Pure Mathematics
Vol.4 No.03(2014), Article ID:13552,10 pages DOI:10.12677/PM.2014.43013

Hui Huang, Zongxuan Chen

School of Mathematical Sciences, South China Normal University, Guangzhou

Email: huanghui201209@126.com, chzx@vip.sina.com

Received: Feb. 25th, 2014; revised: Mar. 30th, 2014; accepted: Apr. 10th, 2014

ABSTRACT

In this paper, we investigate the growth of linear order meromorphic solution of higher order homogeneous and no-homogeneous linear differential equation, and we obtain some precise estimates for their zeros and hyper-orders.

Keywords:Linear Differential Equation, Meromorphic Functions, Hyper-Order, Hyper-Exponent of Convergence

1. 引言和结果

(1.1)

(1.2)

2. 引理

,若是线测度和对数测度均有穷的集合，则存在一列，且，当满足时，有

(2.1)

(2.2)

(2.3)

(2.4)

(2.5)

(2.6)

. (2.7)

(2.8)

. (2.9)

. (2.10)

. (2.11)

.

3. 定理1的证明

(3.1)

(3.2)

(3.3)

(3.4)

(3.5)

(3.6)

(3.7)

. (3.8)

. (3.9)

4. 定理2的证明

(4.1)

(4.2)

(4.3)

(4.4)

. (4.5)

(4.6)

(4.7)

(4.8)

(4.9)

(4.10)

(4.11)

(4.12)

(4.13)

(4.14)

(4.15)

(4.16)

，则对任给的，从而有

(4.17)

(4.18)

(4.19)

(4.20)

5. 定理3的证明

(5.1)

(5.2)

(5.3)

(5.4)

。显然(5.4)在上成立，且的线测度

(5.5)

，当充分大时，有

(5.6)

(5.7)

(5.8)

(5.9)

(5.10)

(5.11)

(5.12)

(5.13)

(5.14)

(5.15)

(5.16)

(5.17)

(5.18)

(5.19)

(5.20)

(5.21)

(5.22)

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