Notes on entire functions sharing an entire function of a smaller order with their difference operators

2012 ◽  
Vol 99 (3) ◽  
pp. 261-270 ◽  
Author(s):  
Xiao-Min Li ◽  
Hong-Xun Yi ◽  
Cong-Yun Kang
Keyword(s):  
Entire Function ◽  
Entire Functions ◽  
Difference Operators

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 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 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 .

The distribution of the values of an entire function whose coefficients are independent random variables (II)

1995 ◽  
Vol 118 (3) ◽  
pp. 527-542 ◽  
Author(s):  
A. C. Offord
Keyword(s):  
Entire Function ◽  
Uniform Distribution ◽  
Entire Functions ◽  
Random Variables ◽  
Independent Random Variables ◽  
Almost All

SummaryThis is a study of entire functions whose coefficients are independent random variables. When the space of such functions is symmetric it is shown that independence of the coefficients alone is sufficient to ensure that almost all such functions will, for large z, be large except in certain small neighbourhoods of the zeros called pits. In each pit the function takes every not too large value and these pits have a certain uniform distribution.

scholarly journals A Class of functional equations which have entire solutions

1988 ◽  
Vol 38 (3) ◽  
pp. 351-356 ◽  
Author(s):  
Peter L. Walker
Keyword(s):  
Functional Equation ◽  
Entire Function ◽  
Wide Class ◽  
Functional Equations ◽  
Entire Functions ◽  
Inverse Function ◽  
Entire Solution ◽  
Entire Solutions ◽  
Image Position

We consider the Abelian functional equationwhere φ is a given entire function and g is to be found. The inverse function f = g−1 (if one exists) must satisfyWe show that for a wide class of entire functions, which includes φ(z) = ez − 1, the latter equation has a non-constant entire solution.

scholarly journals A Remark on Small Values of Entire Functions

1966 ◽  
Vol 15 (2) ◽  
pp. 121-123 ◽  
Author(s):  
S. L. Segal
Keyword(s):  
Entire Function ◽  
Entire Functions ◽  
Image Position

Let f(z) be an entire function, M(r) the maximum of f(z) on ∣z∣=r, and λ>1. Let Eλ=Eλ(f{z:log∣f(z)≦(1-λ)log(M∣z∣)}, and denote the density of Eλbywhere m is planar measure.

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