Abstract
Many researchers’ attentions have been attracted to various growth properties of meromorphic solution f (of finite φ-order) of the following higher order linear difference equation Anzfz+n+...+A1zfz+1+A0zfz=0, where Anz,…,A0z are entire or meromorphic coefficients (of finite φ-order) in the complex plane (φ:[0,∞)→(0,∞) is a non-decreasing unbounded function). In this paper, by introducing a constant b (depending on φ) defined by lim̲r→∞logrlogφ(r)=b<∞, and we show how nicely diverse known results for the meromorphic solution f of finite φ-order of the above difference equation can be modified.
Highlights
Introduction and PreliminariesThroughout this paper, a meromorphic function is meant to be analytic in the whole complex plane C except possibly for poles
By introducing a constant b defined by log r lim log φ(r) = b < ∞, and we show how nicely diverse known results for the meromorphic solution r →∞
The readers are assumed to be familiar with the basic results and standard notations of Nevanlinna’s value distribution theory of meromorphic functions
Summary
Throughout this paper, a meromorphic function is meant to be analytic in the whole complex plane C except possibly for poles. For a more refined growth of meromorphic solutions of the Equation (1), the following (modified) definitions are recalled. The φ-order and the φ-lower order of a meromorphic function f are defined, respectively, as σ ( f , φ) = lim r →∞. If f is a meromorphic function (or an entire function) satisfying 0 < σ ( f , φ) = σ < ∞, φ-type of f is defined, respectively, as. N) be entire functions such that there exists an integer p (0 ≤ p ≤ n) satisfying max σ A j | j = 0, 1, . The following natural question is occurred: When the coefficients of the Equation (1) are entire or meromorphic functions of finite φ-order, what would the growth properties of solutions of the linear difference Equation (1) be like? We show how nicely diverse known results for the meromorphic solution f of finite φ-order of the difference Equation (1) can be amended
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