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.

scholarly journals Ger-type and Hyers-Ulam stabilities for the first-order linear differential operators of entire functions

2004 ◽  
Vol 2004 (22) ◽  
pp. 1151-1158 ◽  
Author(s):  
Takeshi Miura ◽  
Go Hirasawa ◽  
Sin-Ei Takahasi
Keyword(s):  
Entire Function ◽  
Differential Operator ◽  
Stability Problem ◽  
Differential Operators ◽  
Entire Functions ◽  
Ulam Stability ◽  
Nonzero Constant ◽  
First Order ◽  
Linear Differential Operators ◽  
Linear Differential

Lethbe an entire function andTha differential operator defined byThf=f′+hf. We show thatThhas the Hyers-Ulam stability if and only ifhis a nonzero constant. We also consider Ger-type stability problem for|1−f′/hf|≤ϵ.

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 Finiteness of Entire Functions Sharing a Finite Set

2007 ◽  
Vol 185 ◽  
pp. 111-122 ◽  
Author(s):  
Hirotaka Fujimoto
Keyword(s):  
Entire Function ◽  
Entire Functions ◽  
Sufficient Conditions ◽  
Nonzero Constant ◽  
Finite Set

AbstractFor a finite set S = {a1,…, aq}, consider the polynomial PS(w) = (w – a1)(w – a2) … (w – aq) and assume that has distinct k zeros. Suppose that PS(w) is a uniqueness polynomial for entire functions, namely that, for any nonconstant entire functions ɸ and ψ, the equality PS(ɸ) = cPS(ψ) implies ɸ = ψ, where c is a nonzero constant which possibly depends on ɸ and ψ. Then, under the condition q > k + 2, we prove that, for any given nonconstant entire function g, there exist at most (2q-2)/(q – k – 2) nonconstant entire functions f with f*(S) = g*(S), where f*(S) denotes the pull-back of S considered as a divisor. Moreover, we give some sufficient conditions of uniqueness polynomials for entire functions.

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.

scholarly journals Primeable entire functions

1973 ◽  
Vol 51 ◽  
pp. 123-130 ◽  
Author(s):  
Fred Gross ◽  
Chung-Chun Yang ◽  
Charles Osgood
Keyword(s):  
Entire Function ◽  
Periodic Function ◽  
Rational Function ◽  
Entire Functions ◽  
Left And Right

An entire function F(z) = f(g(z)) is said to have f(z) and g(z) as left and right factors respe2tively, provided that f(z) is meromorphic and g(z) is entire (g may be meromorphic when f is rational). F(z) is said to be prime (pseudo-prime) if every factorization of the above form implies that one of the functions f and g is bilinear (a rational function). F is said to be E-prime (E-pseudo prime) if every factorization of the above form into entire factors implies that one of the functions f and g is linear (a polynomial). We recall here that an entire non-periodic function f is prime if and only if it is E-prime [5]. This fact will be useful in the sequel.

scholarly journals A question of Gross and the uniqueness of entire functions

1995 ◽  
Vol 138 ◽  
pp. 169-177 ◽  
Author(s):  
Hong-Xun yi
Keyword(s):  
Entire Function ◽  
Nevanlinna Theory ◽  
Entire Functions ◽  
Linear Measure ◽  
Finite Linear

For any set S and any entire function f letwhere each zero of f — a with multiplicity m is repeated m times in Ef(S) (cf. [1]). It is assumed that the reader is familiar with the notations of the Nevanlinna Theory (see, for example, [2]). It will be convenient to let E denote any set of finite linear measure on 0 < r < ∞, not necessarily the same at each occurrence. We denote by S(r, f) any quantity satisfying .

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