Two meromorphic functions are said to share a set S ⊂ ℂ∪{∞} ignoring multiplicities (IM) if S has the same preimages under both functions. If any two nonconstant meromorphic functions sharing a set IM are identical, then the set is called a “reduced unique-range set for meromorphic functions” [RURSM (or URSM-IM)]. From the existing literature, it is known that there exists a RURSM with 17 elements. We reduce the cardinality of the existing RURSM and show that there exists a RURSM with 15 elements. Our result gives an affirmative answer to the question of L. Z. Yang [Int. Soc. Anal. Appl. Comput., 7, 551–564 (2000)].
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
S. Bartels, “Meromorphic functions sharing a set with 17 elements ignoring multiplicities,” Complex Variables Theory Appl., 39, 85–92 (1999).
M. L. Fang and H. Guo, “On unique range sets for meromorphic or entire functions,” Acta Math. Sinica (N.S.), 14, No. 4, 569–576 (1998).
G. Frank and M. Reinders, “A unique range set for meromorphic function with 11 elements,” Complex Variables Theory Appl., 37, 185–193 (1998).
H. Fujimoto, “On uniqueness of meromorphic functions sharing finite sets,” Amer. J. Math., 122, No. 6, 1175–1203 (2000).
H. Fujimoto, “On uniqueness polynomials for meromorphic functions,” Nagoya Math. J., 170, 33–46 (2003).
F. Gross, “Factorization of meromorphic functions and some open problems,” in: Complex analysis (Proc. Conf., Univ. Kentucky, Lexington, Ky., 1976), Lecture Notes in Math., Vol. 599, Springer, Berlin (1977), pp. 51–67.
F. Gross and C. C. Yang, “On preimage and range sets of meromorphic functions,” Proc. Jap. Acad. Ser. A, Math. Sci., 58, 17–20 (1982).
W. K. Hayman, Meromorphic Functions, Clarendon Press, Oxford (1964).
P. C. Hu, P. Li, and C. C. Yang, Unicity of Meromorphic Mappings, Springer; https://doi.org/10.1007/978-1-4757-3775-2.
P. Li and C. C. Yang, “Some further results on the unique range set of meromorphic functions,” Kodai Math. J., 18, 437–450 (1995).
P. Li and C. C. Yang, “On the unique range set for meromorphic functions,” Proc. Amer. Math. Soc., 124, 177–185 (1996).
M. Reinders, “Unique range sets ignoring multiplicities,” Bull. Hong Kong Math. Soc., 1, No. 2, 339–341 (1997).
C. C. Yang and H. X. Yi, Uniqueness Theory of Meromorphic Functions, Kluwer Academic Publishers, Dordrecht (2003).
L. Z. Yang, “Some recent progress in the uniqueness theory of meromorphic functions,” in: Proc. of the Second ISAAC Congr., Vol. 1 (Fukuoka, 1999), Int. Soc. Anal. Appl. Comput., 7, Kluwer Acad. Publ., Dordrecht (2000), pp. 551–564.
H. X. Yi, “Unicity theorems for meromorphic and entire functions III,” Bull. Austral. Math. Soc., 53, 71–82 (1996).
H. X. Yi, “The reduced unique range sets for entire or meromorphic functions,” Complex Variables Theory Appl., 32, 191–198 (1997).
H. X. Yi, “The reduced range sets for meromorphic functions,” Shandong Daxue Xuebao Ziran Kexue Ban, 33, No. 4, 361–368 (1998).
Author information
Authors and Affiliations
Corresponding author
Additional information
Published in Ukrains’kyi Matematychnyi Zhurnal, Vol. 72, No. 11, pp. 1553–1563, November, 2020. Ukrainian DOI: 10.37863/umzh.v72i11.594.
Rights and permissions
About this article
Cite this article
Chakraborty, B. On the Cardinality of a Reduced Unique-Range Set. Ukr Math J 72, 1794–1806 (2021). https://doi.org/10.1007/s11253-021-01889-z
Received:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s11253-021-01889-z