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# Article

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Keywords:
uniqueness; meromorphic function; differential polynomial; weighted sharing
Summary:
Let $k$ be a nonnegative integer or infinity. For $a\in \mathbb {C}\cup \{\infty \}$ we denote by $E_{k}(a;f)$ the set of all $a$-points of $f$ where an $a$-point of multiplicity $m$ is counted $m$ times if $m\leq k$ and $k+1$ times if $m>k$. If $E_{k}(a;f)= E_{k}(a;g)$ then we say that $f$ and $g$ share the value $a$ with weight $k$. Using this idea of sharing values we study the uniqueness of meromorphic functions whose certain nonlinear differential polynomials share a nonzero polynomial with finite weight. The results of the paper improve and generalize the related results due to Xia and Xu (2011) and the results of Li and Yi (2011).
References:
[1] Banerjee, A.: 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
[2] 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
[3] Bhoosnurmath, S. S., Dyavanal, R. S.: Uniqueness and value-sharing of meromorphic functions. Comput. Math. Appl. 53 (2007), 1191-1205. DOI 10.1016/j.camwa.2006.08.045 | MR 2327673 | Zbl 1170.30011
[4] Fang, M.-L.: Uniqueness and value-sharing of entire functions. Comput. Math. Appl. 44 (2002), 823-831. DOI 10.1016/S0898-1221(02)00194-3 | MR 1925824 | Zbl 1035.30017
[5] Fang, C.-Y., Fang, M.-L.: Uniqueness of meromorphic functions and differential polynomials. Comput. Math. Appl. 44 (2002), 607-617. DOI 10.1016/S0898-1221(02)00175-X | MR 1925805 | Zbl 1035.30018
[6] Hayman, W. K.: Meromorphic Functions. Oxford Mathematical Monographs Clarendon Press, Oxford (1964). MR 0164038 | Zbl 0115.06203
[7] Lahiri, I.: On a question of Hong Xun Yi. Arch. Math., Brno 38 (2002), 119-128. MR 1909593 | Zbl 1087.30028
[8] Lahiri, I.: Value distribution of certain differential polynomials. Int. J. Math. Math. Sci. 28 (2001), 83-91. DOI 10.1155/S0161171201011036 | MR 1885054 | Zbl 0999.30023
[9] 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
[10] Lahiri, I.: Uniqueness of meromorphic functions as governed by their differential polynomials. Yokohama Math. J. 44 (1997), 147-156. MR 1453358 | Zbl 0884.30023
[11] 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
[12] Li, X.-M., Yi, H.-X.: Uniqueness of meromorphic functions whose certain nonlinear differential polynomials share a polynomial. Comput. Math. Appl. 62 (2011), 539-550. DOI 10.1016/j.camwa.2011.04.005 | MR 2817891 | Zbl 1228.30024
[13] Lin, W., Yi, H.: Uniqueness theorems for meromorphic functions concerning fixed-points. Complex Variables, Theory Appl. 49 (2004), 793-806. DOI 10.1080/02781070412331298624 | MR 2097218 | Zbl 1067.30065
[14] Sahoo, P.: Uniqueness and weighted sharing of meromorphic functions. Ann. Pol. Math. 100 (2011), 127-145. DOI 10.4064/ap100-2-3 | MR 2747044 | Zbl 1233.30020
[15] Sahoo, P.: Meromorphic functions that share fixed points with finite weights. Bull. Math. Anal. Appl. (electronic only) 2 (2010), 106-118. MR 2747893 | Zbl 1312.30049
[16] Xia, J., Xu, Y.: Uniqueness and differential polynomials of meromorphic functions sharing one value. Filomat 25 (2011), 185-194. DOI 10.2298/FIL1101185X | MR 2932783 | Zbl 1265.30161
[17] Yang, C. C.: On deficiencies of differential polynomials, II. Math. Z. 125 (1972), 107-112. DOI 10.1007/BF01110921 | MR 0294642 | Zbl 0217.38402
[18] 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
[19] Zhang, J.-L., Yang, L.-Z.: Some results related to a conjecture of R. Brück. JIPAM, J. Inequal. Pure Appl. Math. (electronic only) 8 (2007), Article No. 18, 11 pages. MR 2295712 | Zbl 1136.30009

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