Article
Full entry |
PDF
(0.2 MB)
Feedback
PDF
(0.2 MB)
Feedback
Keywords:
entire function; difference polynomial; uniqueness
entire function; difference polynomial; uniqueness
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.
References:
[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. MR 2354463 | Zbl 1164.30022
[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. DOI 10.1007/s11139-007-9101-1 | MR 2407244 | Zbl 1152.30024
[3] Clunie, J.: On a result of Hayman. J. Lond. Math. Soc. 42 (1967), 389-392. DOI 10.1112/jlms/s1-42.1.389 | MR 0214769 | Zbl 0169.40801
[4] Fang, M. L., Hong, W.: A unicity theorem for entire functions concerning differential polynomials. Indian J. Pure Appl. Math. 32 (2001), 1343-1348. MR 1875450 | Zbl 1005.30023
[5] Hayman, W. K.: Meromorphic Functions. Oxford Mathematical Monographs Clarendon, Oxford (1964). MR 0164038 | Zbl 0115.06203
[6] Hayman, W. K.: Research Problems in Function Theory. University of London The Athlone Press, London (1967). MR 0217268 | Zbl 0158.06301
[7] Hayman, W. K.: Picard values of meromorphic functions and their derivatives. Ann. Math. (2) 70 (1959), 9-42. DOI 10.2307/1969890 | MR 0110807 | Zbl 0088.28505
[8] Lin, S. H., Lin, W. C.: Uniqueness of meromorphic functions concerning weakly weighted-sharing. Kodai Math. J. 29 (2006), 269-280. DOI 10.2996/kmj/1151936441 | MR 2247436 | Zbl 1126.30018
[9] Lin, W. C., Yi, H. X.: 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
[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. DOI 10.1016/S0252-9602(11)60298-1 | MR 2830545 | Zbl 1240.30159
[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). MR 2947727 | Zbl 1247.30047
[12] Yang, C. C., Hua, X. H.: Uniqueness and value-sharing of meromorphic functions. Ann. Acad. Sci. Fenn., Math. 22 (1997), 395-406. MR 1469799 | Zbl 0890.30019
[13] Yang, L.: Value Distribution Theory. Translated and revised from the 1982 Chinese original. Science Press, Beijing Springer, Berlin (1993). MR 1301781 | Zbl 0790.30018
[14] Yi, H. X.: Meromorphic functions that share one or two values. Complex Variables, Theory Appl. 28 (1995), 1-11. DOI 10.1080/17476939508814833 | Zbl 0841.30027
[15] Zhang, J. L.: Value distribution and shared sets of differences of meromorphic functions. J. Math. Anal. Appl. 367 (2010), 401-408. DOI 10.1016/j.jmaa.2010.01.038 | MR 2607267 | Zbl 1188.30044
Search
Partner of
