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Keywords:
uniqueness; meromorphic function; nonlinear differential polynomial
Summary:
In the paper, dealing with a question of Lahiri (1999), we study the uniqueness of meromorphic functions in the case when two certain types of nonlinear differential polynomials, which are the derivatives of some typical linear expression, namely $h^{n}(h-1)^{m}$ ($h=f,g$), share a non-zero polynomial with finite weight. The results obtained in the paper improve, extend, supplement and generalize some recent results due to Sahoo (2013), Li and Gao (2010). In particular, we have shown that under a suitable choice of the sharing non-zero polynomial or when the first derivative is taken under consideration, better conclusions can be obtained.
References:
[1] Alzahary, T. C., Yi, H. X.: Weighted value sharing and a question of I. Lahiri. Complex Variables, Theory Appl. 49 (2004), 1063-1078. DOI 10.1080/02781070410001701074 | MR 2111304 | Zbl 1067.30055
[2] Banerjee, A.: A uniqueness result on some differential polynomials sharing 1-points. Hiroshima Math. J. 37 (2007), 397-408. MR 2376726 | Zbl 1152.30023
[3] Banerjee, A.: On uniqueness for nonlinear differential polynomials sharing the same 1-point. Ann. Pol. Math. 89 (2006), 259-272. DOI 10.4064/ap89-3-3 | MR 2262553 | Zbl 1104.30018
[4] Banerjee, A.: Meromorphic functions sharing one value. Int. J. Math. Math. Sci. 2005 (2005), 3587-3598. DOI 10.1155/IJMMS.2005.3587 | MR 2205158 | Zbl 1093.30024
[5] Fang, M., Qiu, H.: 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
[6] Frank, G.: Eine Vermutung von Hayman über Nullstellen meromorpher Funktionen. Math. Z. 149 German (1976), 29-36. MR 0422615
[7] Hayman, W. K.: Meromorphic Functions. Oxford Mathematical Monographs Clarendon Press, Oxford (1964). MR 0164038 | Zbl 0115.06203
[8] Lahiri, I.: On a question of Hong Xun Yi. Arch. Math., Brno 38 (2002), 119-128. MR 1909593 | Zbl 1087.30028
[9] 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
[10] 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
[11] Lahiri, I.: Uniqueness of meromorphic functions when two linear differential polynomials share the same 1-points. Ann. Pol. Math. 71 (1999), 113-128. MR 1703886 | Zbl 0938.30022
[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] Lahiri, I., Mandal, N.: Uniqueness of nonlinear differential polynomials sharing simple and double 1-points. Int. J. Math. Math. Sci. 2005 (2005), 1933-1942. DOI 10.1155/IJMMS.2005.1933 | MR 2176445 | Zbl 1084.30029
[14] Lahiri, I., Pal, R.: Non-linear differential polynomials sharing 1-points. Bull. Korean Math. Soc. 43 (2006), 161-168. DOI 10.4134/BKMS.2006.43.1.161 | MR 2204868
[15] Lahiri, I., Sahoo, P.: Uniqueness of non-linear differential polynomials sharing 1-points. Georgian Math. J. 12 (2005), 131-138. MR 2136891 | Zbl 1073.30022
[16] Lahiri, I., Sarkar, A.: Nonlinear differential polynomials sharing 1-points with weight two. Chin. J. Contemp. Math. 25 (2004), 325-334. MR 2159311 | Zbl 1069.30051
[17] Li, X.-M., Gao, L.: Meromorphic functions sharing a nonzero polynomial \CM. Bull. Korean Math. Soc. 47 (2010), 319-339. DOI 10.4134/BKMS.2010.47.2.319 | MR 2650701 | Zbl 1189.30066
[18] Lin, W. C.: Uniqueness of differential polynomials and a problem of Lahiri. Pure Appl. Math. 17 Chinese (2001), 104-110. MR 1848848
[19] Lin, W.-C., Yi, H.-X.: Uniqueness theorems for meromorphic function. Indian J. Pure Appl. Math. 35 (2004), 121-132. MR 2040726 | Zbl 1056.30031
[20] Meng, C.: On unicity of meromorphic functions when two differential polynomials share one value. Hiroshima Math. J. 39 (2009), 163-179. MR 2543648 | Zbl 1182.30051
[21] Qiu, H., Fang, M.: On the uniqueness of entire functions. Bull. Korean Math. Soc. 41 (2004), 109-116. DOI 10.4134/BKMS.2004.41.1.109 | MR 2036806 | Zbl 1134.30325
[22] Sahoo, P.: Meromorphic functions sharing a non zero polynomial \IM. Kyungpook Math. J. 53 (2013), 191-205. DOI 10.5666/KMJ.2013.53.2.191 | MR 3078082
[23] Sahoo, P.: Uniqueness of meromorphic functions when two differential polynomials share one value \IM. Mat. Vesn. 62 (2010), 169-182. MR 2639145 | Zbl 1289.30190
[24] 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
[25] Yang, C.-C., Hua, X.: Uniqueness and value-sharing of meromorphic functions. Ann. Acad. Sci. Fenn., Math. 22 (1997), 395-406. MR 1469799 | Zbl 0890.30019
[26] Yang, C.-C., Yi, H.-X.: Uniqueness Theory of Meromorphic Functions. Mathematics and Its Applications 557 Kluwer Academic Publishers, Dordrecht; Science Press, Beijing (2003). MR 2105668 | Zbl 1070.30011
[27] Yi, H. X.: On characteristic function of a meromorphic function and its derivative. Indian J. Math. 33 (1991), 119-133. MR 1140875 | Zbl 0799.30018
[28] Zhang, Q.: Meromorphic function that shares one small function with its derivative. J. Inequal. Pure Appl. Math. (electronic only) 6 (2005), Article No. 116, 13 pages. MR 2178297 | Zbl 1097.30033
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