On the growth of solutions of a class of second-order complex differential equations

Cai Feng Yi,Hong Yan Xu,Xu-Qiang Liu

doi:10.1186/1687-1847-2013-188

Cai Feng Yi, Hong Yan Xu + Show 1 more

Open Access

https://doi.org/10.1186/1687-1847-2013-188

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Abstract

In this paper, we consider the differential equation , where and are meromorphic functions, is a non-constant polynomial. Assume that has an infinite deficient value and finitely many Borel directions. We give some conditions on which guarantee that every solution of the equation has infinite order. MSC:34AD20, 30D35.

Highlights

Introduction and main resultsIn this paper, we shall involve the deficient value and the Borel direction in investigating the growth of solutions of the second-order linear differential equation f + h(z)eP(z)f + Q(z)f =, ( )where h(z) and Q(z) ≡ are meromorphic functions, P(z) is a non-constant polynomial
We assume that the reader is familiar with the Nevanlinna theory of meromorphic functions and the basic notions such as N(r, f ), m(r, f ), T(r, f ) and δ(r, f )
There are some equations of the form ( ) that possess a solution f ≡ which has finite order; for example, f (z) = ez satisfies the equation f + e–zf – (e–z + )f =

Summary

Introduction

Suppose that Q(z) is a finite-order meromorphic function having an infinite deficient value, Q(z) has only finitely many Borel directions: Bj : arg z = θj [ ] Let f (z) be a transcendental meromorphic function with finite order σ , there exists a function λ(r) with the following properties: (i) λ(r) is a non-negative and continuous function for r ≥ with limr−→∞ λ(r) = σ .

Results

Conclusion