Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Some results in the uniqueness of meromorphic function

Xiaohuang Huang

TL;DR

The paper advances the uniqueness theory of meromorphic functions by extending four-value sharing results to the setting of four small functions, and by analyzing how sharing patterns with derivatives, shifts, and difference operators constrain the functions. It establishes a suite of main results (Theorems 1–8) showing that, under various IM/CM sharing configurations of small functions, one obtains strong rigidity: either equality up to derivatives and difference operators or highly constrained growth and structural forms. The work leverages Nevanlinna theory, truncated counting, and a broad collection of auxiliary lemmas to handle both constants and non-constant small functions, including settings with higher-order differences and shifts. It also situates these findings within the landscape of Gundersen-type results, Gol'dberg–Yamanoi type estimates, and current open conjectures, contributing significant new cases where small-function sharing governs uniqueness. Collectively, the results deepen our understanding of how sharing small functional values governs the structure and equality of meromorphic functions in the complex plane and near difference/differential operators.

Abstract

In this research notes, we investigate some remain problems in the uniqueness of meromorphic function. Using some deep results of Yamanoii, we obtain some results in this notes.

Some results in the uniqueness of meromorphic function

TL;DR

The paper advances the uniqueness theory of meromorphic functions by extending four-value sharing results to the setting of four small functions, and by analyzing how sharing patterns with derivatives, shifts, and difference operators constrain the functions. It establishes a suite of main results (Theorems 1–8) showing that, under various IM/CM sharing configurations of small functions, one obtains strong rigidity: either equality up to derivatives and difference operators or highly constrained growth and structural forms. The work leverages Nevanlinna theory, truncated counting, and a broad collection of auxiliary lemmas to handle both constants and non-constant small functions, including settings with higher-order differences and shifts. It also situates these findings within the landscape of Gundersen-type results, Gol'dberg–Yamanoi type estimates, and current open conjectures, contributing significant new cases where small-function sharing governs uniqueness. Collectively, the results deepen our understanding of how sharing small functional values governs the structure and equality of meromorphic functions in the complex plane and near difference/differential operators.

Abstract

In this research notes, we investigate some remain problems in the uniqueness of meromorphic function. Using some deep results of Yamanoii, we obtain some results in this notes.

Paper Structure

This paper contains 16 sections, 238 equations.

Table of Contents

  1. Some Notations
  2. Meromorphic functions that share four small functions
  3. Meromorphic functions that share small functions with their kth order derivatives
  4. Uniqueness of Meromorphic and their shift or n-exact order difference operators
  5. Meromorphic function that share pairs of small functions
  6. Meromorphic functions that four small functions with their derivatives
  7. Main Results
  8. Lemmas
  9. The proof of Theorem 1
  10. Proof of Theorem 2
  11. The proof of Theorem 3
  12. The proof of Theorem 4
  13. The proof of Theorem 5
  14. The proof of Theorem 6
  15. The proof of Theorem 7
  16. ...and 1 more sections

Theorems & Definitions (3)

  • proof
  • proof
  • proof