On hyper-order of solutions of higher order linear differential equations with meromorphic coefficients

Jianren Long,Jun Zhu

doi:10.1186/s13662-016-0806-6

Abstract

In this paper, we investigate the growth of meromorphic solutions of the differential equations $$f^{(k)}+A_{k-1}(z)f^{(k-1)}+\cdots+A_{1}(z)f'+A_{0}(z)f=0 $$ and $$f^{(k)}+A_{k-1}(z)f^{(k-1)}+\cdots+A_{1}(z)f'+A_{0}(z)f=F(z), $$ where $A_{0}(z)\not\equiv0, A_{1}(z), \ldots, A_{k-1}(z)$ and $F(z)\not \equiv0$ are meromorphic functions. A precise estimation of the hyper-order of meromorphic solutions of the above equations is given provided that there exists one dominant coefficient, which improves and extends previous results given by Belaidi, Chen, etc.

Highlights

Introduction and main resultsFor a meromorphic function f in the complex plane C, the order of growth and the lower order of growth are defined as log+ T(r, f ) ρ(f ) = lim sup, r→∞ log r log+ T(r, f ) μ(f ) = lim inf, r→∞ log r respectively
In this paper, we investigate the growth of meromorphic solutions of the differential equations f (k) + Ak–1(z)f (k–1) + · · · + A1(z)f + A0(z)f = 0 and f (k) + Ak–1(z)f (k–1) + · · · + A1(z)f + A0(z)f = F(z), where A0(z) ≡ 0, A1(z), . . . , Ak–1(z) and F(z) ≡ 0 are meromorphic functions
1 Introduction and main results For a meromorphic function f in the complex plane C, the order of growth and the lower order of growth are defined as log+ T(r, f )

Summary

Introduction

Every solution of f (k) + Ak– (z)f (k– ) + · · · + A (z)f + A (z)f = F(z) is either a polynomial or an entire function of infinite order. Every nontrivial meromorphic solution f whose poles are of uniformly bounded multiplicities of equation

Results

Conclusion