Mathematica Bohemica, Vol. 143, No. 3, pp. 251-275, 2018
Further investigation "On an open problem of Zhang and Xu"
Sujoy Majumder
Received February 3, 2016. First published October 18, 2017.
Abstract: The purpose of the paper is to study the uniqueness of meromorphic functions sharing a nonzero polynomial. The result of the paper improves and generalizes the recent results due to X. B. Zhang and J. F. Xu (2011). We also solve an open problem posed in the last section of X. B. Zhang and J. F. Xu (2011).
Keywords: uniqueness; meromorphic function; small function; nonlinear differential polynomial
References: [1] A. Banerjee: On a question of Gross. J. Math. Anal. Appl. 327 (2007), 1273-1283. DOI 10.1016/j.jmaa.2006.04.078 | MR 2280003 | Zbl 1115.30029 [2] A. Banerjee: Uniqueness of certain non-linear differential polynomials sharing $1$-points. Kyungpook Math. J. 51 (2011), 43-58. DOI 10.5666/KMJ.2011.51.1.043 | MR 2784645 | Zbl 1218.30073 [3] M. Fang, H. Qiu: Meromorphic functions that share fixed-points. J. Math. Anal. Appl. 268 (2002), 426-439. DOI 10.1006/jmaa.2000.7270 | MR 1896207 | Zbl 1030.30028 [4] G. Frank: Eine Vermutung von Hayman über Nullstlellen meromorpher Funktionen. Math. Z. 149 (1976), 29-36. DOI 10.1007/BF01301627 | MR 0422615 | Zbl 0312.30032 [5] W. K. Hayman: Picard values of meromorphic functions and their derivatives. Ann. Math. (2) 70 (1959), 9-42. DOI 10.2307/1969890 | MR 0110807 | Zbl 0088.28505 [6] W. K. Hayman: Meromorphic Functions. Oxford Mathematical Monographs. Clarendon Press, Oxford (1964). MR 0164038 | Zbl 0115.06203 [7] I. Lahiri: Value distribution of certain differential polynomials. Int. J. Math. Math. Sci. 28 (2001), 83-91. DOI 10.1155/S0161171201011036 | MR 1885054 | Zbl 0999.30023 [8] I. Lahiri: Weighted sharing and uniqueness of meromorphic functions. Nagoya Math. J. 161 (2001), 193-206. MR 1820218 | Zbl 0981.30023 [9] I. Lahiri: Weighted value sharing and uniqueness of meromorphic functions. Complex Variables, Theory Appl. 46 (2001), 241-253. MR 1869738 | Zbl 1025.30027 [10] I. Lahiri, A. Banerjee: Weighted sharing of two sets. Kyungpook Math. J. 46 (2006), 79-87. MR 2214802 | Zbl 1103.30017 [11] I. Lahiri, S. Dewan: Value distribution of the product of a meromorphic function and its derivative. Kodai Math. J. 26 (2003), 95-100. DOI 10.2996/kmj/1050496651 | MR 1966685 | Zbl 1077.30025 [12] S. Majumder: On an open problem of Xiao-Bin Zhang and Jun-Feng Xu. Demonstr. Math. 49 (2016), 161-182. DOI 10.1515/dema-2016-0015 | MR 350793 | Zbl 1338.30022 [13] J.-F. Xu, F. Lü, H.-X. Yi: Fixed points and uniqueness of meromorphic functions. Comput. Math. Appl. 59 (2010), 9-17. DOI 10.1016/j.camwa.2009.07.024 | MR 2575487 | Zbl 1189.30069 [14] K. Yamanoi: The second main theorem for small functions and related problems. Acta Math. 192 (2004), 225-294. DOI 10.1007/BF02392741 | MR 2096455 | Zbl 1203.30035 [15] C. C. Yang: On deficiencies of differential polynomials II. Math. Z. 125 (1972), 107-112. DOI 10.1007/BF01110921 | MR 0294642 | Zbl 0217.38402 [16] C.-C. Yang, X. Hua: Uniqueness and value-sharing of meromorphic functions. Ann. Acad. Sci. Fenn. Math. 22 (1997), 395-406. MR 1469799 | Zbl 0890.30019 [17] C.-C. Yang, H.-X. Yi: Uniqueness Theory of Meromorphic Functions. Mathematics and Its Applications 557. Kluwer Academic Publishers, Dordrecht; Science Press, Beijing (2003). MR 2105668 | Zbl 1070.30011 [18] H.-X. Yi: Meromorphic functions that share one or two values II. Kodai Math. J. 22 (1999), 264-272. DOI 10.2996/kmj/1138044046 | MR 1700596 | Zbl 0939.30020 [19] Q. Zhang: Meromorphic function that shares one small function with its derivative. JIPAM, J. Inequal. Pure Appl. Math. (electronic only) 6 (2005), 13 pages. Article No. 116. MR 2178297 | Zbl 1097.30033 [20] X.-B. Zhang, J.-F. Xu: Uniqueness of meromorphic functions sharing a small function and its applications. Comput. Math. Appl. 61 (2011), 722-730. DOI 10.1016/j.camwa.2010.12.022 | MR 2764068 | Zbl 1217.30031
Affiliations: Sujoy Majumder, Department of Mathematics, Katwa College, University of Burdwan, Katwa College Rd., Katwa, Burdwan, C.P. 713130, West Bengal, India; Present Address: Department of Mathematics, Raiganj University, Raiganj, West Bengal, 733134, India, e-mail: sujoy.katwa@gmail.com, sr.koshigram@gmail.com
