scholarly journals Equipolar meromorphic functions sharing a set

2021 ◽  
Vol 66 (3) ◽  
pp. 507-520
Author(s):  
Arindam Sarkar
Keyword(s):  
Meromorphic Functions ◽  
Uniqueness Results ◽  
Counting Multiplicity ◽  
Finite Set

"Two meromorphic functions f and g having the same set of poles are known as equipolar. In this paper we study some uniqueness results of equi-polar meromorphic functions sharing a finite set and improve some recent results of Bhoosnurmath-Dyavanal [4] and Banerjee-Mallick [3] by removing some unnec- essary conditions on rami cation indices as well as relaxing the condition on the nature of sharing of the value infinity by f and g from counting multiplicity to ignoring multiplicity."

Borel Directions and the Uniqueness of Algebroid Functions Dealing with Multiple Values

2021 ◽  
Vol 58 (1) ◽  
pp. 104-118
Author(s):  
Yang Tan ◽  
Qingcai Zhang
Keyword(s):  
Conformal Mapping ◽  
Meromorphic Functions ◽  
Angular Domain ◽  
Uniqueness Results ◽  
Algebroid Functions

In this paper, we investigate the uniqueness of algebroid functions in angular domain by the method of conformal mapping. We discuss the relations between the Borel directions and uniquenss with the multiple values of algebroid functions and obtain some results which extend some uniqueness results of meromorphic functions to that of algebroid functions.

TRANSCENDENCE OVER MEROMORPHIC FUNCTIONS

2017 ◽  
Vol 95 (3) ◽  
pp. 393-399 ◽  
Author(s):  
MICHAEL COONS ◽  
YOHEI TACHIYA
Keyword(s):  
Power Series ◽  
General Result ◽  
Meromorphic Functions ◽  
Short Note ◽  
Finite Set

In this short note, considering functions, we show that taking an asymptotic viewpoint allows one to prove strong transcendence statements in many general situations. In particular, as a consequence of a more general result, we show that if$F(z)\in \mathbb{C}[[z]]$is a power series with coefficients from a finite set, then$F(z)$is either rational or it is transcendental over the field of meromorphic functions.

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.

Sharing sets of q-difference of meromorphic functions

2014 ◽  
Vol 64 (1) ◽  
Author(s):  
Xiaoguang Qi ◽  
Lianzhong Yang
Keyword(s):  
Meromorphic Function ◽  
Meromorphic Functions ◽  
Zero Order ◽  
Partial Answer ◽  
Uniqueness Results

AbstractThis paper is devoted to proving some uniqueness results for meromorphic functions f(z) share sets with f(qz). We give a partial answer to a question of Gross concerning a zero-order meromorphic function f(z) and its q-difference f(qz).

scholarly journals On the cardinality of unique range sets with weight one

2020 ◽  
Vol 72 (7) ◽  
pp. 997-1005
Author(s):  
B. Chakraborty ◽  
S. Chakraborty
Keyword(s):  
Meromorphic Functions ◽  
Weight One ◽  
Finite Set ◽  
Appl Math

UDC 517.9 Two meromorphic functions f and g are said to share the set S ⊂ ℂ ∪ { ∞ } with weight l ∈ ℕ ∪ { 0 } ∪ { ∞ } , if E f ( S , l ) = E g ( S , l ) , where where t = p if p ≤ l and t = p + 1 if p > l . In this paper, we improve and supplement the result of L. W. Liao and C. C. Yang [Indian J.  Pure and Appl.  Math., <strong>31</strong>, No~4, 431–440 (2000)] by showing that there exist a finite set S with 13 elements such that E f ( S ,1 ) = E g ( S ,1 ) implies f ≡ g .

Uniqueness of Differential Polynomials of Meromorphic Functions Sharing a Small Function Without Counting Multiplicity

2016 ◽  
Vol 57 (1) ◽  
pp. 121-135
Author(s):  
Bui Thi Kieu Oanh ◽  
Ngo Thi Thu Thuy
Keyword(s):  
Nevanlinna Theory ◽  
Complex Analysis ◽  
Meromorphic Functions ◽  
Small Function ◽  
Uniqueness Problem ◽  
Differential Polynomials ◽  
Counting Multiplicity

Abstract The paper concerns interesting problems related to the field of Complex Analysis, in particular Nevanlinna theory of meromorphic functions. The author have studied certain uniqueness problem on differential polynomials of meromorphic functions sharing a small function without counting multiplicity. The results of this paper are extension of some problems studied by K. Boussaf et. al. in [2] and generalization of some results of S.S. Bhoosnurmath et. al. in [4].

Two semigroup rings associated to a finite set of meromorphic functions

2020 ◽  
Vol 70 (5) ◽  
pp. 1249-1257
Author(s):  
Mircea Cimpoeaş
Keyword(s):  
Finite Number ◽  
Finite Order ◽  
Meromorphic Functions ◽  
Semigroup Rings ◽  
General Properties ◽  
Number Of Zeros ◽  
Finite Set ◽  
Zeros And Poles

AbstractWe fix z0 ∈ ℂ and a field 𝔽 with ℂ ⊂ 𝔽 ⊂ 𝓜z0 := the field of germs of meromorphic functions at z0. We fix f1, …, fr ∈ 𝓜z0 and we consider the 𝔽-algebras S := 𝔽[f1, …, fr] and $\begin{array}{} \overline S: = \mathbb F[f_1^{\pm 1},\ldots,f_r^{\pm 1}]. \end{array} $ We present the general properties of the semigroup rings$$\begin{array}{} \displaystyle S^{hol}: = \mathbb F[f^{\mathbf a}: = f_1^{a_1}\cdots f_r^{a_r}: (a_1,\ldots,a_r)\in\mathbb N^r \text{ and }f^{\mathbf a}\text{ is holomorphic at }z_0],\\\overline S^{hol}: = \mathbb F[f^{\mathbf a}: = f_1^{a_1}\cdots f_r^{a_r}: (a_1,\ldots,a_r)\in\mathbb Z^r \text{ and }f^{\mathbf a}\text{ is holomorphic at }z_0], \end{array} $$and we tackle in detail the case 𝔽 = 𝓜<1, the field of meromorphic functions of order < 1, and fj’s are meromorphic functions over ℂ of finite order with a finite number of zeros and poles.

The geometry of open continuous mappings having two valences between Riemann surfaces

1996 ◽  
Vol 120 (2) ◽  
pp. 309-329 ◽  
Author(s):  
Abdallah Lyzzaik
Keyword(s):  
Riemann Surface ◽  
Finite Number ◽  
Riemann Surfaces ◽  
Branch Points ◽  
Continuous Mappings ◽  
Counting Multiplicity ◽  
Covering Properties ◽  
Finite Set ◽  
Closed Riemann Surface ◽  
Finite Genus

AbstractAn open continuous function from an open Riemann surface with finite genus and finite number of boundary components into a closed Riemann surface is termed a (p, q)-map, 0 < q < p, if it has a finite number of branch points and assumes every point in the image surface either p or q times, counting multiplicity, with possibly a finite number of exceptions.The object of this paper is to prove that the geometry of any (p, q)-map resembles that of a (p, q)-map whose q-set (the set of image points of f that are taken on exactly q times, counting multiplicity), constitutes a finite set of Jordan arcs or curves (loops). This leads to interesting geometrie results regarding (p, q)-maps without exceptional points. Further, it yields that every (p, q)-map is homotopic to a simplicial (p, q)-map having the same covering properties.

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