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>

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).

scholarly journals Value distribution of meromorphic functions concerning rational functions and differences

2020 ◽  
Vol 2020 (1) ◽  
Author(s):  
Mingliang Fang ◽  
Degui Yang ◽  
Dan Liu
Keyword(s):  
Positive Integer ◽  
Meromorphic Function ◽  
Rational Function ◽  
Finite Order ◽  
Meromorphic Functions ◽  
Rational Functions ◽  
Value Distribution ◽  
Exponent Of Convergence ◽  
Nonzero Constant ◽  
Zero Points

AbstractLet c be a nonzero constant and n a positive integer, let f be a transcendental meromorphic function of finite order, and let R be a nonconstant rational function. Under some conditions, we study the relationships between the exponent of convergence of zero points of $f-R$ f − R , its shift $f(z+nc)$ f ( z + n c ) and the differences $\Delta _{c}^{n} f$ Δ c n f .

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.

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 On the common right factors of meromorphic functions

1997 ◽  
Vol 55 (3) ◽  
pp. 395-403 ◽  
Author(s):  
Tuen-Wai Ng ◽  
Chung-Chun Yang
Keyword(s):  
Entire Function ◽  
Meromorphic Function ◽  
Rational Function ◽  
Finite Order ◽  
Meromorphic Functions ◽  
Transcendental Meromorphic Function ◽  
The Common ◽  
Periodic Entire Function

In this paper, common right factors (in the sense of composition) of p1 + p2F and p3 + p4F are investigated. Here, F is a transcendental meromorphic function and pi's are non-zero polynomials. Moreover, we also prove that the quotient (p1 + p2F)/(p3 + p4F) is pseudo-prime under some restrictions on F and the pi's. As an application of our results, we have proved that R (z) H (z)is pseudo-prime for any nonconstant rational function R (z) and finite order periodic entire function H (z).

scholarly journals A Positive Answer for 3IM+1CM Problem with a General Difference Polynomial

2021 ◽  
Vol 2021 ◽  
pp. 1-9
Author(s):  
Huicai Xu ◽  
Shugui Kang ◽  
Qingcai Zhang
Keyword(s):  
Meromorphic Function ◽  
Finite Order ◽  
Analytic Functions ◽  
Positive Answer ◽  
The Other ◽  
Main Difficulty ◽  
New Methods ◽  
Other Hand ◽  
Difference Polynomial ◽  
General Difference

In this paper, the 3IM+1CM theorem with a general difference polynomial L z , f will be established by using new methods and technologies. Note that the obtained result is valid when the sum of the coefficient of L z , f is equal to zero or not. Thus, the theorem with the condition that the sum of the coefficient of L z , f is equal to zero is also a good extension for recent results. However, it is new for the case that the sum of the coefficient of L z , f is not equal to zero. In fact, the main difficulty of proof is also from this case, which causes the traditional theorem invalid. On the other hand, it is more interesting that the nonconstant finite-order meromorphic function f can be exactly expressed for the case f ≡ − L z , f . Furthermore, the sharpness of our conditions and the existence of the main result are illustrated by examples. In particular, the main result is also valid for the discrete analytic functions.

Entire and meromorphic solutions of linear difference equations

2016 ◽  
Vol 56 (1) ◽  
pp. 43-59
Author(s):  
Renukadevi S. Dyavanal ◽  
Madhura M. Mathai
Keyword(s):  
Meromorphic Function ◽  
Difference Equation ◽  
Finite Order ◽  
Difference Equations ◽  
Entire Solution ◽  
Linear Difference Equation ◽  
Meromorphic Solutions ◽  
Linear Difference Equations ◽  
Difference Polynomial

Abstract In this paper, we shall investigate the existence of finite order entire and meromorphic solutions of linear difference equation of the form $$f^n (z) + p(z)f^{n - 2} (z) + L(z,f) = h(z)$$ where L(z, f) is linear difference polynomial in f(z), p(z) is non-zero polynomial and h(z) is a meromorphic function of finite order. We also consider finite order entire solution of linear difference equation of the form $$f^n (z) + p(z)L(z,f) = r(z)e^{q(z)}$$ where r(z) and q(z) are polynomials.

scholarly journals Fixed points of differences of meromorphic functions

2019 ◽  
Vol 2019 (1) ◽  
Author(s):  
Zhaojun Wu ◽  
Jia Wu
Keyword(s):  
Meromorphic Function ◽  
Fixed Points ◽  
Finite Order ◽  
Complex Number ◽  
Meromorphic Functions ◽  
Acta Math ◽  
Transcendental Meromorphic Function ◽  
Nonzero Complex Number ◽  
Discrete Analogues

Abstract Let f be a transcendental meromorphic function of finite order and c be a nonzero complex number. Define $\Delta _{c}f=f(z+c)-f(z)$ Δ c f = f ( z + c ) − f ( z ) . The authors investigate the existence on the fixed points of $\Delta _{c}f$ Δ c f . The results obtained in this paper may be viewed as discrete analogues on the existing theorem on the fixed points of $f'$ f ′ . The existing theorem on the fixed points of $\Delta _{c}f$ Δ c f generalizes the relevant results obtained by (Chen in Ann. Pol. Math. 109(2):153–163, 2013; Zhang and Chen in Acta Math. Sin. New Ser. 32(10):1189–1202, 2016; Cui and Yang in Acta Math. Sci. 33B(3):773–780, 2013) et al.

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 $$\mathop {\lim \,{\rm sup}}\limits_{r \to + \infty } {{T(r,\,f)} \over {T(r,\,f')}}{\rm{ &#x003C; }} + \infty ,$$ 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 A result on a question of Lü, Li and Yang

Filomat ◽  
2019 ◽  
Vol 33 (9) ◽  
pp. 2893-2906
Author(s):  
Sujoy Majumder ◽  
Somnath Saha
Keyword(s):  
Meromorphic Function ◽  
Finite Order ◽  
Transcendental Meromorphic Function

Let f be a transcendental meromorphic function of finite order with finitely many poles, c ? C\{0} and n,k ? N. Suppose fn(z)-Q1(z) and (fn(z+c))(k)- Q2(z) share (0,1) and f(z), f(z+c) share 0 CM. If n ? k + 1, then (fn(z+c))(k) ? Q2(z)/Q1(z)fn(z), where Q1, Q2 are polynomials with Q1Q2 ?/ 0. Furthermore, if Q1 = Q2, then f(z)=c1e?/n z, where c1 and ? are non-zero constants such that e?c = 1 and ?k = 1. Also we exhibit some examples to show that the conditions of our result are the best possible.

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