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 Results on the deficiencies of some differential-difference polynomials of meromorphic functions

2016 ◽  
Vol 14 (1) ◽  
pp. 100-108 ◽  
Author(s):  
Xiu-Min Zheng ◽  
Hong-Yan Xu
Keyword(s):  
Meromorphic Function ◽  
Finite Order ◽  
Meromorphic Functions ◽  
Value Distribution ◽  
Small Function ◽  
Complex Constant ◽  
Difference Polynomials

Abstract In this paper, we study the relation between the deficiencies concerning a meromorphic function f(z), its derivative f′(z) and differential-difference monomials f(z)mf(z+c)f′(z), f(z+c)nf′(z), f(z)mf(z+c). The main results of this paper are listed as follows: Let f(z) be a meromorphic function of finite order satisfying $$\mathop {\lim \,{\rm sup}}\limits_{r \to + \infty } {{T(r,\,f)} \over {T(r,\,f')}}{\rm{ < }} + \infty ,$$ and c be a non-zero complex constant, then δ(∞, f(z)m f(z+c)f′(z))≥δ(∞, f′) and δ(∞,f(z+c)nf′(z))≥ δ(∞, f′). We also investigate the value distribution of some differential-difference polynomials taking small function a(z) with respect to f(z).

scholarly journals Zeros and Value Sharing Results for q-Shifts Difference and Differential Polynomials

2018 ◽  
Author(s):  
Rajshree Dhar
Keyword(s):  
Meromorphic Functions ◽  
Zero Order ◽  
Differential Polynomials ◽  
Common Value ◽  
Value Sharing ◽  
Difference Polynomials

In this paper, we consider the zero distributions of q-shift monomi-als and difference polynomials of meromorphic functions with zero order, that extends the classical Hayman results on the zeros of differential poly-nomials to q-shift difference polynomials. We also investigate problem of q-shift difference polynomials that share a common value.

scholarly journals Value Distribution for a Class of Small Functions in the Unit Disk

2011 ◽  
Vol 2011 ◽  
pp. 1-24
Author(s):  
Paul A. Gunsul
Keyword(s):  
Positive Integer ◽  
Meromorphic Function ◽  
Unit Disk ◽  
Nevanlinna Theory ◽  
Meromorphic Functions ◽  
Finite Measure ◽  
Differential Polynomials ◽  
Proximity Function ◽  
Small Functions ◽  
Linear Differential

If is a meromorphic function in the complex plane, R. Nevanlinna noted that its characteristic function could be used to categorize according to its rate of growth as . Later H. Milloux showed for a transcendental meromorphic function in the plane that for each positive integer , as , possibly outside a set of finite measure where denotes the proximity function of Nevanlinna theory. If is a meromorphic function in the unit disk , analogous results to the previous equation exist when . In this paper, we consider the class of meromorphic functions in for which , , and as . We explore characteristics of the class and some places where functions in the class behave in a significantly different manner than those for which holds. We also explore connections between the class and linear differential equations and values of differential polynomials and give an analogue to Nevanlinna's five-value theorem.

Uniqueness of meromorphic functions of differential polynomials sharing a small function with finite weight

2019 ◽  
Vol 25 (2) ◽  
pp. 141-153
Author(s):  
Harina P. Waghamore ◽  
Vijaylaxmi Bhoosnurmath
Keyword(s):  
Meromorphic Function ◽  
Meromorphic Functions ◽  
Differential Polynomial ◽  
Small Function ◽  
Differential Polynomials ◽  
Weighted Sharing ◽  
Linear Differential Polynomials ◽  
Linear Differential

Abstract Let f be a non-constant meromorphic function and {a=a(z)} ( {\not\equiv 0,\infty} ) a small function of f. Here, we obtain results similar to the results due to Indrajit Lahiri and Bipul Pal [Uniqueness of meromorphic functions with their homogeneous and linear differential polynomials sharing a small function, Bull. Korean Math. Soc. 54 2017, 3, 825–838] for a more general differential polynomial by introducing the concept of weighted sharing.

scholarly journals ZEROS OF DIFFERENTIAL POLYNOMIALS IN REAL MEROMORPHIC FUNCTIONS

2005 ◽  
Vol 48 (2) ◽  
pp. 279-293 ◽  
Author(s):  
Walter Bergweiler ◽  
Alex Eremenko ◽  
Jim K. Langley
Keyword(s):  
Meromorphic Function ◽  
Meromorphic Functions ◽  
Rational Functions ◽  
Differential Polynomials ◽  
Real Zeros ◽  
Transcendental Meromorphic Function

