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Keywords:
normal families; uniqueness; meromorphic function; small functions
Summary:
The motivation of this paper is to study the uniqueness of meromorphic functions sharing a nonzero polynomial with the help of the idea of normal family. The result of the paper improves and generalizes the recent result due to Zhang and Xu [24]. Our another remarkable aim is to solve an open problem as posed in the last section of [24].
References:
[1] Banerjee, A.: On a question of Gross. J. Math. Anal. Appl. 327 (2) (2007), 1273–1283. DOI 10.1016/j.jmaa.2006.04.078 | MR 2280003 | Zbl 1115.30029
[2] Banerjee, A.: Uniqueness of certain nonlinear differential polynomials sharing 1-points. Kyungpook Math. J. 51 (2011), 43–58. DOI 10.5666/KMJ.2011.51.1.043 | MR 2784645
[3] Chang, J.M., Zalcman, L.: Meromorphic functions that share a set with their derivatives. J. Math. Anal. Appl. 338 (2008), 1191–1205. DOI 10.1016/j.jmaa.2007.05.079 | MR 2386477
[4] Fang, M.L., Qiu, H.L.: 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
[5] Frank, G.: Eine Vermutung Von Hayman über Nullslellen meromorphic Funktion. Math. Z. 149 (1976), 29–36. DOI 10.1007/BF01301627
[6] Hayman, W.K.: Picard values of meromorphic functions and their derivatives. Ann. of Math. (2) 70 (1959). Zbl 0088.28505
[7] Hayman, W.K.: Meromorphic Functions. The Clarendon Press, Oxford, 1964. Zbl 0115.06203
[8] Lahir, I., Banerjee, A.: Weighted sharing of two sets. Kyungpook Math. J. 46 (2006). MR 2214802
[9] Lahiri, I.: Value distribution of certain differential polynomials. Internat. J. Math. Math. Sci. 28 (2001), 83–91. DOI 10.1155/S0161171201011036 | MR 1885054 | Zbl 0999.30023
[10] Lahiri, I.: Weighted sharing and uniqueness of meromorphic functions. Nagoya Math. J. 161 (2001), 193–206. DOI 10.1017/S0027763000027215 | MR 1820218 | Zbl 0981.30023
[11] Lahiri, I.: Weighted value sharing and uniqueness of meromorphic functions. Complex Variables Theory Appl. 46 (2001), 241–253. DOI 10.1080/17476930108815411 | MR 1869738 | Zbl 1025.30027
[12] Lahiri, I., Dewan, S.: 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
[13] Majumder, S.: On an open problem of Xiao-Bin Zhang and Jun-Feng Xu. Demonstratio Math. 49 (2) (2016), 161–182. MR 3507931
[14] Pang, X.C.: Normality conditions for differential polynomials. Kexue Tongbao (Chinese) 33 (1988), 1690–1693.
[15] Schiff, J.: Normal families. Berlin, 1993.
[16] Xu, J.F., Lü, F., Yi, H.X.: Fixed points and uniqueness of meromorphic functions. Comput. Math. Appl. 59 (2010), 9–17. DOI 10.1016/j.camwa.2009.07.024 | MR 2575487
[17] Yamanoi, K.: 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
[18] Yang, C.C.: On deficiencies of differential polynomials II. Math. Z. 125 (1972), 107–112. DOI 10.1007/BF01110921 | Zbl 0217.38402
[19] Yang, C.C., Hua, X.H.: Uniqueness and value sharing of meromorphic functions. Ann. Acad. Sci. Fenn. Math. 22 (1997), 395–406. Zbl 0890.30019
[20] Yang, C.C., Yi, H.X.: Uniqueness Theory of Meromorphic Functions. Kluwer Academic Publishers, Dordrecht/Boston/London, 2003. MR 2105668 | Zbl 1070.30011
[21] Zalcman, L.: A heuristic principle in complex theory. Amer. Math. Monthly 82 (1975), 813–817. DOI 10.1080/00029890.1975.11993942
[22] Zalcman, L.: Normal families, new perspectives. Bull. Amer. Math. Soc. 35 (1998), 215–230. DOI 10.1090/S0273-0979-98-00755-1
[23] Zhang, Q.C.: Meromorphic function that shares one small function with its derivative. J. Inequal. Pure Appl. Math. 6 (4) (2005), Art.116 online http://jipam.vu.edu.au/ MR 2178297 | Zbl 1097.30033
[24] Zhang, X.B., Xu, J.F.: 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
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