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 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 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.

ON THE BRÜCK CONJECTURE

2015 ◽  
Vol 93 (2) ◽  
pp. 248-259 ◽  
Author(s):  
TINGBIN CAO
Keyword(s):  
Entire Function ◽  
Differential Equations ◽  
Entire Functions ◽  
Linear Differential Equations ◽  
Small Function ◽  
Shared Value ◽  
Nonzero Constant ◽  
Hyper Order ◽  
Linear Differential ◽  
Brück Conjecture

The Brück conjecture states that if a nonconstant entire function $f$ with hyper-order ${\it\sigma}_{2}(f)\in [0,+\infty )\setminus \mathbb{N}$ shares one finite value $a$ (counting multiplicities) with its derivative $f^{\prime }$, then $f^{\prime }-a=c(f-a)$, where $c$ is a nonzero constant. The conjecture has been established for entire functions with order ${\it\sigma}(f)<+\infty$ and hyper-order ${\it\sigma}_{2}(f)<{\textstyle \frac{1}{2}}$. The purpose of this paper is to prove the Brück conjecture for the case ${\it\sigma}_{2}(f)=\frac{1}{2}$ by studying the infinite hyper-order solutions of the linear differential equations $f^{(k)}+A(z)f=Q(z)$. The shared value $a$ is extended to be a ‘small’ function with respect to the entire function $f$.

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 Relationship between entire functions and their derivatives sharing small function except a set

Filomat ◽  
2018 ◽  
Vol 32 (19) ◽  
pp. 6845-6855
Author(s):  
Feng Lü
Keyword(s):  
Differential Equation ◽  
Entire Function ◽  
Entire Functions ◽  
Infinite Product ◽  
Small Function ◽  
Entire Solutions ◽  
Infinite Products ◽  
Growth Property ◽  
The Relationship

The paper is mainly devoted to deriving the relationship between an entire function and its derivative when they share one small function except possibly a set, which is related to the famous Br?ck conjecture. In addition, two propositions of infinite products are obtained. The first one is the growth property of a certain infinite product. The second one is the property of entire solutions of the differential equation which concerns infinite products.

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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