A Uniqueness Polynomial Generating a Unique Range Set and Vice Versa

2012 ◽  
Vol 12 (2) ◽  
pp. 527-539 ◽  
Author(s):  
Abhijit Banerjee ◽  
Indrajit Lahiri
Keyword(s):  
Unique Range Set

scholarly journals The Unique Range Set of Meromorphic Functions in an Angular Domain

2014 ◽  
Vol 2014 ◽  
pp. 1-10
Author(s):  
Hong-Yan Xu ◽  
Zu-Xing Xuan ◽  
Hua Wang
Keyword(s):  
Meromorphic Functions ◽  
Angular Domain ◽  
Shared Set ◽  
Unique Range Set

By using Tsuji's characteristic, we investigate uniqueness of meromorphic functions in an angular domain dealing with the shared set, which is different from the set of the paper (Lin et al., 2006) and obtain a series of results about the unique range set of meromorphic functions in angular domain.

scholarly journals On the unique range set of meromorphic functions

1996 ◽  
Vol 124 (1) ◽  
pp. 177-185 ◽  
Author(s):  
Ping Li ◽  
Chung-Chun Yang
Keyword(s):  
Meromorphic Functions ◽  
Unique Range Set

scholarly journals Further Results about a Special Fermat-Type Difference Equation

2020 ◽  
Vol 2020 ◽  
pp. 1-6
Author(s):  
Hongwei Ma ◽  
Jianming Qi ◽  
Zhenjie Zhang
Keyword(s):  
Difference Equation ◽  
Finite Order ◽  
Complex Plane ◽  
Meromorphic Solution ◽  
Unique Range Set ◽  
The Difference

In this paper, we prove the difference equation Fz3+ΔcFz+c3=1 does not have meromorphic solution of finite order over the complex plane C. We also discuss an application to the unique range set problem.

A unique range set for meromorphic functions with 11 elements

1998 ◽  
Vol 37 (1-4) ◽  
pp. 185-193 ◽  
Author(s):  
Günter Frank ◽  
Martin Reinders
Keyword(s):  
Meromorphic Functions ◽  
Unique Range Set

scholarly journals Unique range sets - a further study

Filomat ◽  
2020 ◽  
Vol 34 (5) ◽  
pp. 1499-1516
Author(s):  
Sanjay Mallick
Keyword(s):  
Sufficient Conditions ◽  
General Setting ◽  
General Polynomial ◽  
Unique Range Set ◽  
First Time

The purpose of the paper is to investigate the problems of unique range sets in the most general setting. Accordingly, we have studied sufficient conditions for a general polynomial to generate a unique range set which put all the variants of unique range sets into one structure. Most importantly, as an application of the main result we have been able to accommodate not only examples of critically injective polynomials but also examples of non-critically injective polynomials generating unique range sets which are for the first time being exemplified in the literature. Furthermore, some of these examples show that characterization of unique range sets generated by non-critically injective polynomials does not always demand gap polynomials which also complements the recent results by An and Banerjee-Lahiri in this direction. Moreover, one of the lemmas proved in this paper improves and generalizes some results due to Frank-Reinders and Lahiri respectively.

scholarly journals URS AND URSIMS FOR P-ADIC MEROMORPHIC FUNCTIONS INSIDE A DISC

2001 ◽  
Vol 44 (3) ◽  
pp. 485-504 ◽  
Author(s):  
Abdelbaki Boutabaa ◽  
Alain Escassut
Keyword(s):  
Characteristic Function ◽  
Analytic Functions ◽  
Nevanlinna Theory ◽  
Meromorphic Functions ◽  
Closed Field ◽  
Open Disc ◽  
Absolute Value ◽  
Unique Range Set ◽  
Bounded Characteristic ◽  
Primary 12H25

AbstractLet $K$ be an algebraically closed field of characteristic zero, complete for an ultrametric absolute value. We show that the $p$-adic main Nevanlinna Theorem holds for meromorphic functions inside an ‘open’ disc in $K$. Let $P_{n,c}$ be the Frank–Reinders’s polynomial$$ (n-1)(n-2)X^n-2n(n-2)X^{n-1}+ n(n-1)X^{n-2}-c\qq (c\neq0,\ c\neq1,\ c\neq2) $$and let $S_{n,c}$ be the set of its $n$ distinct zeros. For every $n\geq 7$, we show that $S_{n,c}$ is an $n$-points unique range set (counting multiplicities) for unbounded analytic functions inside an ‘open disc’, and for every $n\geq10$, we show that $S_{n,c}$ is an $n$-points unique range set ignoring multiplicities for the same set of functions. Similar results are obtained for meromorphic functions whose characteristic function is unbounded: we obtain unique range sets ignoring multiplicities of $17$ points. A better result is obtained for an analytic or a meromorphic function $f$ when its derivative is ‘small’ comparatively to $f$. In particular, for every $n\geq5$ we show that $S_{n,c}$ is an $n$-points unique range set ignoring multiplicities for unbounded analytic functions with small derivative. Actually, in each case, results also apply to pairs of analytic functions when just one of them is supposed unbounded. The method we use is based upon the $p$-adic Nevanlinna Theory, and Frank–Reinders’s and Fujimoto’s methods used for meromorphic functions in $\mathbb{C}$. Among other results, we show that the set of functions having a bounded characteristic function is just the field of fractions of the ring of bounded analytic functions in the disc.AMS 2000 Mathematics subject classification: Primary 12H25. Secondary 12J25; 46S10

scholarly journals A new type of unique range set with deficient values

2014 ◽  
Vol 26 (7-8) ◽  
pp. 1561-1572
Author(s):  
Abhijit Banerjee ◽  
Bikash Chakraborty
Keyword(s):  
Unique Range Set ◽  
New Type

scholarly journals Unique Range Set for Meromorphic Functions with Deficient Values

2013 ◽  
Vol 2013 ◽  
pp. 1-8
Author(s):  
Abhijit Banerjee ◽  
Sujoy Majumder
Keyword(s):  
Meromorphic Functions ◽  
Weighted Sharing ◽  
Unique Range Set

We employ the idea of weighted sharing of sets to find a unique range set for meromorphic functions with deficient values. Our result improves, generalises, and extends the result of Lahiri. Examples are exhibited that a condition in one of our results is the best possible one.

scholarly journals A note on a new unique range set with truncated multiplicity

2017 ◽  
Vol 21 (2) ◽  
pp. 195-205
Author(s):  
Abhijit Banerjee ◽  
Santanu Dhar
Keyword(s):  
Meromorphic Function ◽  
Zero Set ◽  
Unique Range Set

We introduce a new polynomial whose zero set forms a unique range set for meromorphic function with 11 elements under relaxed sharing hypothesis.

scholarly journals Unicity theorems for meromorphic or entire functions III

1996 ◽  
Vol 53 (1) ◽  
pp. 71-82 ◽  
Author(s):  
Hong-Xun Yi
Keyword(s):  
Meromorphic Functions ◽  
Entire Functions ◽  
Unique Range Set

This paper studies the unique range set of meromorphic functions and shows that the set S = {w | w13 + w11 + 1 = 0} is unique range set of meromorphic functions with 13 elements.

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