Hostname: page-component-78c5997874-s2hrs Total loading time: 0 Render date: 2024-11-10T20:10:45.014Z Has data issue: false hasContentIssue false
  • English
  • Français

REMARKS ON VALUE SHARING OF CERTAIN DIFFERENTIAL POLYNOMIALS OF MEROMORPHIC FUNCTIONS

Part of: Entire and meromorphic functions, and related topics

Published online by Cambridge University Press:  28 August 2014

XIAO-MIN LI  and
HONG-XUN YI
XIAO-MIN LI*
Affiliation:
Department of Mathematics, Ocean University of China, Qingdao, Shandong 266100, PR China email lixiaomin@ouc.edu.cn
HONG-XUN YI
Affiliation:
Department of Mathematics, Shandong University, Jinan, Shandong 250100, PR China email hxyi@sdu.edu.cn
*
✉lixiaomin@ouc.edu.cn
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

We use Zalcman’s lemma to study a uniqueness question for meromorphic functions where certain associated nonlinear differential polynomials share a nonzero value. The results in this paper extend Theorem 1 in Yang and Hua [‘Uniqueness and value-sharing of meromorphic functions’, Ann. Acad. Sci. Fenn. Math. 22 (1997), 395–406] and Theorem 1 in Fang [‘Uniqueness and value sharing of entire functions’, Comput. Math. Appl. 44 (2002), 823–831]. Our reasoning in this paper also corrects a defect in the reasoning in the proof of Theorem 4 in Bhoosnurmath and Dyavanal [‘Uniqueness and value sharing of meromorphic functions’, Comput. Math. Appl. 53 (2007), 1191–1205].

Keywords

meromorphic functionsshared valuesdifferential polynomialsuniqueness theorems

MSC classification

Secondary: 30D35: Distribution of values, Nevanlinna theory 30D30: Meromorphic functions, general theory
Type
Research Article
Information
Bulletin of the Australian Mathematical Society , Volume 90 , Issue 3 , December 2014 , pp. 427 - 443
Copyright
Copyright © 2014 Australian Mathematical Publishing Association Inc. 

References

Bhoosnurmath, S. S. and Dyavanal, R. S., ‘Uniqueness and value sharing of meromorphic functions’, Comput. Math. Appl. 53 (2007), 1191–1205.Google Scholar
Chang, J. M. and Zalcman, L., ‘Meromorphic functions that share a set with their derivatives’, J. Math. Anal. Appl. 338 (2008), 1020–1028.CrossRefGoogle Scholar
Fang, M. L., ‘Uniqueness and value sharing of entire functions’, Comput. Math. Appl. 44 (2002), 823–831.CrossRefGoogle Scholar
Fujimoto, H., ‘On uniqueness of meromorphic functions sharing finite sets’, Amer. J. Math. 122 (2000), 1175–1203.CrossRefGoogle Scholar
Fujimoto, H., ‘On uniqueness polynomials for meromorphic functions’, Nagoya Math. J. 170 (2003), 33–46.CrossRefGoogle Scholar
Fujimoto, H., ‘Finiteness of entire functions sharing a finite set’, Nagoya Math. J. 185 (2007), 111–122.Google Scholar
Hayman, W. K., Meromorphic Functions (Clarendon Press, Oxford, 1964).Google Scholar
Lahiri, I., ‘Uniqueness of meromorphic functions as governed by their differential polynomials’, Yokohama Math. J. 44 (1997), 147–156.Google Scholar
Lahiri, I., ‘Weighted sharing of three values and uniqueness of meromorphic functions’, Kodai. Math. J. 24 (2001), 421–435.CrossRefGoogle Scholar
Laine, I., Nevanlinna Theory and Complex Differential Equations (Walter de Gruyter, Berlin, 1993).Google Scholar
Li, X. M. and Yi, H. X., ‘Uniqueness of meromorphic functions whose certain nonlinear differential polynomials share a polynomial’, Comput. Math. Appl. 62 (2011), 539–550.Google Scholar
Mokhon’ko, A. Z., ‘On the Nevanlinna characteristics of some meromorphic functions’, in: Theory of Functions, Functional Analysis and Their Applications, 14 (Izd-vo Khar’kovsk. Un-ta, 1971), 83–87.Google Scholar
Pang, X. C., ‘Normality conditions for differential polynomials’, Kexue Tongbao 33 (1988), 1690–1693 (in Chinese).Google Scholar
Whittaker, J. M., ‘The order of the derivative of a meromorphic function’, J. Lond. Math. Soc. (2) 11 (1936), 82–87.Google Scholar
Yang, C. C. and Yi, H. X., Uniqueness Theory of Meromorphic Functions (Kluwer Academic Publishers, Dordrecht, 2003).Google Scholar
Yang, L., Value Distribution Theory (Springer, Berlin, 1993).Google Scholar
Yang, C. C. and Hua, X. H., ‘Uniqueness and value-sharing of meromorphic functions’, Ann. Acad. Sci. Fenn. Math. 22 (1997), 395–406.Google Scholar
Yi, H. X., ‘Uniqueness theorems for meromorphic functions whose n-th derivatives share the same 1-points’, Complex Var. Theory Appl. 34 (1997), 421–436.Google Scholar
Zalcman, L., ‘A heuristic principle in complex function theory’, Amer. Math. Monthly 82 (1975), 813–817.Google Scholar
Zhang, Q. C., ‘Meromorphic functions sharing three values’, Indian J. Pure Appl. Math. 30 (1999), 667–682.Google Scholar