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无零点亚纯函数族的正规与拟正规定则
程春暖，徐焱
南京师范大学数学科学学院，南京 210023
摘要：设 k, K 为正整数，φ(z)(≡ 0) 为解析函数，F 为在区域 D 内没有零点的亚纯函数族,
每个函数极点均为重级. 如果对任意的 f ∈ F，f
(k)
(z) − φ(z) 至多有 K 个不同零点, 则 F 在
D 内为至多 ν 阶的拟正规族，这里 ν = [
K
k+2
] 为不超过
K
k+2
的最大整数. 特别地，若
K = k + 1，则 F 在 D 内正规.
关键词：亚纯函数，正规族，拟正规族
中图分类号： O174.52
Normality and Quasinormality Criteria of
Zero-free Meromorphic Functions
Cheng Chunnuan, Xu Yan
School of Mathematics, Nanjing Normal University, Nanjing 210023
Abstract: Let k, K be positive integers, φ(z)(≡ 0) be an analytic function, and F be a
family of zero-free meromorphic functions on a domain D, all of whose poles are multiple. If
for each f ∈ F, f
(k)
(z) − φ(z) has at most K distinct zeros(ignoring multiplicity), then F is
quasinormal of order at most ν on D, where ν = [
K
k+2
] is equal to the largest integer not
exceeding
K
k+2
. In particular, if K = k + 1, then F is normal on D.
Key words: meromorphic function, normal family, quasinormal family.
0 Introduction
Let D be a domain in the complex plane C, and F be a family of meromorphic functions
defined on D. F is said to be normal (quasinormal of order ν) on D, in the sense of Montel, if for
any sequence {f
n
} ⊂ F there exists a subsequence {f
n
j
}, such that {f
n
j
} converges spherically
locally uniformly on D (minus a set containing at most ν points), to a meromorphic function
or ∞. The subtracted set may depend on the subsequence. Clearly, the family is quasinormal
of order 0 on D if and only if it is normal on D. See [1, 2, 3, 4].
In 1959, Hayman[5] proved that if f is meromorphic in C and if f = 0 and f
(k)
= 1 for a
positive integer k, then f is constant. The corresponding normality criterion is due to Gu[6],
Foundations: Research supported by NSFC(Grant Nos.11171045, 11471163) and SRFDP(Grant No.20123207110003).
Author Introduction: Cheng Chunnuan（1985-），female，Ph.D student，major research direction：complex analysis.
Correspondence author：Xu Yan（1964-），male，professor，major research direction：complex analysis.
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