AbstractWe investigate whether differential polynomials in real transcendental meromorphic functions have non-real zeros. For example, we show that if $g$ is a real transcendental meromorphic function, $c\in\mathbb{R}\setminus\{0\}$ and $n\geq3$ is an integer, then $g'g^n-c$ has infinitely many non-real zeros. If $g$ has only finitely many poles, then this holds for $n\geq2$. Related results for rational functions $g$ are also considered.

scholarly journals Characteristic estimation of differential polynomials

2021 ◽  
Vol 2021 (1) ◽  
Author(s):  
Min-Feng Chen ◽  
Zhi-Bo Huang
Keyword(s):  
Meromorphic Function ◽  
Quantitative Estimate ◽  
Meromorphic Functions ◽  
Value Distribution ◽  
Differential Polynomials ◽  
Complete Extension

AbstractIn this paper, we give the characteristic estimation of a meromorphic function f with the differential polynomials $f^{l}(f^{(k)})^{n}$ f l ( f ( k ) ) n and obtain that $$\begin{aligned} T(r,f)\leq M\overline{N} \biggl(r,\frac{1}{f^{l}(f^{(k)})^{n}-a} \biggr)+S(r,f) \end{aligned}$$ T ( r , f ) ≤ M N ‾ ( r , 1 f l ( f ( k ) ) n − a ) + S ( r , f ) holds for $M=\min \{\frac{1}{l-2},6\}$ M = min { 1 l − 2 , 6 } , integers $l(\geq 2)$ l ( ≥ 2 ) , $n(\geq 1)$ n ( ≥ 1 ) , $k(\geq 1)$ k ( ≥ 1 ) , and a non-zero constant a. This quantitative estimate is an interesting and complete extension of earlier results. The value distribution of a differential monomial of meromorphic functions is also investigated.

Uniqueness of Entire or Meromorphic Functions Concerning Differential Polynomials

2009 ◽  
Vol 16 (3) ◽  
pp. 449-463
Author(s):  
Jun-Fan Chen ◽  
Wei-Chuan Lin ◽  
Xiao-Yu Zhang
Keyword(s):  
Meromorphic Function ◽  
Meromorphic Functions ◽  
Differential Polynomials

Abstract We study the uniqueness problems on entire or meromorphic functions concerning differential polynomials that share one slowly growing meromorphic function. Furthermore, we greatly improve and generalize some former results obtained by Fang and Hong, Lin and Yi, Xiong and Lin.

scholarly journals Meromorphic or Holomorphic Completion of a Reinhardt Domain

1970 ◽  
Vol 38 ◽  
pp. 1-12 ◽  
Author(s):  
Eiichi Sakai
Keyword(s):  
Meromorphic Function ◽  
Power Series ◽  
Meromorphic Functions ◽  
Integral Formula ◽  
Complex Variables ◽  
Several Complex Variables ◽  
Reinhardt Domain ◽  
Continuity Theorem ◽  
Cauchy’S Integral Formula ◽  
Long Time

In the theory of functions of several complex variables, the problem about the continuation of meromorphic functions has not been much investigated for a long time in spite of its importance except the deeper result of the continuity theorem due to E. E. Levi [4] and H. Kneser [3], The difficulty of its investigation is based on the following reasons: we can not use the tools of not only Cauchy’s integral formula but also the power series and there are indetermination points for the meromorphic function of many variables different from one variable. Therefore we shall also follow the Levi and Kneser’s method and seek for the aspect of meromorphic completion of a Reinhardt domain in Cn.

scholarly journals On Properties of Differences Polynomials about Meromorphic Functions

2012 ◽  
Vol 2012 ◽  
pp. 1-11
Author(s):  
Jianming Qi ◽  
Jie Ding ◽  
Wenjun Yuan
Keyword(s):  
Meromorphic Function ◽  
Finite Order ◽  
Special Class ◽  
Meromorphic Functions ◽  
Value Distribution ◽  
Class Difference ◽  
Difference Polynomial

We study the value distribution of a special class difference polynomial about finite order meromorphic function. Our methods of the proof are also different from ones in the previous results by Chen (2011), Liu and Laine (2010), and Liu and Yang (2009).

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