Growth of Meromorphic Function Sharing Functions and Some Uniqueness Problems

Jianming Qi,Fanning Meng,Wenjun Yuan

doi:10.1155/2016/6250867

Abstract

Estimating the growth of meromorphic solutions has been an important topic of research in complex differential equations. In this paper, we devoted to considering uniqueness problems by estimating the growth of meromorphic functions. Further, some examples are given to show that the conclusions are meaningful.

Highlights

Introduction and Main ResultsAssuming that the reader is familiar with the notations and results on Nevanlinna theory [1] and the applications of normal family theory on estimating the growth of meromorphic functions, it is an interesting attempt to consider the growth properties of meromorphic functions under the condition involved sharing value or some complex differential equations.For a meromorphic function f, the order ρ(f) and hyperorder σ(f) of f are defined as follows [1]: ρ (f) fl lim sup r→∞log T (r, log r f) (1)σ log log T (r, log r f)Let f(z) and g(z) be two nonconstant meromorphic functions in the complex plane C, and let α(z) be a meromorphic function or a finite complex number
If g(z) − α(z) = 0 whenever f(z) − α(z) = 0 and the multiplicity of the zero z0 of g − α is greater than or equal to that of the zero z0 of f − α, we denote this condition by f(z) − α(z) = 0 󳨃→ g(z) − α(z) = 0
Let R be a rational function which behaves asymptotically crβ as r → ∞, where c ≠ 0, β are constants

Summary

Introduction and Main Results

Assuming that the reader is familiar with the notations and results on Nevanlinna theory [1] and the applications of normal family theory on estimating the growth of meromorphic functions (see [2,3,4]), it is an interesting attempt to consider the growth properties of meromorphic functions under the condition involved sharing value or some complex differential (or difference) equations (see [5,6,7,8,9]). Let f be a nonconstant meromorphic function with finitely many poles, and let R be a nonzero rational function. Let f be a nonconstant meromorphic function with finitely many poles, and let R be a nonzero rational function such that f and R have no common poles. Let f be a meromorphic function with at most finitely many poles, and let α = ReQ (α ≢ α󸀠), where R ( ≢ 0) is a rational function and Q is a nonconstant polynomial. Let f be a nonconstant transcendental meromorphic function with finitely many poles. Let α = ReQ (here Q is a polynomial and α ≢ α󸀠) be a function and let R be a nonzero rational function such that f and R have no common poles, and let L(f) be defined as (2).

Some Lemmas

Proof of Theorem 4

Proof of Theorem 9

Proof of Theorem 13