# Article

 Title: Uniqueness of entire functions concerning difference polynomials (English) Author: Meng, Chao Language: English Journal: Mathematica Bohemica ISSN: 0862-7959 (print) ISSN: 2464-7136 (online) Volume: 139 Issue: 1 Year: 2014 Pages: 89-97 Summary lang: English . Category: math . Summary: In this paper, we investigate the uniqueness problem of difference polynomials sharing a small function. With the notions of weakly weighted sharing and relaxed weighted sharing we prove the following: Let $f(z)$ and $g(z)$ be two transcendental entire functions of finite order, and $\alpha (z)$ a small function with respect to both $f(z)$ and $g(z)$. Suppose that $c$ is a non-zero complex constant and $n\geq 7$ (or $n\geq 10$) is an integer. If $f^{n}(z)(f(z)-1)f(z+c)$ and $g^{n}(z)(g(z)-1)g(z+c)$ share “$(\alpha (z),2)$” (or $(\alpha (z),2)^{*}$), then $f(z)\equiv g(z)$. Our results extend and generalize some well known previous results. (English) Keyword: entire function Keyword: difference polynomial Keyword: uniqueness MSC: 30D35 MSC: 39A05 idZBL: Zbl 06362244 idMR: MR3231431 DOI: 10.21136/MB.2014.143638 . Date available: 2014-03-20T08:31:35Z Last updated: 2020-07-29 Stable URL: http://hdl.handle.net/10338.dmlcz/143638 . Reference: [1] Banerjee, A., Mukherjee, S.: Uniqueness of meromorphic functions concerning differential monomials sharing the same value.Bull. Math. Soc. Sci. Math. Roum., Nouv. Sér. 50 (2007), 191-206. Zbl 1164.30022, MR 2354463 Reference: [2] Chiang, Y. M., Feng, S. J.: On the Nevanlinna characteristic of $f(z+\eta)$ and difference equations in the complex plane.Ramanujan J. 16 (2008), 105-129. Zbl 1152.30024, MR 2407244, 10.1007/s11139-007-9101-1 Reference: [3] Clunie, J.: On a result of Hayman.J. Lond. Math. Soc. 42 (1967), 389-392. Zbl 0169.40801, MR 0214769, 10.1112/jlms/s1-42.1.389 Reference: [4] Fang, M. L., Hong, W.: A unicity theorem for entire functions concerning differential polynomials.Indian J. Pure Appl. Math. 32 (2001), 1343-1348. Zbl 1005.30023, MR 1875450 Reference: [5] Hayman, W. K.: Meromorphic Functions.Oxford Mathematical Monographs Clarendon, Oxford (1964). Zbl 0115.06203, MR 0164038 Reference: [6] Hayman, W. K.: Research Problems in Function Theory.University of London The Athlone Press, London (1967). Zbl 0158.06301, MR 0217268 Reference: [7] Hayman, W. K.: Picard values of meromorphic functions and their derivatives.Ann. Math. (2) 70 (1959), 9-42. Zbl 0088.28505, MR 0110807, 10.2307/1969890 Reference: [8] Lin, S. H., Lin, W. C.: Uniqueness of meromorphic functions concerning weakly weighted-sharing.Kodai Math. J. 29 (2006), 269-280. Zbl 1126.30018, MR 2247436, 10.2996/kmj/1151936441 Reference: [9] Lin, W. C., Yi, H. X.: Uniqueness theorems for meromorphic functions concerning fixed-points.Complex Variables, Theory Appl. 49 (2004), 793-806. Zbl 1067.30065, MR 2097218, 10.1080/02781070412331298624 Reference: [10] Lin, X. Q., Lin, W. C.: Uniqueness of entire functions sharing one value.Acta Math. Sci., Ser. B, Engl. Ed. 31 (2011), 1062-1076. Zbl 1240.30159, MR 2830545, 10.1016/S0252-9602(11)60298-1 Reference: [11] Wang, G., Han, D. L., Wen, Z. T.: Uniqueness theorems on difference monomials of entire functions.Abstr. Appl. Anal. 2012 ID 407351, 8 pages (2012). Zbl 1247.30047, MR 2947727 Reference: [12] 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, MR 1469799 Reference: [13] Yang, L.: Value Distribution Theory.Translated and revised from the 1982 Chinese original. Science Press, Beijing Springer, Berlin (1993). Zbl 0790.30018, MR 1301781 Reference: [14] Yi, H. X.: Meromorphic functions that share one or two values.Complex Variables, Theory Appl. 28 (1995), 1-11. Zbl 0841.30027, 10.1080/17476939508814833 Reference: [15] Zhang, J. L.: Value distribution and shared sets of differences of meromorphic functions.J. Math. Anal. Appl. 367 (2010), 401-408. Zbl 1188.30044, MR 2607267, 10.1016/j.jmaa.2010.01.038 .

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