Home OALib Journal OALib PrePrints Submit Ranking News My Lib FAQ About Us Follow Us+ All Title Author Keywords Abstract
 Publish in OALib Journal ISSN: 2333-9721 APC: Only \$99

 Relative Articles UNIQUENESS OF MEROMORPHIC OR ENTIRE FUNCTIONS AND THEIR DIFFERENTIAL POLYNOMIALS Uniqueness of Transcendental Meromorphic Functions with Their Nonlinear Differential Polynomials Sharing the Small Function Uniqueness of Meromorphic Functions of Differential Polynomials Sharing Two Values IM Uniqueness of Meromorphic Functions with Their Nonlinear Differential Polynomials Share a Small Function Uniqueness of Meromorphic Functions Whose Differential Polynomials Share One Value Uniqueness of meromorphic functions concerning differential polynomials Uniqueness of Some Differential Polynomials of Meromorphic Functions On the uniqueness problems of entire functions and their linear differential polynomials Zeros of differential polynomials in real meromorphic functions Uniqueness of Meromorphic Functions Concerning Differential Monomials More...

# Uniqueness of Meromorphic Functions and Differential Polynomials

 Full-Text   Cite this paper

Abstract:

We study the uniqueness of meromorphic functions and differential polynomials sharing one value with weight and prove two main theorems which generalize and improve some results earlier given by M. L. Fang, S. S. Bhoosnurmath and R. S. Dyavanal, and so forth. 1. Introduction and Results Let be a nonconstant meromorphic function defined in the whole complex plane . It is assumed that the reader is familiar with the notations of the Nevanlinna theory such as , , , and , that can be found, for instance, in [1–3]. Let and be two nonconstant meromorphic functions. Let be a finite complex number. We say that and share the value CM (counting multiplicities) if and have the same zeros with the same multiplicities, and we say that and share the value IM (ignoring multiplicities) if we do not consider the multiplicities. When and share 1 IM, let be a 1-point of of order and a 1-points of of order ; we denote by the counting function of those 1-points of and , where and by the counting function of those 1-points of and , where . is the counting function of those 1-points of both and , where . In the same way, we can define , , and . If and share 1 IM, it is easy to see that Let be a nonconstant meromorphic function. Let be a finite complex number and a positive integer; we denote by (or ) the counting function for zeros of with multiplicity (ignoring multiplicities) and by (or ) the counting function for zeros of with multiplicity at least (ignoring multiplicities). Set We further define In 2002, C. Y. Fang and M. L. Fang  proved the following result. Theorem A (see ). Let and be two nonconstant entire functions, and let (≥8) be a positive integer. If and share 1 CM, then . Fang  proved the following result. Theorem B (see ). Let and be two nonconstant entire functions, and let , be two positive integers with . If and share 1 CM, then . In , for some general differential polynomials such as , Liu proved the following result. Theorem C (see ). Let and be two nonconstant entire functions, and let be three positive integers such that . If and share 1 IM, then either or and satisfy the algebraic equation , where . The following example shows that Theorem A is not valid when and are two meromorphic functions. Example 1.1. Let , , where . Then and share 1 CM, but . Lin and Yi  and Bhoosnurmath and Dyavanal  generalized the above results and obtained the following results. Theorem D (see ). Let and be two nonconstant meromorphic functions with , and let be a positive integer. If and share 1 CM, then . Theorem E (see ). Let and be two

References

  W. K. Hayman, Meromorphic Functions, Oxford Mathematical Monographs, Clarendon Press, Oxford, UK, 1964.  L. Yang, Value Distribution Theory, Springer, Berlin, Germany, 1993.  H. X. Yi and C. C. Yang, Uniqueness Theory of Meromorphic Functions, Science Press, Beijing, China, 1995.  C.-Y. Fang and M.-L. Fang, “Uniqueness of meromorphic functions and differential polynomials,” Computers & Mathematics with Applications, vol. 44, no. 5-6, pp. 607–617, 2002.  M.-L. Fang, “Uniqueness and value-sharing of entire functions,” Computers & Mathematics with Applications, vol. 44, no. 5-6, pp. 823–831, 2002.  L. Liu, “Uniqueness of meromorphic functions and differential polynomials,” Computers & Mathematics with Applications, vol. 56, no. 12, pp. 3236–3245, 2008.  W.-C. Lin and H.-X. Yi, “Uniqueness theorems for meromorphic function,” Indian Journal of Pure and Applied Mathematics, vol. 35, no. 2, pp. 121–132, 2004.  S. S. Bhoosnurmath and R. S. Dyavanal, “Uniqueness and value-sharing of meromorphic functions,” Computers & Mathematics with Applications, vol. 53, no. 8, pp. 1191–1205, 2007.  I. Lahiri, “Weighted sharing and uniqueness of meromorphic functions,” Nagoya Mathematical Journal, vol. 161, pp. 193–206, 2001.  H.-Y. Xu, C.-F. Yi, and T.-B. Cao, “Uniqueness of meromorphic functions and differential polynomials sharing one value with finite weight,” Annales Polonici Mathematici, vol. 95, no. 1, pp. 51–66, 2009.  T. Zhang and W. Lü, “Uniqueness theorems on meromorphic functions sharing one value,” Computers & Mathematics with Applications, vol. 55, no. 12, pp. 2981–2992, 2008.

Full-Text