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Multiple values and uniqueness problem of meromorphic mappings sharing hypersurfaces  [PDF]
Ting-Bin Cao,Hong-Zhe Cao
Mathematics , 2013,
Abstract: The purpose of this article is to deal with the multiple values and uniqueness problem of meromorphic mappings from $\mathbb{C}^{m}$ into the complex projective space $\mathbb{P}^{n}(\mathbb{C})$ sharing fixed and moving hypersurfaces. We obtain several uniqueness theorems which improve and extend some known results.
Uniqueness of Meromorphic Functions Sharing Fixed Point  [PDF]
Subhas S. Bhoosnurmath,Veena L. Pujari
International Journal of Analysis , 2013, DOI: 10.1155/2013/538027
Abstract: We study the uniqueness of meromorphic functions concerning differential polynomials sharing fixed point and obtain some significant results , which improve the results due to Lin and Yi (2004). 1. Introduction and Main Results Let be a nonconstant meromorphic function in the whole complex plane . We will use the following standard notations of value distribution theory: , (see [1, 2]). We denote by any function satisfying possibly outside of a set with finite measure. Let be a finite complex number and a positive integer. We denote by the counting function for the zeros of in with multiplicity and by the corresponding one for which multiplicity is not counted. Let be the counting function for the zeros of in with multiplicity and the corresponding one for which multiplicity is not counted. Set Let be a nonconstant meromorphic function. We denote by the counting function for -points of both and about which has larger multiplicity than , where multiplicity is not counted. Similarly, we have notation . We say that and share CM (counting multiplicity) if and have same zeros with the same multiplicities. Similarly, we say that and share IM (ignoring multiplicity) if and have same zeros with ignoring multiplicities. In 2004, Lin and Yi [3] obtained the following results. Theorem A. Let and be two transcendental meromorphic functions, an integer. If and share CM, then either or where is a nonconstant meromorphic function. Theorem B. Let and be two transcendental meromorphic functions, an integer. If and share CM, then . In this paper, we study the uniqueness problems of entire or meromorphic functions concerning differential polynomials sharing fixed point, which improves Theorems A and B. 1.1. Main Results Theorem 1. Let and be two nonconstant meromorphic functions, a positive integer. If and share CM, and share IM, then either or where is a nonconstant meromorphic function. Theorem 2. Let and be two nonconstant meromorphic functions, a positive integer. If and share CM, and share IM, then . Theorem 3. Let and be two nonconstant entire functions, an integer. If and share CM, then . 2. Some Lemmas Lemma 4 (see [4]). Let , , and be nonconstant meromorphic functions such that . If , , and are linearly independent, then where and . Lemma 5 (see [1]). Let and be two nonconstant meromorphic functions. If , where , , and are non-zero constants, then Lemmas 4 and 5 play a very important role in proving our theorems. Lemma 6 (see [1]). Let be a nonconstant meromorphic function and let be a nonnegative integer, then The following lemmas play a cardinal role in proving
Generalization of Uniqueness of Meromorphic Functions Sharing Fixed Point  [PDF]
Harina P. Waghamore, Sangeetha Anand
Applied Mathematics (AM) , 2016, DOI: 10.4236/am.2016.79084
Abstract: In this paper, we study the uniqueness problems of entire and meromorphic functions concerning differential polynomials sharing fixed point and obtain some results which generalize the results due to Subhas S. Bhoosnurmath and Veena L. Pujari [1].
Weakly Weighted Sharing and Uniqueness of Meromorphic Functions  [PDF]
Thamir Alzahary
Journal of Complex Analysis , 2014, DOI: 10.1155/2014/378958
Abstract: With the aid of the notion of weakly weighted sharing, we study the uniqueness of meromorphic functions sharing four pairs of small functions. Our results improve and generalize some results given by Czubiak and Gundersen, Li and Yang, and other authors. 1. Introduction and Main Results In this paper, a meromorphic function means meromorphic in the whole complex plane . We assume the reader is familiar with the standard notion used in the Nevanlinna value distribution theory such as , , , and (see [1, 2]). For any nonconstant meromorphic function , the term denotes any quantity that satisfies as outside a possible exceptional set of finite linear measure. Let be a nonconstant meromorphic function. A meromorphic function is called a small function of , if . If is a positive integer, we denote by the reduced counting function of the poles of whose multiplicities are less than or equal to and denote by the reduced counting function of the poles of whose multiplicities are greater than or equal to . Let and be nonconstant meromorphic functions, and let , be two values in . We say that and share the value IM provided that and have the same zeros ignoring multiplicities. In addition, we say that and share the value IM, if and share IM. We say that and share the pair of values IM provided that and have the same zeros ignoring multiplicities. The following theorem is a well-known and significant result in the uniqueness theory of meromorphic functions and has been proved by Czubiak and Gundersen. Theorem A (see [3]). Let and be two nonconstant meromorphic functions that share six pairs of values , IM, where whenever and whenever . Then is a M?bius transformation of . The following example, found by Gundersen, shows that the number “six” in Theorem A cannot be replaced with “five.” Example 1 (see [4]). Let , . We see that , share , , , and IM, and is not a M?bius transformation of . Let and be nonconstant meromorphic functions and let be two small meromorphic functions of and . We denote by the reduced counting function of the common zeros of and . We say that and share , if As in Theorem A and throughout this paper, when and are nonconstant meromorphic functions, we let denote the term which is both and simultaneously. We denote by the reduced counting function of those -points of , which are not the -points of . We note that and share if and only if and . According to this note, we generalize the definitions of IM and to the weakly weighted IM sharing which is given by the following definition. Definition 2 (see [5]). Let be a positive integer or infinity, and
Uniqueness of Transcendental Meromorphic Functions with Their Nonlinear Differential Polynomials Sharing the Small Function  [PDF]
Hong-Yan Xu
International Journal of Mathematics and Mathematical Sciences , 2007, DOI: 10.1155/2007/84961
Abstract: We deal with some uniqueness theorems of two transcendental meromorphic functions with their nonlinear differential polynomials sharing a small function. These results in this paper improve those given by C.-Y. Fang and M.-L. Fang (2002), by Lahiri and Pal (2006), and by Lin and Yi (2004).
Uniqueness of Meromorphic Functions and Differential Polynomials  [PDF]
Jin-Dong Li
International Journal of Mathematics and Mathematical Sciences , 2011, DOI: 10.1155/2011/514218
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 [4] proved the following result. Theorem A (see [4]). Let and be two nonconstant entire functions, and let (≥8) be a positive integer. If and share 1 CM, then . Fang [5] proved the following result. Theorem B (see [5]). Let and be two nonconstant entire functions, and let , be two positive integers with . If and share 1 CM, then . In [6], for some general differential polynomials such as , Liu proved the following result. Theorem C (see [6]). 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 [7] and Bhoosnurmath and Dyavanal [8] generalized the above results and obtained the following results. Theorem D (see [7]). Let and be two nonconstant meromorphic functions with , and let be a positive integer. If and share 1 CM, then . Theorem E (see [8]). Let and be two
Meromorphic Functions Sharing Three Values  [PDF]
Changjun Li, Limei Wang
Applied Mathematics (AM) , 2011, DOI: 10.4236/am.2011.26095
Abstract: In this paper, we prove a result on the uniqueness of meromorphic functions sharing three values counting multiplicity and improve a result obtained by Xiaomin Li and Hongxun Yi.
亚纯函数微分多项式分担两个值的唯一性
Uniqueness of Meromorphic Function of Differential Polynomials Sharing Two Values
 [PDF]

