scholarly journals On slow escaping and non-escaping points of quasimeromorphic mappings

2020 ◽  
pp. 1-27
Author(s):  
LUKE WARREN
Keyword(s):  
Growth Rate ◽  
Complex Plane ◽  
Meromorphic Functions ◽  
Essential Singularity ◽  
Quasiregular Mappings ◽  
Quasimeromorphic Mapping ◽  
Quasimeromorphic Mappings

We show that for any quasimeromorphic mapping with an essential singularity at infinity, there exist points whose iterates tend to infinity arbitrarily slowly. This extends a result by Nicks for quasiregular mappings, and Rippon and Stallard for transcendental meromorphic functions on the complex plane. We further establish a new result for the growth rate of quasiregular mappings near an essential singularity, and briefly extend some results regarding the bounded orbit set and the bungee set to the quasimeromorphic setting.

scholarly journals On the iteration of quasimeromorphic mappings

2018 ◽  
Vol 168 (1) ◽  
pp. 1-11
Author(s):  
LUKE WARREN
Keyword(s):  
Meromorphic Functions ◽  
Julia Set ◽  
Rational Functions ◽  
Higher Dimensions ◽  
Essential Singularity ◽  
Quasiregular Mappings ◽  
Different Types ◽  
Quasimeromorphic Mappings

AbstractThe Fatou–Julia theory for rational functions has been extended both to transcendental meromorphic functions and more recently to several different types of quasiregular mappings in higher dimensions. We extend the iterative theory to quasimeromorphic mappings with an essential singularity at infinity and at least one pole, constructing the Julia set for these maps. We show that this Julia set shares many properties with those for transcendental meromorphic functions and for quasiregular mappings of punctured space.

On Uniqueness of Dirichlet Series

2020 ◽  
Author(s):  
Bao Qin Li
Keyword(s):  
Analytic Continuation ◽  
Dirichlet Series ◽  
Finite Order ◽  
Half Plane ◽  
Complex Plane ◽  
Meromorphic Functions ◽  
Uniqueness Problem ◽  
Uniqueness Theorems

Abstract We give a characterization of the ratio of two Dirichelt series convergent in a right half-plane under an analytic condition, which is applicable to a uniqueness problem for Dirichlet series admitting analytic continuation in the complex plane as meromorphic functions of finite order; uniqueness theorems are given in terms of a-points of the functions.

Entire solutions of certain type of non-linear differential equations

2020 ◽  
Vol 70 (1) ◽  
pp. 87-94
Author(s):  
Bo Xue
Keyword(s):  
Differential Equations ◽  
Complex Plane ◽  
Meromorphic Functions ◽  
Nonlinear Differential Equations ◽  
Differential Polynomial ◽  
Value Distribution ◽  
Linear Differential Equations ◽  
Entire Solutions ◽  
Value Distribution Theory ◽  
Linear Differential

AbstractUtilizing Nevanlinna’s value distribution theory of meromorphic functions, we study transcendental entire solutions of the following type nonlinear differential equations in the complex plane$$\begin{array}{} \displaystyle f^{n}(z)+P(z,f,f',\ldots,f^{(t)})=P_{1}\text{e}^{\alpha_{1}z}+P_{2}\text{e}^{\alpha_{2}z}+P_{3}\text{e}^{\alpha_{3}z}, \end{array}$$where Pj and αi are nonzero constants for j = 1, 2, 3, such that |α1| > |α2| > |α3| and P(z, f, f′, …, f(t) is an algebraic differential polynomial in f(z) of degree no greater than n – 1.

scholarly journals Reflection Identities of Harmonic Sums of Weight Four

Universe ◽  
2019 ◽  
Vol 5 (3) ◽  
pp. 77 ◽  
Author(s):  
Alexander Prygarin
Keyword(s):  
Analytic Continuation ◽  
Linear Combination ◽  
Complex Plane ◽  
Meromorphic Functions ◽  
Bfkl Equation ◽  
Color Singlet ◽  
Pole Structure

In attempt to find a proper space of function expressing the eigenvalue of the color-singlet BFKL equation in N = 4 SYM, we consider an analytic continuation of harmonic sums from positive even integer values of the argument to the complex plane. The resulting meromorphic functions have pole singularities at negative integers. We derive the reflection identities for harmonic sums at weight four decomposing a product of two harmonic sums with mixed pole structure into a linear combination of terms each having a pole at either negative or non-negative values of the argument. The pole decomposition demonstrates how the product of two simpler harmonic sums can build more complicated harmonic sums at higher weight. We list a minimal irreducible set of bilinear reflection identities at weight four, which represents the main result of the paper. We also discuss how other trilinear and quadlinear reflection identities can be constructed from our result with the use of well known quasi-shuffle relations for harmonic sums.

