Subsequent investigations of the least cardinalities of unique range set for two minimum weights over a non-Archimedean field
Abstract
UDC 517.53
First of all, we indicate a severe error in the analysis of the main results of both Chakraborty [Ukr. Math. J., 72, No. 11, 1794–1806 (2021)] and Chakraborty–Chakraborty [Ukr. Math. J., 72, No. 7, 1164–1174 (2020)], to show that both these papers cease to be true. Further, pertinent to the results of these two papers, we deal with the unique range set of a meromorphic function over a non-Archimedean field with the smallest possible weights 0 and 1 under the aegis of its most generalized form to improve the existing result.
References
V. H. An, H. H. Khoai, Value sharing problems for p-adic meromorphic functions and their difference polynomials, Ukranian Math. J., 64, No. 2, 147–164 (2012). DOI: https://doi.org/10.1007/s11253-012-0636-y
V. H. An, P. N. Hoa, H. H. Khoai, Value sharing problems for differential and difference polynomials of meromorphic function in a non-Archimedean field, p-Adic Number, Ultrametric Anal. and Appl., 9, No. 1, 1–14 (2017). DOI: https://doi.org/10.1134/S2070046617010010
A. Banerjee, Fujimoto's theorem – a further study, J. Contemp. Math. Anal., 51, No. 4, 199–207 (2016). DOI: https://doi.org/10.3103/S1068362316040051
A. Banerjee, Uniqueness of meromorphic functions sharing two sets with finite weight II, Tamkang J. Math., 41, No. 4, 379–392 (2010). DOI: https://doi.org/10.5556/j.tkjm.41.2010.787
A. Banerjee, I. Lahiri, A uniqueness polynomial generating a unique range set and vise versa, Comput. Methods Funct. Theory, 12, No. 2, 527–539 (2012). DOI: https://doi.org/10.1007/BF03321842
S. Bartels, Meromorphic functions sharing a set with 17 elements ignoring multiplicities, Complex Var. and Elliptic Equat., 39, No. 1, 85–92 (1999). DOI: https://doi.org/10.1080/17476939908815183
K. Boussaf, A. Escassut, J. Ojeda, P-adic meromorphic functions $f'P'(f),$ $g'P'(g)$ sharing a small function, Bull. Sci. Math., 136, 72–200 (2012).
B. Chakraborty, On the cardinality of a reduced unique range set, Ukranian Math. J., 72, No. 11, 1794–1806 (2021). DOI: https://doi.org/10.1007/s11253-021-01889-z
B. Chakraborty, S. Chakraborty, On the cardinality of unique range sets with weight one, Ukranian Math. J., 72, No. 7, 1164–1174 (2020). DOI: https://doi.org/10.1007/s11253-020-01849-z
G. Frank, M. Reinders, A unique range set for meromorphic functions with 11 elements, Complex Var., Theory and Appl., 37, No. 1, 185–193 (1998). DOI: https://doi.org/10.1080/17476939808815132
H. Fujimoto, On uniqueness of meromorphic functions sharing finite sets, Amer. J. Math., 122, 1175–1203 (2000). DOI: https://doi.org/10.1353/ajm.2000.0045
W. K. Hayman, Meromorphic functions, Clarendon Press, Oxford (1964).
P. C. Hu, C. C. Yang, Meromorphic functions over non-Archimedean fields, Kluwer Acad. Publ. (2000).
H. H. Khoai, V. H. An, URS and biURS for meromorphic functions in a non-Archimedean field, $p$-Adic Number, Ultrametric Anal. and Appl., 12, No. 4, 276–284 (2020). DOI: https://doi.org/10.1134/S2070046620040020
I. Lahiri, Weighted sharing and uniqueness of meromorphic functions, Nagoya Math. J., 161, 193–206 (2001). DOI: https://doi.org/10.1017/S0027763000027215
P. Li, Uniqueness and value sharing of meromorphic functions, Thesis, The Hong Kong Univ. Sci. Tech. (1996).
P. Li, C. C. Yang, Some further results on the unique range sets of meromorphic functions, Kodai Math. J., 18, 437–450 (1995). DOI: https://doi.org/10.2996/kmj/1138043482
C. Meng, G. Liu, Uniqueness for the difference monomials of $p$-adic entire functions, Tbilisi Math. J., 11, No. 2, 67–76 (2018). DOI: https://doi.org/10.32513/tbilisi/1529460023
J. T-Y. Wang, Uniqueness polynomials and bi-unique range sets for rational functions and non-Archimedean meromorphic functions, Acta Arith., 104, No. 2, 183–200 (2002). DOI: https://doi.org/10.4064/aa104-2-7
C. C. Yang, H. X. Yi, Uniqueness theory of meromorphic functions, Sci. Press and Kluwer Acad. Publ. (2003). DOI: https://doi.org/10.1007/978-94-017-3626-8
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