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Unicity on meromorphic function sharing three small functions CM with its higher-order difference operators

XiaoHuang Huang

Abstract

In this paper, we study the uniqueness of the shift of meromorphic functions. We prove: Let $f$ be a non-constant meromorphic function satisfying $ρ_{2}(f)<1$, let $η$ be a non-zero complex number, and let $a,b,c\in\hat{S}(f)$ be three distinct small functions. If $f$ and $Δ^{n}_ηf$ share $a,b,c$ CM, then $f\equiv Δ^{n}_ηf$.

Unicity on meromorphic function sharing three small functions CM with its higher-order difference operators

Abstract

In this paper, we study the uniqueness of the shift of meromorphic functions. We prove: Let be a non-constant meromorphic function satisfying , let be a non-zero complex number, and let be three distinct small functions. If and share CM, then .

Paper Structure

This paper contains 3 sections, 13 theorems, 121 equations.

Table of Contents

  1. Introduction and main results
  2. Some Lemmas
  3. The proof of Theorem 1

Key Result

Lemma 2.1

h3 Let $f$ be a non-constant meromorphic function of $\rho_{2}(f)<1$, and let $\eta$ be a non-zero complex number. Then for all r outside of a possible exceptional set $E_{1}$ with finite logarithmic measure.

Theorems & Definitions (13)

  • Lemma 2.1
  • Lemma 2.2
  • Lemma 2.3
  • Lemma 2.4
  • Lemma 2.5
  • Lemma 2.6
  • Lemma 2.7
  • Lemma 2.8
  • Lemma 2.9
  • Lemma 2.10
  • ...and 3 more