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 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 Unicity theorems for meromorphic or entire functions II

1995 ◽  
Vol 52 (2) ◽  
pp. 215-224 ◽  
Author(s):  
Hong-Xun Yi
Keyword(s):  
Meromorphic Functions ◽  
Entire Functions ◽  
Inverse Image ◽  
Finite Sets ◽  
Finite Set ◽  
Special Case

In 1976, Gross posed the question “can one find two (or possibly even one) finite sets Sj (j = 1, 2) such that any two entire functions f and g satisfying Ef(Sj) = Eg(Sj) for j = 1,2 must be identical?”, where Ef(Sj) stands for the inverse image of Sj under f. In this paper, we show that there exists a finite set S with 11 elements such that for any two non-constant meromorphic functions f and g the conditions Ef(S) = Eg(S) and Ef({∞}) = Eg({∞}) imply f ≡ g. As a special case this also answers the question posed by Gross.

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.

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