scholarly journals Uniqueness Theorems on Entire Functions and Their Difference Operators or Shifts

2012 ◽  
Vol 2012 ◽  
pp. 1-8 ◽  
Author(s):  
Baoqin Chen ◽  
Zongxuan Chen ◽  
Sheng Li
Keyword(s):  
Entire Function ◽  
Entire Functions ◽  
Difference Operators ◽  
Uniqueness Theorems ◽  
Difference Analogue ◽  
Function Sharing

We study the uniqueness problems on entire functions and their difference operators or shifts. Our main result is a difference analogue of a result of Jank-Mues-Volkmann, which is concerned with the uniqueness of the entire function sharing one finite value with its derivatives. Two relative results are proved, and examples are provided for our results.

scholarly journals Difference Equations and Sharing Values Concerning Entire Functions and Their Difference

2014 ◽  
Vol 2014 ◽  
pp. 1-6
Author(s):  
Zhiqiang Mao ◽  
Huifang Liu
Keyword(s):  
Difference Equations ◽  
Entire Functions ◽  
Value Distribution ◽  
Difference Operators ◽  
Sharing Values ◽  
Difference Analogue ◽  
The Difference ◽  
Brück Conjecture

The value distribution of solutions of certain difference equations is investigated. As its applications, we investigate the difference analogue of the Brück conjecture. We obtain some results on entire functions sharing a finite value with their difference operators. Examples are provided to show that our results are the best possible.

scholarly journals Entire functions that share a small function with their linear difference polynomial

2021 ◽  
Vol 7 (3) ◽  
pp. 3731-3744
Author(s):  
Minghui Zhange ◽  
◽  
Jianbin Xiao ◽  
Mingliang Fang
Keyword(s):  
Entire Function ◽  
Entire Functions ◽  
Small Function ◽  
Difference Polynomial ◽  
Function Sharing

<abstract><p>In this paper, we investigate the uniqueness of an entire function sharing a small function with its linear difference polynomial. Our results improve some results due to Li and Yi <sup>[<xref ref-type="bibr" rid="b11">11</xref>]</sup>, Zhang, Chen and Huang <sup>[<xref ref-type="bibr" rid="b17">17</xref>]</sup>, Zhang, Kang and Liao <sup>[<xref ref-type="bibr" rid="b18">18</xref>,<xref ref-type="bibr" rid="b19">19</xref>]</sup> etc.</p></abstract>

scholarly journals Some unity results on entire functions and their difference operators related to 4 CM theorem

2020 ◽  
Vol 2020 (1) ◽  
Author(s):  
BaoQin Chen ◽  
Sheng Li
Keyword(s):  
Entire Function ◽  
Finite Order ◽  
Entire Functions ◽  
Difference Operators ◽  
Finite Complex

Abstract This paper is to consider the unity results on entire functions sharing two values with their difference operators and to prove some results related to 4 CM theorem. The main result reads as follows: Let $f(z)$ f ( z ) be a nonconstant entire function of finite order, and let $a_{1}$ a 1 , $a_{2}$ a 2 be two distinct finite complex constants. If $f(z)$ f ( z ) and $\Delta _{\eta }^{n}f(z)$ Δ η n f ( z ) share $a_{1}$ a 1 and $a_{2}$ a 2 “CM”, then $f(z)\equiv \Delta _{\eta }^{n} f(z)$ f ( z ) ≡ Δ η n f ( z ) , and hence $f(z)$ f ( z ) and $\Delta _{\eta }^{n}f(z)$ Δ η n f ( z ) share $a_{1}$ a 1 and $a_{2}$ a 2 CM.

scholarly journals An entire function sharing fixed points with its linear differential polynomial

2018 ◽  
Vol 22 (1) ◽  
pp. 125-136
Author(s):  
Imrul Kaish ◽  
Indrajit Lahiri
Keyword(s):  
Entire Function ◽  
Fixed Points ◽  
Entire Functions ◽  
Differential Polynomial ◽  
Differential Polynomials ◽  
Linear Polynomial ◽  
Linear Differential Polynomial ◽  
Linear Differential Polynomials ◽  
Linear Differential ◽  
Function Sharing

We study the uniqueness of entire functions, when they share a linear polynomial, in particular, fixed points, with their linear differential polynomials.

scholarly journals Some results on the entire function sharing problem

2014 ◽  
Vol 64 (5) ◽  
Author(s):  
BaoQin Chen ◽  
Sheng Li
Keyword(s):  
Entire Function ◽  
Entire Functions ◽  
Nonzero Constant ◽  
Function Sharing

AbstractOur main result is as follows: let f and a be two entire functions such that $$\max \{ \rho _2 (f),\rho _2 (a)\} < \tfrac{1} {2}$$. If f and f (k) a CM, and if ρ(a (k) − a) < ρ(f − a), then f (k) − a = c(f − a) for some nonzero constant c. This result is applied to improve a result of Gundersen and Yang.

Entire function sharing a small function with its mixed-operators

2019 ◽  
Vol 26 (1) ◽  
pp. 47-62 ◽  
Author(s):  
Xianjing Dong ◽  
Kai Liu
Keyword(s):  
Entire Function ◽  
Linear Combination ◽  
Small Function ◽  
Uniqueness Problem ◽  
Difference Operators ◽  
Transcendental Entire Function ◽  
Shift Operators ◽  
Function Sharing

Abstract In this article, we investigate the uniqueness problem on a transcendental entire function {f(z)} with its linear mixed-operators Tf, where T is a linear combination of differential-difference operators {D^{\nu}_{\eta}:=f^{(\nu)}(z+\eta)} and shift operators {E_{\zeta}:=f(z+\zeta\/)} , where {\eta,\nu,\zeta} are constants. We obtain that if a transcendental entire function {f(z)} satisfies {\lambda(f-\alpha)<\sigma(f\/)<+\infty} , where {\alpha(z)} is an entire function with {\sigma(\alpha)<1} , and if f and Tf share one small entire function {a(z)} with {\sigma(a)<\sigma(f\/)} , then {\frac{Tf-a(z)}{f(z)-a(z)}=\tau,} where τ is a non-zero constant. Furthermore, we obtain the value τ and the expression of f by imposing additional conditions.

scholarly journals Uniqueness on entire functions and their nth order exact differences with two shared values

2020 ◽  
Vol 18 (1) ◽  
pp. 211-215
Author(s):  
Shengjiang Chen ◽  
Aizhu Xu
Keyword(s):  
Entire Function ◽  
Entire Functions ◽  
Shared Values ◽  
Simple Method ◽  
Exact Difference ◽  
Hyper Order

Abstract Let f(z) be an entire function of hyper order strictly less than 1. We prove that if f(z) and its nth exact difference {\Delta }_{c}^{n}f(z) share 0 CM and 1 IM, then {\Delta }_{c}^{n}f(z)\equiv f(z) . Our result improves the related results of Zhang and Liao [Sci. China A, 2014] and Gao et al. [Anal. Math., 2019] by using a simple method.

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