Browse > Article
http://dx.doi.org/10.4134/JKMS.2011.48.3.499

THE BRÜCK CONJECTURE AND ENTIRE FUNCTIONS SHARING POLYNOMIALS WITH THEIR κ-TH DERIVATIVES  

Lu, Feng (School of Mathematics and Computational Science China University of Petroleum)
Yi, Hongxun (Department of Mathematics Shandong University)
Publication Information
Journal of the Korean Mathematical Society / v.48, no.3, 2011 , pp. 499-512 More about this Journal
Abstract
The purpose of this paper is twofold. The first is to establish a uniqueness theorem for entire function sharing two polynomials with its ${\kappa}$-th derivative, by using the theory of normal families. Meanwhile, the theorem generalizes some related results of Rubel and Yang and of Li and Yi. Several examples are provided to show the conditions are necessary. The second is to generalize the Br$\"{u}$-ck conjecture with the idea of sharing polynomial.
Keywords
Vandermonde determinant; entire functions; Nevanlinna theory; uniqueness; normal family; differential equation;
Citations & Related Records

Times Cited By Web Of Science : 0  (Related Records In Web of Science)
Times Cited By SCOPUS : 0
연도 인용수 순위
  • Reference
1 J. L. Zhang, Researches on Bruck Conjecture and Fermat Diophantine Equations over function fields, Doctoral Dissertation (2008), 16-18.
2 J. L. Zhang and L. Z. Yang, Some results related to a conjecture of R. Bruck concerning meromorphic functions sharing one small function with their derivatives, Ann. Acad. Sci. Fenn. Math. 32 (2007), no. 1, 141-149.
3 J. L. Zhang and L. Z. Yang, A power of a meromorphic function sharing a small function with its derivative, Ann. Acad. Sci. Fenn. Math. 34 (2009), no. 1, 249-260.
4 W. Hayman, Meromorphic Functions, Clarendon Press, Oxford, 1964.
5 X. J. Liu, S. Nevo, and X. C. Pang, On the kth derivative of meromorphic functions with zeros of multiplicity at least k+1, J. Math. Anal. Appl. 348 (2008), no. 1, 516-529.   DOI   ScienceOn
6 F. Lu, J. F. Xu, and A. Chen, Entire functions sharing polynomials with their first derivatives, Arch. Math. (Basel) 92 (2009), no. 6, 593-601.   DOI   ScienceOn
7 J. T. Li and H. X. Yi, Normal families and uniqueness of entire functions and their derivatives, Arch. Math. (Basel) 87 (2006), no. 1, 52-59.   DOI   ScienceOn
8 X. M. Li and C. C. Gao, Entire functions sharing one polynomial with their derivatives, Proc. Indian Acad. Sci. Math. Sci. 118 (2008), no. 1, 13-26.   DOI
9 E. Mues and N. Steinmetz, Meromorphe Funktionen, die mit ihrer Ableitung Werte teilen, Manuscripta Math. 29 (1979), no. 2-4, 195-206.   DOI
10 L. A. Rubel and C. C. Yang, Values shared by an entire function and its derivative, Complex analysis (Proc. Conf., Univ. Kentucky, Lexington, Ky., 1976), pp. 101-103. Lecture Notes in Math., Vol. 599, Springer, Berlin, 1977.   DOI
11 J. Wang and H. X. Yi, The uniqueness of entire functions that share a small function with its differential polynomials, Indian J. Pure Appl. Math. 35 (2004), no. 9, 1119-1129.
12 G. G. Gundersen and L. Z. Yang, Entire functions that share one value with one or two of their derivatives, J. Math. Anal. Appl. 223 (1998), no. 1, 88-95.   DOI   ScienceOn
13 C. C. Yang and H. X. Yi, Uniqueness Theory of Meromorphic Functions, Mathematics and its Applications, 557. Kluwer Academic Publishers Group, Dordrecht, 2003.
14 L. Z. Yang and J. L. Zhang, Non-existence of meromorphic solutions of a Fermat type functional equation, Aequationes Math. 76 (2008), no. 1-2, 140-150.   DOI
15 L. Zalcman, A heuristic principle in complex function theory, Amer. Math. Monthly 82 (1975), no. 8, 813-817.   DOI   ScienceOn
16 R. Bruck, On entire functions which share one value CM with their first derivative, Results Math. 30 (1996), no. 1-2, 21-24.   DOI
17 Z. X. Chen and K. H. Shon, On conjecture of R. Bruck concerning the entire function sharing one value CM with its derivative, Taiwanese J. Math. 8 (2004), no. 2, 235-244.   DOI
18 J. Grahl and C. Meng, Entire functions sharing a polynomial with their derivatives and normal families, Analysis (Munich) 28 (2008), no. 1, 51-61.