Unique range sets without Fujimoto's hypothesis

TL;DR

This paper investigates the conditions under which two non-constant meromorphic functions are uniquely determined by sharing a finite set, and introduces new unique range sets not constrained by Fujimoto's hypothesis.

Contribution

It presents new results on the uniqueness of meromorphic functions sharing finite sets and constructs unique range sets without relying on Fujimoto's hypothesis.

Findings

Established new criteria for meromorphic function uniqueness

Constructed unique range sets beyond Fujimoto's hypothesis

Extended the understanding of shared value problems in complex analysis

Abstract

This paper studies the uniqueness of two non-constant meromorphic functions when they share a finite set. Moreover, we will give the existence of unique range sets for meromorphic functions that are zero sets of polynomials that do not necessarily satisfy the Fujimoto's hypothesis.