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UNIQUENESS OF MEROMORPHIC OR ENTIRE FUNCTIONS AND THEIR DIFFERENTIAL POLYNOMIALS
Li Jiangtao,Li Chunghong,
李江涛
,李纯红

数学物理学报(A辑) , 2006,
Abstract: This article studies the problem of uniqueness of two entire or meromorphic functions whose differential polynomials share a finite set. The results extendand improve on some theorems given in 3].
On Meromorphic Functions That Share One Small Function of Differential Polynomials with Their Derivatives  [PDF]
Harina P. Waghamore, S. Rajeshwari
Applied Mathematics (AM) , 2015, DOI: 10.4236/am.2015.612178
Abstract: In this paper, we study the problem of meromorphic functions that share one small function of differential polynomial with their derivatives and prove one theorem. The theorem improves the results of Jin-Dong Li and Guang-Xin Huang [1].
Uniqueness of Meromorphic Functions with Their Nonlinear Differential Polynomials Share a Small Function  [PDF]
Harina P. Waghamore, Tanuja Adaviswamy
Applied Mathematics (AM) , 2011, DOI: 10.4236/am.2011.211191
Abstract: In this paper we deal with the uniqueness of meromorphic functions when two nonlinear differential polynomials generated by two meromorphic functions share a small function. We consider the case for some general differential polynomials [fnP(f)f,] where P(f) is a polynomial which generalize some result due to Abhijit Banerjee and Sonali Mukherjee [1].
Uniqueness of Meromorphic Functions Whose Differential Polynomials Share One Value  [PDF]
Jin Tao, Xinli Wang
Journal of Applied Mathematics and Physics (JAMP) , 2018, DOI: 10.4236/jamp.2018.611188
Abstract: In this paper, we prove a uniqueness theorem of meromorphic functions whose some nonlinear differential shares 1 IM with powers of the meromorphic functions, where the degrees of the powers are equal to those of the nonlinear differential polynomials. This result improves the corresponding one given by Zhang and Yang, and other authors.
Generalization of Uniqueness Theorems for Entire and Meromorphic Functions  [PDF]
Harina P. Waghamore, N. Shilpa
Applied Mathematics (AM) , 2014, DOI: 10.4236/am.2014.58118
Abstract:

In this paper, we deal with the uniqueness problems on entire and meromorphic functions con- cerning differential polynomials that share fixed-points. Moreover, we generalise and improve some results of Weichuan Lin, Hongxun Yi, Meng Chao, C. Y. Fang, M. L. Fang and Junfeng xu.

On the uniqueness problems of entire functions and their linear differential polynomials  [PDF]
Qi Han,Hongxun Yi
Mathematics , 2014,
Abstract: The uniqueness problems on transcendental meromorphic or entire functions sharing at least two values with their derivatives or linear differential polynomials have been studied and many results have been obtained. In this paper, we study a transcendental entire function f that shares a non-zero polynomial a with f', together with its linear differential polynomials of the form with rational function coefficients.
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
具有权弱分担值的差分或微分多项式的亚纯函数的唯一性
The uniqueness of meromorphic functions of the differential or differential polynomial with the weakly weighed sharing value
 [PDF]

林珊华,林伟川
福州大学学报(自然科学版) , 2018, DOI: 10.7631/issn.1000-2243.16270
Abstract: 利用权弱分担的定义以及唯一性理论方法讨论亚纯函数的k阶导数与差分或微分多项式的k阶导数分担值的问题. 分析结果表明,当分担“(1,m)”且Θ(∞,f)介于一定范围的情况下,两个函数的k阶导数相等,其中对应不同的m值,n,m需满足不同的不等式.
We mainly use the definition of the weakly weighted sharing and the theory of uniqueness to discuss the problem that the k derivative of meromorphic function and the k derivative of the differential or differential polynomial share one value. The result is that k derivative of the two functions are equal when they share “(1,m)” and Θ(∞,f) is in a certain range,and different m value needs to satisfy different inequalities
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
Meromorphic functions share three values with their difference operators  [PDF]
Feng Lü,Weiran Lü
Mathematics , 2015,
Abstract: In the work, we focus on a conjecture due to Z.X. Chen and H.X. Yi[1] which is concerning the uniqueness problem of meromorphic functions share three distinct values with their difference operators. We prove that the conjecture is right for meromorphic function of finite order. Meanwhile, a result of J. Zhang and L.W. Liao[10] is generalized from entire functions to meromorphic functions.
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