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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.
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].
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
Zeros of differential polynomials in real meromorphic functions  [PDF]
W. Bergweiler,A. Eremenko,J. Langley
Mathematics , 2004,
Abstract: We show that for a real transcendental meromorphic function f, the differential polynomial f'+f^m with m > 4 has infinitely many non-real zeros. Similar results are obtained for differential polynomials f'f^m-1. We specially investigate the case of meromorphic functions with finitely many poles. We show by examples the precision of our results. One of our main tools is the Fatou theorem from complex dynamics.
On Entire and Meromorphic Functions That Share One Small Function with Their Differential Polynomial  [PDF]
Subhas S. Bhoosnurmath,Smita R. Kabbur
International Journal of Analysis , 2013, DOI: 10.1155/2013/926340
Abstract: We study the uniqueness of meromorphic functions that share one small function with more general differential polynomial . As corollaries, we obtain results which answer open questions posed by Yu (2003). 1. Introduction and Main Results In this paper, a meromorphic functions mean meromorphic in the whole complex plane. We use the standard notations of Nevanlinna theory (see [1]). A meromorphic function is called a small function with respect to if , that is, as possibly outside a set of finite linear measure. If and have the same zeros with same multiplicities (ignoring multiplicities), then we say that and share ?CM (IM). For any constant , we denote by the counting function for zeros of with multiplicity no more than and the corresponding for which multiplicity is not counted. Let be the counting function for zeros of with multiplicity at least and the corresponding for which the multiplicity is not counted. Let and be two nonconstant meromorphic functions sharing value 1?IM. Let be common one point of and with multiplicity and , respectively. We denote by ?? the counting (reduced) function of those 1 points of where ; by the counting function of those 1-points of where ; by the counting function of those 1-points of where . In the same way, we can define , and (see [2]). In 1996, Brück [3] posed the following conjecture. Conjecture 1. Let be a nonconstant entire function such that the hyper-order of is not a positive integer and . If and share a finite value ?CM, then , where is a nonzero constant. In [3], under an additional hypothesis, Brück proved that the conjecture holds when . Theorem A. Let be a nonconstant entire function. If and share the value 1?CM and if , then , for some constant . Many people extended this theorem and obtained many results. In 2003, Yu [4] proved the following theorem. Theorem B. Let . Let be a nonconstant meromorphic function and a meromorphic function such that , and do not have any common pole and as . If and share the value 0?CM and then . Theorem C. Let . Let be a nonconstant entire function and be a meromorphic function such that and as . If and share the value 0?CM and then . In the same paper, the author posed the following questions. Question 1. Can ?CM shared value be replaced by an IM shared value in Theorem C? Question 2. Is the condition sharp in Theorem C? Question 3. Is the condition sharp in Theorem B? In 2004, Liu and Gu [5] applied different method and obtained the following theorem which answers some questions posed in [4]. Theorem D. Let . Let be a nonconstant meromorphic function and a meromorphic
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).
Value Distribution of Meromorphic Solutions and Their Derivatives of Complex Differential Equations  [PDF]
Abdallah El Farissi
ISRN Mathematical Analysis , 2013, DOI: 10.1155/2013/497921
Abstract: We deal with the relationship between the small functions and the derivatives of solutions of higher-order linear differential equations , ?? , where ?? are meromorphic functions. The theorems of this paper improve the previous results given by El Farissi, Bela?di, Wang, Lu, Liu, and Zhang. 1. Introduction and Statement of Result Throughout this paper, we assume that the reader is familiar with the fundamental results and the standard notations of Nevanlinna's value distribution theory (see [1, 2]). In addition, we will use and to denote, respectively, the exponents of convergence of the zero sequence and the pole sequence of a meromorphic function , to denote the order of growth of , to denote the type of the entire function with , and and to denote, respectively, the exponents of convergence of the sequence of distinct zeros and distinct poles of . A meromorphic function is called a small function of a meromorphic function if as , where is the Nevanlinna characteristic function of . In order to express the rate of growth of meromorphic solutions of infinite order, we recall the following definitions. Definition 1 (see [2–4]). Let be a meromorphic function, and let , such that , be the sequence of the fixed points of , with each point being repeated only once. The exponent of convergence of the sequence of distinct fixed points of is defined by the following: Clearly, where is the counting function of distinct fixed points of in . Definition 2 (see [4–6]). Let be a meromorphic function. Then the hyperorder of is defined by the following: Definition 3 (see [4, 5]). Let be a meromorphic function. Then the hyperexponent of convergence of the sequence of distinct zeros of is defined by the following: where is the counting function of distinct zeros of in . For , we consider the following linear differential equation: where is a transcendental meromorphic function of finite order . Many important results have been obtained on the fixed points of general transcendental meromorphic functions for almost four decades (see [7]). However, there are a few studies on the fixed points of solutions of differential equations. In [8], Wang and Lü have investigated the fixed points and hyperorder of solutions of second-order linear differential equations with meromorphic coefficients and their derivatives, and they have obtained the following result. Theorem A (see [8]). Suppose that is a transcendental meromorphic function satisfying , . Then, every meromorphic solution of the equation satisfies that and , all have infinitely many fixed points and Theorem A has been
Uniqueness of Meromorphic Functions of Differential Polynomials Sharing Two Values IM  [PDF]
Xinhua Shi
Advances in Pure Mathematics (APM) , 2014, DOI: 10.4236/apm.2014.42005
Abstract:

In this paper, we shall study the uniqueness problems of meromorphic functions of differential polynomials sharing two values IM. Our results improve or generalize many previous results on value sharing of meromorphic functions.

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.
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