石新华
Pure Mathematics (PM) , 2014, DOI: 10.12677/PM.2014.42011
Abstract:
本文运用Nevanlinna理论讨论了一类亚纯函数微分多项式分担两个值的唯一性问题,得到的结果改进或推广了亚纯函数分担值的许多唯一性结果。
We will use Nevanlinna distribution theory to discuss a power of meromorphic functions of differential polynomials sharing two values. The results generalize many results on value sharing of meromorphic functions.
一类线性差分方程的亚纯解与一个亚纯函数分担3个值的唯一性
Uniqueness for Meromorphic Solutions Sharing Three Values with a Meromorphic Function to Some Linear Difference Equations
 [PDF]

崔宁,陈宗煊
- , 2017, DOI: 10.16205/j.cnki.cama.2017.0002
Abstract: 主要研究差分方程~$a_{1}(z)f(z+1)+a_{0}(z)f(z)=F(z)$~的一个有穷级超越亚纯解 $f(z)$~与亚纯函数~$g(z)$~分担~$0, 1, \infty$~CM~时的唯一性问题~(其中~$a_{1}(z)$, $a_{0}(z), F(z)$~为非零多项式, 且满足~$a_{1}(z)+a_{0}(z)\not\equiv0$), 得到~$f(z)\equiv g(z)$, 或$f(z)+g(z)\equiv f(z)g(z)$, 或存在一个多项式 $\beta(z)=az+b_{0}$~和一个常数~$a_{0}$~满足~$\rme^{a_{0}}\neq \rme^{b_{0}}$,~使得 $f(z)=\frac{1-\rme^{\beta(z)}}{\rme^{\beta(z)}(\rme^{a_{0}-b_{0}}-1)}$~与 $g(z)=\frac{1-\rme^{\beta(z)}}{1-\rme^{b_{0}-a_{0}}}$, 其中~$a(\neq0), b_{0}$~为常数.
This paper deals with the uniqueness of a finite-order meromorphic solution $f(z)$ of some linear difference equation $a_{1}(z)f(z+1)+a_{0}(z)f(z)=F(z)$ sharing $0, 1, \infty$ CM with meromorphic function $g(z)$ (where $a_{1}(z)$, $a_{0}(z)$ and $ F(z)$ are nonzero polynomials satisfying $a_{1}(z)+a_{0}(z)\not\equiv0$), and obtain either $f(z)\equiv g(z)$ or $f(z)+g(z)\equiv f(z)g(z)$ or there exists a polynomial $\beta(z)=az+b_{0}$ and a constant $a_{0}$ satisfying $\rme^{a_{0}}\neq \rme^{b_{0}}$, such that $f(z)=\frac{1-\rme^{\beta(z)}}{\rme^{\beta(z)}(\rme^{a_{0}-b_{0}}-1)}$ and $g(z)=\frac{1-\rme^{\beta(z)}}{1-\rme^{b_{0}-a_{0}}}$, where $a(\neq0), b_{0}$ are constants.
权分担一个值的亚纯函数的唯一性
UNIQUENESS ON MEROMORPHIC FUNCTIONS SHARING ONE VALUE WITH WEIGHT
 [PDF]

作者,李延玲,刘慧芳
- , 2015,
Abstract: 本文研究了亚纯函数关于微分多项式权分担一个值的唯一性问题.利用权分担值的思想,获得了一个亚纯函数唯一性定理,完善了Bhoosnurmath和Dyavanal所得的结果.
In this paper, we study the uniqueness problem of meromorphic functions concerning their differential polynomials sharing one value with weight. By using the weighted sharing method, we obtain one uniqueness theorem of meromorphic functions, which improves the result given by Bhoosnurmath and Dyavanal
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