Value Distribution of Differential Polynomials in Meromorphic Functions

Let f（z） be a meromorphic function and ψ be the differential polynomial of f which satisfies the condition of -↑N（r, f）＋-↑N （r, 1/f） = S（r, f）. We obtain several results about the zero point of the ψ and those results extend and improve the results of Yang and Yi in this paper.

On Meromorphic Functions That Share One Small Function of Differential Polynomials with Their Derivatives

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].

Shared Values of Certain Nonlinear Differential Polynomials of Meromorphic Functions

In this paper, by using the idea of truncated counting functions of meromorphic functions, we deal with the problem of uniqueness of the meromorphic functions whose certain nonlinear differential polynomials share one finite nonzero value.

Uniqueness of Meromorphic Functions with Their Nonlinear Differential Polynomials Share a Small Function

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 of Differential Polynomials Sharing Two Values IM

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.

Uniqueness of Meromorphic Functions Whose Differential Polynomials Share One Value

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

This article studies the problem of uniqueness of two entire or meromorphic functions whose differential polynomials share a finite set. The results extend and improve on some theorems given in [3].

Singular Directions and Differential Polynomials of Meromorphic Functions

We prove the existance of a kind of singular directions concerning the differentialpolynomials f（k）+sum from j=0 to k-1 ajfj-afn

Quasi-normal Family of Meromorphic Functions Whose Certain Type of Differential Polynomials Have No Zeros

Define the differential operators φ_(n) for n∈N inductively by φ_(1)[f](z)=f(z)and φ_(n+1)[f](z)=f(z)φ_(n)[f](z)+d/daφ_(n)[f](z).For a positive integer k≥2 and a positive number δ,let F be the family of functions f meromorphic on domain D■C such that φ_(k)[f](z)≠0 and|Res(f,a)-j|≥δ for all j∈{0,1,...,k-1}and all simple poles α of f in D.Then F is quasi-normal on D of order 1.

Generalization of Uniqueness of Meromorphic Functions Sharing Fixed Point

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].

On Homogeneous Differential Polynomials of Meromorphic Functions

In this paper, we study one conjecture proposed by W. Bergweiler and showthat any transcendental meromorphic functions f(z) have the form exp(αz + β) if f(z)f″(z) —a(f′(z))~2 7≠ 0, where a ≠ 1, (n±1)/n, n ∈ N. Moreover, an analogous normality criterion isobtained.

Real meromorphic functions and linear differential polynomials

We determine all real meromorphic functions f in the plane such that f has finitely many zeros, the poles of f have bounded multiplicities, and f and F have finitely many non-real zeros, where F is a linear differential polynomial given by F = f (k) +Σk-1j=0ajf(j) , in which k≥2 and the coefficients aj are real numbers with a0≠0.

Value-Sharing of Meromorphic Functions and Differential Polynomials

This paper is devoted to studying the relationship between meromorphic functions f（z） and g（z） when their differential polynomials satisfy sharing condition weaker than sharing one value IM.

Value Distribution of a Class of Differential Polynomials

Let f（z） be a transcendental meromorphic function in the complex plane and a ≠0 be a constant, for any positive integer m, n, k, satisfy m ≥ nk＋n＋2, ψ= f^m ＋a（f^（κ））^n has infinitely many zeros. The corresponding normal criterion also is proved.

NORMAL FAMILY OF MEROMORPHIC FUNCTIONS SHARING HOLOMORPHIC FUNCTIONS AND THE CONVERSE OF THE BLOCH PRINCIPLE

In 1996, C. C. Yang and P. C. Hu [8] showed that： Let f be a transcendental meromorphic function on the complex plane, and a ≠ 0 be a complex number; then assume that n 〉 2, n1,… , nk are nonnegative integers such that n1＋… ＋ nk ≥1; thus fn（f′）n1…（f（k））nk-a has infinitely zeros. The aim of this article is to study the value distribution of differential polynomial, which is an extension of the result of Yang and Hu for small function and all zeros of f having multiplicity at least k ≥2. Namely, we prove that fn（f′）n1…（f（k））nk-a（z） has infinitely zeros, where f is a transcendental meromorphic function on the complex plane whose all zeros have multiplicity at least k≥ 2, and a（z） 0 is a small function of f and n ≥ 2, n1,… ,nk are nonnegative integers satisfying n1＋ …＋ nk ≥1. Using it, we establish some normality criterias for a family of meromorphic functions under a condition where differential polynomials generated by the members of the family share a holomorphic function with zero points. The results of this article are supplement of some problems studied by d. Yunbo and G. Zongsheng [6], and extension of some problems studied X. Wu and Y. Xu [10]. The main result of this article also leads to a counterexample to the converse of Bloeh＇s principle.

Generalization of Uniqueness Theorems for Entire and Meromorphic Functions

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.

Normal Criteria and Shared Values by Differential Polynomials

For a family of meromorphic functions on a domain D, it is discussed whether F is normal on D if for every pair functions f(z),g∈F , f'–afnand g'–agn share value d on D when n=2,3, where a, b are two complex numbers, a≠0,∞,b≠∞.Finally, the following result is obtained:Let F be a family of meromorphic functions in D, all of whose poles have multiplicity at least 4 , all of whose zeros have multiplicity at least 2. Suppose that there exist two functions a(z) not idendtically equal to zero, d(z) analytic in D, such that for each pair of functions f and in F , f'–a(z)f2 and g'–a(z)g2 share the function d(z) . If a(z) has only a multiple zeros and f(z)≠∞ whenever a(z)=0 , then F is normal in D.

Notes on Zeros of Differential Polynomials

The value distribution of differential polynomials is studied. The results in this paper improve and generalize some previous theorems given by Yang Chungchun （On deficiencies of differential polynomials, Math. Z., 116（1970）, 197- 204）, H. S. Gopalakrishna and S. S. Bhoosnurmath （On distribution of values of differential polynomials, Indian 3. Pure Appl. Math., 17（1986）, 367-372）, I. Lahiri （A note on distribution of nonhomogeneous differential polynomials, Hokkaido Math. J., 31（2002）, 453-458） and Yi Hongxun （On zeros of differential polynomials, Adv. in Math., 18（1989）, 335-351） et al. Examples show that the results in this paper are sharu.

DIFFERENTIAL POLYNOMIALS SHARING ONE VALUE

In this paper, we investigate uniqueness problems of differential polynomials of meromorphic functions. Let a, b be non-zero constants and let n, k be positive integers satisfying n ≥ 3k ＋ 12. If f^n＋ af^（k）and g^n＋ ag^（k）share b CM and the b-points of f^n＋ af^（k）are not the zeros of f and g, then f and g are either equal or closely related.

Normality Family and Shared Functions by Meromorphic Functions and Its Differential Polynomials

In this paper, we obtain the following normal criterion： Let F be a family of mero-morphic functions in domain D belong to C, all of whose zeros have multiplicity k ＋ 1 at least. If there exist holomorphic functions α（z） not vanishing on D, such that for every function f（z） ∈F, f（z） shares α（z） IM with L（f） on D, then F is normal on D, where L（f） is linear differential polynomials of f（z） with holomorphic coefficients, and k is some positive numbers. We also proved coressponding results on normal functions.