scholarly journals Some Results on the Deficiencies of Some Differential-Difference Polynomials of Meromorphic Function

Mathematics ◽  
2018 ◽  
Vol 6 (10) ◽  
pp. 207
Author(s):  
Hong-Yan Xu ◽  
Xiu-Min Zheng ◽  
Hua Wang
Keyword(s):  
Meromorphic Function ◽  
Transcendental Meromorphic Function ◽  
Difference Polynomials ◽  
Nevanlinna Deficiency

For a transcendental meromorphic function f ( z ) , the main aim of this paper is to investigate the properties on the zeros and deficiencies of some differential-difference polynomials. Some results about the deficiencies of some differential-difference polynomials concerning Nevanlinna deficiency and Valiron deficiency are obtained, which are a generalization of and improvement on previous theorems given by Liu, Lan and Zheng, etc.

scholarly journals Fixed Points of Difference Operator of Meromorphic Functions

2014 ◽  
Vol 2014 ◽  
pp. 1-4
Author(s):  
Zhaojun Wu ◽  
Hongyan Xu
Keyword(s):  
Meromorphic Function ◽  
Fixed Points ◽  
Meromorphic Functions ◽  
Difference Operator ◽  
Transcendental Meromorphic Function ◽  
Exact Difference ◽  
Borel Exceptional Values ◽  
Nevanlinna Deficiency

Letfbe a transcendental meromorphic function of order less than one. The authors prove that the exact differenceΔf=fz+1-fzhas infinitely many fixed points, ifa∈ℂand∞are Borel exceptional values (or Nevanlinna deficiency values) off. These results extend the related results obtained by Chen and Shon.

scholarly journals Uniqueness of difference polynomials

2021 ◽  
Vol 6 (10) ◽  
pp. 10485-10494
Author(s):  
Xiaomei Zhang ◽  
◽  
Xiang Chen ◽  
Keyword(s):  
Meromorphic Function ◽  
Finite Order ◽  
Complex Numbers ◽  
Nonzero Constant ◽  
Transcendental Meromorphic Function ◽  
Difference Polynomial ◽  
Difference Polynomials

<abstract><p>Let $ f(z) $ be a transcendental meromorphic function of finite order and $ c\in\Bbb{C} $ be a nonzero constant. For any $ n\in\Bbb{N}^{+} $, suppose that $ P(z, f) $ is a difference polynomial in $ f(z) $ such as $ P(z, f) = a_{n}f(z+nc)+a_{n-1}f(z+(n-1)c)+\cdots+a_{1}f(z+c)+a_{0}f(z) $, where $ a_{k} (k = 0, 1, 2, \cdots, n) $ are not all zero complex numbers. In this paper, the authors investigate the uniqueness problems of $ P(z, f) $.</p></abstract>

Power of meromorphic function sharing polynomials with derivative of it’s combination with it’s shift

2019 ◽  
Vol 69 (5) ◽  
pp. 1037-1052
Author(s):  
Sujoy Majumder ◽  
Somnath Saha
Keyword(s):  
Meromorphic Function ◽  
Transcendental Meromorphic Function ◽  
Function Sharing

Abstract In this paper we consider the situation when a power of a transcendental meromorphic function shares non-zero polynomials with derivative of it’s combination with it’s shift. Also we exhibit some examples to fortify the conditions of our results.

scholarly journals Value distribution of certain monomials of algebroid functions

1997 ◽  
Vol 62 (3) ◽  
pp. 398-404
Author(s):  
Kari Katajamäki
Keyword(s):  
Meromorphic Function ◽  
Value Distribution ◽  
Algebroid Function ◽  
Transcendental Meromorphic Function ◽  
Algebroid Functions

AbstractHayman has shown that if f is a transcendental meromorphic function and n ≽ 3, then fn f′ assumes all finite values except possibly zero infinitely often. We extend his result in three directions by considering an algebroid function ω, its monomial ωn0 ω′n1, and by estimating the growth of the number of α-points of the monomial.

scholarly journals The Zeros of Difference Polynomials of Meromorphic Functions

2012 ◽  
Vol 2012 ◽  
pp. 1-13
Author(s):  
Junfeng Xu ◽  
Xiaobin Zhang
Keyword(s):  
Meromorphic Function ◽  
Meromorphic Functions ◽  
Differential Polynomials ◽  
Difference Polynomials

We investigate the value distributions of difference polynomialsΔf(z)-af(z)nandf(z)nf(z+c)which related to two well-known differential polynomials, wheref(z)is a meromorphic function.

scholarly journals A note on a result of Singh and Kulkarni

2000 ◽  
Vol 23 (4) ◽  
pp. 285-288 ◽  
Author(s):  
Mingliang Fang
Keyword(s):  
Meromorphic Function ◽  
Finite Order ◽  
Transcendental Meromorphic Function

We prove that iffis a transcendental meromorphic function of finite order and∑a≠∞δ(a,f)+δ(∞,f)=2, thenK(f(k))=2k(1−δ(∞,f))1+k−kδ(∞,f), whereK(f(k))=limr→∞N(r,1/f(k))+N(r,f(k))T(r,f(k))This result improves a result by Singh and Kulkarni.

Normality and fixpoints of analytic functions

2003 ◽  
Vol 133 (6) ◽  
pp. 1335-1339 ◽  
Author(s):  
J. D. Hinchliffe
Keyword(s):  
Meromorphic Function ◽  
Analytic Functions ◽  
Transcendental Meromorphic Function ◽  
The Family

Let D be a domain in C and let f be a transcendental meromorphic function on C such that C* \f(C) = ∅, {∞} or {α, β}, where α and β are two distinct values in C* = C ∪ {∞}. Then the family of functions g that are analytic on D and such that f ∘ g has no fixpoints on D is normal.

On the value distribution of f2f(k)

2005 ◽  
Vol 78 (1) ◽  
pp. 17-26 ◽  
Author(s):  
Xiaojun Huang ◽  
Yongxing Gu
Keyword(s):  
Positive Integer ◽  
Meromorphic Function ◽  
Complex Plane ◽  
Meromorphic Functions ◽  
Value Distribution ◽  
Multiple Zeros ◽  
Transcendental Meromorphic Function

AbstractIn this paper, we prove that for a transcendental meromorphic function f(z) on the complex plane, the inequality T(r, f) < 6N (r, 1/(f2 f(k)−1)) + S(r, f) holds, where k is a positive integer. Moreover, we prove the following normality criterion: Let ℱ be a family of meromorphic functions on a domain D and let k be a positive integer. If for each ℱ ∈ ℱ, all zeros of ℱ are of multiplicity at least k, and f2 f(k) ≠ 1 for z ∈ D, then ℱ is normal in the domain D. At the same time we also show that the condition on multiple zeros of f in the normality criterion is necessary.

Fatou Components and a Problem of Bergweiler

1998 ◽  
Vol 08 (07) ◽  
pp. 1613-1616
Author(s):  
Xinhou Hua ◽  
Chung-Chun Yang
Keyword(s):  
Meromorphic Function ◽  
Fatou Set ◽  
Transcendental Meromorphic Function ◽  
Fatou Components

For any transcendental meromorphic function, we shall study the relations between the components of the Fatou set and their images. This relates to a problem studied by Bergweiler [1993].

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