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Entire functions sharing a small function with their two difference operators.

Feng Lü1, Yanfeng Wang1, Junfeng Xu2

  • 1College of Science, China University of Petroleum, Qingdao, Shandong 266580 P.R. China.

Advances in Difference Equations
|August 22, 2017
PubMed
Summary
This summary is machine-generated.

This study proves a uniqueness theorem for entire functions sharing a small function with two difference operators. This generalizes prior results by removing the periodicity condition for the shared function.

Keywords:
Nevanlinna theorydifference operatorsentire functionsuniqueness

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Area of Science:

  • Complex Analysis
  • Nevanlinna Theory
  • Analytic Number Theory

Background:

  • Uniqueness theory in complex analysis investigates conditions under which functions are identical.
  • Previous research established theorems on entire functions sharing values or functions with differential operators.
  • Existing results often required the shared small function to be periodic, limiting broader applicability.

Purpose of the Study:

  • To establish a new uniqueness result for entire functions sharing a small entire function with their difference operators.
  • To generalize and extend existing theorems by removing the periodicity assumption on the shared small entire function.
  • To contribute to the understanding of value distribution theory for functions and their difference operators.

Main Methods:

  • Utilizing the theory of Nevanlinna sharing values.
  • Applying techniques related to difference operators acting on entire functions.
  • Developing new inequalities and estimations based on the characteristic function and its related magnitudes.

Main Results:

  • A novel uniqueness theorem is presented for entire functions sharing a small entire function with their two difference operators.
  • The theorem holds without the prior restriction that the shared small entire function must be periodic.
  • The result extends and improves upon significant theorems by Farissi et al. (2015) and Chen and Li (2014).

Conclusions:

  • The removal of the periodicity condition broadens the scope of uniqueness theorems in complex analysis.
  • The findings offer a more flexible framework for analyzing the behavior of entire functions and their difference operators.
  • This work deepens the understanding of function sharing problems within the context of Nevanlinna theory.