Abstract
The purpose of this paper is to study the problems of meromorphic functions sharing unique range sets with one or two elements, which can be seen as a new answer to a question due to Gross. The examples also illustrate the accuracy of all of the conditions.
Funding source: Natural Science Foundation of Fujian Province
Award Identifier / Grant number: 2018J01658
Funding statement: The project was supported by the Natural Science Foundation of Fujian Province, China (Grant no. 2018J01658).
References
[1] J.-F. Chen, Uniqueness of meromorphic functions sharing two finite sets, Open Math. 15 (2017), no. 1, 1244–1250. 10.1515/math-2017-0102Search in Google Scholar
[2] J. B. Conway, Functions of one Complex Variable, Grad. Texts in Math. 11, Springer, New York, 1973. 10.1007/978-1-4615-9972-2Search in Google Scholar
[3] W. Doeringer, Exceptional values of differential polynomials, Pacific J. Math. 98 (1982), no. 1, 55–62. 10.2140/pjm.1982.98.55Search in Google Scholar
[4] M. Fang and W. Xu, On the uniqueness of entire functions, Bull. Malaysian Math. Soc. (2) 19 (1996), no. 1, 29–37. Search in Google Scholar
[5] M.-L. Fang and H. Guo, On meromorphic functions sharing two values, Analysis 17 (1997), no. 4, 355–366. 10.1524/anly.1997.17.4.355Search in Google Scholar
[6] G. Frank and M. Reinders, A unique range set for meromorphic functions with 11 elements, Complex Variables Theory Appl. 37 (1998), no. 1–4, 185–193. 10.1080/17476939808815132Search in Google Scholar
[7] H. Fujimoto, On uniqueness of meromorphic functions sharing finite sets, Amer. J. Math. 122 (2000), no. 6, 1175–1203. 10.1007/978-1-4613-0269-8_34Search in Google Scholar
[8] F. Gross, Factorization of meromorphic functions and some open problems, Complex Analysis (Lexington 1976), Lecture Notes in Math. 599, Springer, Berlin (1977), 51–67. 10.1007/BFb0096825Search in Google Scholar
[9] W. K. Hayman, Meromorphic Functions, Oxford Math. Monogr., Clarendon Press, Oxford, 1964. Search in Google Scholar
[10] P. Li and C.-C. Yang, Some further results on the unique range sets of meromorphic functions, Kodai Math. J. 18 (1995), no. 3, 437–450. 10.2996/kmj/1138043482Search in Google Scholar
[11] P. Li and C.-C. Yang, On the unique range set of meromorphic functions, Proc. Amer. Math. Soc. 124 (1996), no. 1, 177–185. 10.1090/S0002-9939-96-03045-6Search in Google Scholar
[12] E. Mues and M. Reinders, Meromorphic functions sharing one value and unique range sets, Kodai Math. J. 18 (1995), no. 3, 515–522. 10.2996/kmj/1138043489Search in Google Scholar
[13] C.-C. Yang and H.-X. Yi, Uniqueness Theory of Meromorphic Functions, Math. Appl. 557, Kluwer Academicp, Dordrecht, 2003. 10.1007/978-94-017-3626-8Search in Google Scholar
[14] L.-Z. Yang, Meromorphic functions that share two values, J. Math. Anal. Appl. 209 (1997), no. 2, 542–550. 10.1006/jmaa.1997.5329Search in Google Scholar
[15] B. Yi and Y. H. Li, Uniqueness of meromorphic functions that share two sets with CM, Acta Math. Sinica (Chin. Ser.) 55 (2012), no. 2, 363–368. Search in Google Scholar
[16] H. X. Yi, Uniqueness of meromorphic functions and question of Gross, Sci. China Ser. A 37 (1994), no. 7, 802–813. Search in Google Scholar
[17] H. X. Yi, Unicity theorems for meromorphic or entire functions. II, Bull. Aust. Math. Soc. 52 (1995), no. 2, 215–224. 10.1017/S0004972700014635Search in Google Scholar
[18] H.-X. Yi, On a question of Gross concerning uniqueness of entire functions, Bull. Aust. Math. Soc. 57 (1998), no. 2, 343–349. 10.1017/S0004972700031701Search in Google Scholar
[19] H. X. Yi, Meromorphic functions that share two sets, Acta Math. Sinica (Chin. Ser.) 45 (2002), no. 1, 75–82. 10.2996/kmj/1138043751Search in Google Scholar
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