Unicity of Meromorphic Solutions of the Pielou Logistic Equation

Sheng Li,Baoqin Chen

doi:10.1155/2020/4253967

Abstract

This paper mainly considers the unicity of meromorphic solutions of the Pielou logistic equation yz+1=Rzyz/Qz+Pzyz, where Pz,Qz, and Rz are nonzero polynomials. It shows that the finite order transcendental meromorphic solution of the Pielou logistic equation is mainly determined by its poles and 1-value points. Examples are given for the sharpness of our result.

Highlights

For a meromorphic function f(z), we use standard notations of the Nevanlinna theory, such as T(r, f), m(r, f), and N(r, f)
We know that the unicity of solutions of a given equation is always one of its most essential properties. is paper is to discuss the unicity of meromorphic solutions of the Pielou logistic equation
Li and Chen [16] turned to consider the following question: What can we say about the unicity of finite order transcendental meromorphic solutions of the equation

Summary

Introduction

For a meromorphic function f(z), we use standard notations of the Nevanlinna theory, such as T(r, f), m(r, f), and N(r, f) (see, e.g., [1,2,3]). If a meromorphic function g(z) shares 0, 1, ∞ CM with f(z), one of the following cases holds: Discrete Dynamics in Nature and Society (i) f(z) ≡ g(z), (ii) f(z) + g(z) f(z)g(z), (iii) ere exists a polynomial β(z) az + b0 and a constant a0 satisfying ea0 ≠ eb0 such that 1 − eβ(z) f(z) eβ(z) ea0− b0 −. Li and Chen [16] turned to consider the following question: What can we say about the unicity of finite order transcendental meromorphic solutions of the equation. For the unicity of finite order transcendental meromorphic solutions equation (2), we only need to consider the case that two CM shared values are 1, ∞. We conjecture that the conclusions in eorem 5 still hold when the shared condition “CM” is replaced by “IM.”

Proof of Theorem 5

Conclusion