﻿ 涉及fnf(k)+H(f)-b 的零点重级的正规定则 A Normality Criterion Concerning the Zeros’ Multiplicity of fnf(k)+H(f)-b

Pure Mathematics
Vol.06 No.06(2016), Article ID:19053,10 pages
10.12677/PM.2016.66067

A Normality Criterion Concerning the Zeros’ Multiplicity of

Jing Li1, Jun’an Zhao2, Bingmao Deng1

1Institute of Applied Mathematics, South China Agricultural University, Guangzhou Guangdong

2Department of Mathematics, Jinan University, Guangzhou Guangdong

Received: Nov. 5th, 2016; accepted: Nov. 20th, 2016; published: Nov. 28th, 2016

ABSTRACT

In this paper, we study the normality of holomorphic functions and prove the following results: Let be three positive integers satisfying when and when, is a finite complex number; let be a family of holomorphic functions in a domain and be a differential polynomial of and satisfy, if for each, satisfies (1) all zeros of have multiplicity at least; (2) all zeros of have multiplicity, then is normal in.

Keywords:Meromorphic Function, Normal Family, Zalcman Lemma, Differential Polynomial

1华南农业大学数学研究所，广东 广州

2暨南大学数学系，广东 广州

1. 引言及主要结果

1965年，杨乐和张广厚 [3] 证明了

1982年，Oshkin [4] 进一步证明了

1993年，方明亮和徐万松 [5] 推广了上述定理，把换成了的线性微分多项式，证明了

2. 几个引理

a) 实数

b) 点列

c) 正数列

d) 函数列

(1)

(2)

(3)

(4)

(5)

，则有

(6)

(7)

(8)

(9)

(10)

(11)

(12)

(13)

(14)

(15)

(16)

，且

(17)

(18)

(19)

(20)

(21)

(22)

(23)

(24)

(25)

(26)

(27)

(28)

，这与是超越整函数矛盾。

(29)

(30)

，即。矛盾。

，则(为一个常数)。如果，则为次数的多项式，又因为的零点重级，则恒为常数，矛盾；如果，显然没有零点也没有极点，且，由Nevanlinna第一基本定理得

3. 定理1的证明

a) 实数,

b) 点列,

c) 正数列

d) 函数列

4. 推论2的证明

A Normality Criterion Concerning the Zeros’ Multiplicity of fnf(k)+H(f)-b[J]. 理论数学, 2016, 06(06): 486-495. http://dx.doi.org/10.12677/PM.2016.66067

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3. 3. Yang, L. and Zhang, G.H. (1965) Recherchessur la normalit des familles de fonctionsanalytiquesa des valeurs multiples. Un nouveau crit re etquelques applications. Sci. Sinica, 14, 1258-1271.

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