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On the cardinality of unique range sets with weight one

Bikash Chakraborty, Sagar Chakraborty

Abstract

Two meromorphic functions $f$ and $g$ are said to share the set $S\subset \mathbb{C}\cup\{\infty\}$ with weight $l\in\mathbb{N}\cup\{0\}\cup\{\infty\}$, if $E_{f}(S,l)=E_{g}(S,l)$ where $$E_{f}(S,l)=\bigcup\limits_{a \in S}\{(z,t) \in \mathbb{C}\times\mathbb{N}~ |~ f(z)=a ~\text{with~ multiplicity}~ p\},$$ where $t=p$ if $p\leq l$ and $t=p+1$ if $p>l$. In this paper, we improve and supplement the result of L. W. Liao and C. C. Yang (On the cardinality of the unique range sets for meromorphic and entire functions, Indian J. Pure appl. Math., 31 (2000), no. 4, 431-440) by showing that there exist a finite set $S$ with cardinality $\geq 13$ such that $E_{f}(S,1)=E_{g}(S,1)$ implies $f\equiv g$.

On the cardinality of unique range sets with weight one

Abstract

Two meromorphic functions and are said to share the set with weight , if where where if and if . In this paper, we improve and supplement the result of L. W. Liao and C. C. Yang (On the cardinality of the unique range sets for meromorphic and entire functions, Indian J. Pure appl. Math., 31 (2000), no. 4, 431-440) by showing that there exist a finite set with cardinality such that implies .

Paper Structure

This paper contains 4 sections, 7 theorems, 41 equations.

Table of Contents

  1. Introduction and Definitions
  2. Main Results
  3. Lemmas
  4. Proof of the theorem

Key Result

Theorem 2.1

Suppose that $n(\geq1)$ be a positive integer. Further suppose that $S=\{z :P(z)=0\}$ where the polynomial $P(z)$ of degree $n$ defined by (abcp1). Let $f$ and $g$ be two non-constant meromorphic functions satisfying $E_{f}(S,1)=E_{g}(S,1)$. If $n\geq13$, then $f\equiv g$.

Theorems & Definitions (17)

  • Definition 1.1
  • Definition 1.2
  • Definition 1.3
  • Definition 1.4
  • Definition 1.5
  • Definition 1.6
  • Definition 1.7
  • Definition 1.8
  • Definition 1.9
  • Theorem 2.1
  • ...and 7 more