scholarly journals Uniqueness Theorems on Difference Monomials of Entire Functions

2012 ◽  
Vol 2012 ◽  
pp. 1-8
Author(s):  
Gang Wang ◽  
Deng-li Han ◽  
Zhi-Tao Wen
Keyword(s):  
Finite Order ◽  
Entire Functions ◽  
Uniqueness Theorems ◽  
Complex Constant ◽  
Transcendental Entire Functions ◽  
The Difference

The aim of this paper is to discuss the uniqueness of the difference monomialsfnf(z+c). It assumed thatfandgare transcendental entire functions with finite order andEk)(1,fnf(z+c))=Ek)(1,gng(z+c)), wherecis a nonzero complex constant andn,kare integers. It is proved that if one of the following holds (i)n≥6andk=3, (ii)n≥7andk=2, and (iii)n≥10andk=1, thenfg=t1orf=t2gfor some constantst2andt3which satisfyt2n+1=1andt3n+1=1. It is an improvement of the result of Qi, Yang and Liu.

scholarly journals Entire Functions with Separated Zeros and 1-Points

2020 ◽  
Vol 20 (3-4) ◽  
pp. 729-746
Author(s):  
Walter Bergweiler ◽  
Alexandre Eremenko
Keyword(s):  
Finite Order ◽  
Entire Functions ◽  
Specific Form ◽  
Transcendental Entire Functions

AbstractWe consider transcendental entire functions of finite order for which the zeros and 1-points are in disjoint sectors. Under suitable hypotheses on the sizes of these sectors we show that such functions must have a specific form, or that such functions do not exist at all.

scholarly journals Hausdorff measure of escaping and Julia sets for bounded-type functions of finite order

2011 ◽  
Vol 33 (1) ◽  
pp. 284-302 ◽  
Author(s):  
JÖRN PETER
Keyword(s):  
Finite Order ◽  
Hausdorff Measure ◽  
Entire Functions ◽  
Julia Set ◽  
Julia Sets ◽  
Gauge Function ◽  
Exponential Map ◽  
Hausdorff Measures ◽  
Transcendental Entire Functions ◽  
Bounded Type Functions

AbstractWe show that the escaping sets and the Julia sets of bounded-type transcendental entire functions of order ρ become ‘smaller’ as ρ→∞. More precisely, their Hausdorff measures are infinite with respect to the gauge function hγ(t)=t2g(1/t)γ, where g is the inverse of a linearizer of some exponential map and γ≥(log ρ(f)+K1)/c, but for ρ large enough, there exists a function fρ of bounded type with order ρ such that the Hausdorff measures of the escaping set and the Julia set of fρ with respect to hγ′ are zero whenever γ′ ≤(log ρ−K2)/c.

scholarly journals Uniqueness Problems about Entire Functions with Their Difference Operator Sharing Sets

2020 ◽  
Vol 2020 ◽  
pp. 1-6
Author(s):  
Fan Niu ◽  
Jianming Qi ◽  
Zhiyong Zhou
Keyword(s):  
Finite Order ◽  
Difference Operator ◽  
Entire Functions ◽  
Difference Operators ◽  
Transcendental Entire Functions

In this paper, we study the uniqueness questions of finite order transcendental entire functions and their difference operators sharing a set consisting of two distinct entire functions of finite smaller order. Our results in this paper improve the corresponding results from Liu (2009) and Li (2012).

scholarly journals Uniqueness Theorems of Difference Operator on Entire Functions

2013 ◽  
Vol 2013 ◽  
pp. 1-6
Author(s):  
Jie Ding
Keyword(s):  
Difference Operator ◽  
Entire Functions ◽  
Uniqueness Theorems ◽  
The Difference

We investigate the uniqueness questions of the difference operator on entire functions and obtain three uniqueness theorems using the idea of weight sharing.

On the fixpoints of composite entire functions of finite order

1994 ◽  
Vol 124 (5) ◽  
pp. 995-1001 ◽  
Author(s):  
J. K. Langley
Keyword(s):  
Rational Function ◽  
Finite Order ◽  
Entire Functions ◽  
Transcendental Entire Functions

Suppose that f and g are transcendental entire functions such that the composition F = f(g) has finite order, and suppose that Q is a nonconstant rational function. We show that N(r, 1/(F – Q)) ≠ o(T(r, F)). The theorem is related to results of Bergweiler, Goldstein and others.

scholarly journals Eremenko’s Conjecture for Functions with Real Zeros: The Role of the Minimum Modulus

2020 ◽  
Author(s):  
D A Nicks ◽  
P J Rippon ◽  
G M Stallard
Keyword(s):  
Finite Order ◽  
Entire Functions ◽  
Minimum Modulus ◽  
Real Zeros ◽  
Modulus Condition ◽  
Transcendental Entire Functions

Abstract We consider the class of real transcendental entire functions $f$ of finite order with only real zeros and show that if the iterated minimum modulus tends to $\infty $, then the escaping set $I(\,f)$ of $f$ has the structure of a spider’s web, in which case Eremenko’s conjecture holds. This minimum modulus condition is much weaker than that used in previous work on Eremenko’s conjecture. For functions in this class, we analyse the possible behaviours of the iterated minimum modulus in relation to the order of the function $f$.

scholarly journals ON PROPERTIES OF FINITE-ORDER MEROMORPHIC SOLUTIONS OF A CERTAIN DIFFERENCE PAINLEVÉ I EQUATION

2011 ◽  
Vol 85 (3) ◽  
pp. 463-475 ◽  
Author(s):  
MEI-RU CHEN ◽  
ZONG-XUAN CHEN
Keyword(s):  
Finite Order ◽  
Meromorphic Solutions ◽  
Rational Solutions ◽  
Polynomial Solutions ◽  
The Difference ◽  
Image Position

AbstractIn this paper, we investigate properties of finite-order transcendental meromorphic solutions, rational solutions and polynomial solutions of the difference Painlevé I equation where a, b and c are constants, ∣a∣+∣b∣≠0.

ON EXCEPTIONAL SETS: THE SOLUTION OF A PROBLEM POSED BY K. MAHLER

2016 ◽  
Vol 94 (1) ◽  
pp. 15-19 ◽  
Author(s):  
DIEGO MARQUES ◽  
JOSIMAR RAMIREZ
Keyword(s):  
Entire Functions ◽  
Complex Conjugation ◽  
Exceptional Set ◽  
Exceptional Sets ◽  
Transcendental Numbers ◽  
Lecture Notes ◽  
Transcendental Entire Functions

In this paper, we shall prove that any subset of $\overline{\mathbb{Q}}$, which is closed under complex conjugation, is the exceptional set of uncountably many transcendental entire functions with rational coefficients. This solves an old question proposed by Mahler [Lectures on Transcendental Numbers, Lecture Notes in Mathematics, 546 (Springer, Berlin, 1976)].

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