. Ed., 24 (2004), No1, 83–90.