On uniqueness of meromorphic functions sharing finite sets

2000 ◽  
Vol 122 (6) ◽  
pp. 1175-1203 ◽  
Author(s):  
Hirotaka Fujimoto
Keyword(s):  
Meromorphic Functions ◽  
Finite Sets

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.

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).

scholarly journals Uniqueness of meromorphic functions sharing two sets with least possible cardinalities

2019 ◽  
Vol 57 (2) ◽  
pp. 112-130
Author(s):  
Arindam Sarkar
Keyword(s):  
Meromorphic Functions ◽  
Uniqueness Theorems ◽  
Finite Sets ◽  
Shared Set ◽  
Weaker Conditions

Abstract Let f and g be two nonconstant meromorphic functions sharing two finite sets, namely S ⊂ ℂ and {∞}. We prove two uniqueness theorems under weaker conditions on ramification indices, reducing the cardinality of the shared set S and weakening the nature of sharing of the set {∞} which improve results of Fang-Lahiri [7], Lahiri [17], Banerjee -Majumder-Mukherjee [5] and others.

Meromorphic functions sharing two finite sets

2007 ◽  
Vol 27 (4) ◽  
pp. 845-851
Author(s):  
Lin Weichuan ◽  
Yi Hongxun
Keyword(s):  
Meromorphic Functions ◽  
Finite Sets

scholarly journals Unicity theorems for meromorphic or entire functions

1994 ◽  
Vol 49 (2) ◽  
pp. 257-265 ◽  
Author(s):  
Hong-Xun Yi
Keyword(s):  
Meromorphic Functions ◽  
Entire Functions ◽  
Finite Sets

In this paper, we prove that there exist three finite sets Sj (j = 1, 2, 3) such that any two non-constant meromorphic functions f and g satisfying Ef(Sj) = Eg(Sj) for j = 1,2,3 must be identical. As a particular case of the above result, we obtain that there exist two finite sets Sj (j = 1, 2) such that any two non-constant entire functions f and g satisfying Ej(Sj) = Eg(Sj) for j = 1, 2 must be identical, which answers a question posed by Gross.

scholarly journals Uniqueness of meromorphic functions sharing two finite sets

2017 ◽  
Vol 15 (1) ◽  
pp. 1244-1250 ◽  
Author(s):  
Jun-Fan Chen
Keyword(s):  
Meromorphic Functions ◽  
Uniqueness Theorems ◽  
Finite Sets ◽  
Shared Sets

Abstract We prove uniqueness theorems of meromorphic functions, which show how two meromorphic functions are uniquely determined by their two finite shared sets. This answers a question posed by Gross. Moreover, some examples are provided to demonstrate that all the conditions are necessary.

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