scholarly journals The uniqueness of meromorphic functions in k-punctured complex plane

2017 ◽  
Vol 15 (1) ◽  
pp. 724-733 ◽  
Author(s):  
Hong Yan Xu ◽  
San Yang Liu
Keyword(s):  
Complex Plane ◽  
Meromorphic Functions ◽  
Finite Sets

Abstract The main purpose of this paper is to investigate the uniqueness of meromorphic functions that share two finite sets in the k-punctured complex plane. It is proved that there exist two sets S1, S2 with ♯S1 = 2 and ♯S2 = 5, such that any two admissible meromorphic functions f and g in Ω must be identical if EΩ(Sj, f) = EΩ(Sj, g)(j = 1,2).

On the value distribution of f2f(k)

2005 ◽  
Vol 78 (1) ◽  
pp. 17-26 ◽  
Author(s):  
Xiaojun Huang ◽  
Yongxing Gu
Keyword(s):  
Positive Integer ◽  
Meromorphic Function ◽  
Complex Plane ◽  
Meromorphic Functions ◽  
Value Distribution ◽  
Multiple Zeros ◽  
Transcendental Meromorphic Function

AbstractIn this paper, we prove that for a transcendental meromorphic function f(z) on the complex plane, the inequality T(r, f) < 6N (r, 1/(f2 f(k)−1)) + S(r, f) holds, where k is a positive integer. Moreover, we prove the following normality criterion: Let ℱ be a family of meromorphic functions on a domain D and let k be a positive integer. If for each ℱ ∈ ℱ, all zeros of ℱ are of multiplicity at least k, and f2 f(k) ≠ 1 for z ∈ D, then ℱ is normal in the domain D. At the same time we also show that the condition on multiple zeros of f in the normality criterion is necessary.

The distribution of finite values of meromorphic functions with deficient poles

2002 ◽  
Vol 132 (2) ◽  
pp. 311-317
Author(s):  
J. K. LANGLEY
Keyword(s):  
Complex Plane ◽  
Meromorphic Functions ◽  
Nevanlinna Deficiency

We prove the existence of unbounded open subsets S of the complex plane with the following property. If f is a function transcendental and meromorphic in the plane, the poles of which have positive Nevanlinna deficiency, then f takes every finite value, with at most one exception, infinitely often in the complement of S.

scholarly journals Hausdorff Dimension of Escaping Sets of Nevanlinna Functions

2019 ◽  
Author(s):  
Weiwei Cui
Keyword(s):  
Hausdorff Dimension ◽  
Differential Equations ◽  
Complex Plane ◽  
Meromorphic Functions ◽  
Second Order ◽  
Hausdorff Dimensions ◽  
Second Order Differential Equations ◽  
Schwarzian Derivatives ◽  
Nevanlinna Functions

Abstract We determine the exact values of Hausdorff dimensions of escaping sets of meromorphic functions with polynomial Schwarzian derivatives. This will follow from the relation between these functions and the second-order differential equations in the complex plane.

scholarly journals Continuity of Cima and Rung's extension and non normal meromorphic functions

1979 ◽  
Vol 20 (1) ◽  
pp. 139-143
Author(s):  
Douglas M. Campbell
Keyword(s):  
Meromorphic Function ◽  
Complex Plane ◽  
Open Problem ◽  
Meromorphic Functions ◽  
Continuous Extension ◽  
Entire Complex ◽  
Long Time ◽  
Limit Points

A function meromorphic in |z| < 1 is constructed such that on every curve in |z| < 1 which goes to |z| = 1 the set of limit points of the function is the entire complex plane. This example is used to prove the existence of non-normal meromorphic functions in |z| < 1 which have continuous set valued extensions. Cima and Rung had introduced a set valued extension for meromorphic functions and proved that all normal meromorphic functions have a continuous extension while all functions with a continuous extension have the Lindelöf property. For a long time it was thought that this might characterize normal meromorphic functions. This paper proves that it is not possible to determine the normality of a meromorphic function by the continuity of Cima and Rung's set valued extension. The paper closes with the open problem: do there exist non-normal analytic functions for which Cima and Rung's set valued extension is continuous?

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