与零点个数有关的亚纯函数正规定则

数学物理学报 ›› 2017, Vol. 37 ›› Issue (1): 7-17.

与零点个数有关的亚纯函数正规定则

王琼¹, 陈玮², 袁文俊³, 田宏根¹

1. 新疆师范大学数学科学学院乌鲁木齐 830054;
2. 山东大学数学学院济南 250100;
3. 广东大学数学与信息科学学院广州 510006

收稿日期:2016-06-22 修回日期:2016-12-02 出版日期:2017-02-26 发布日期:2017-02-26
通讯作者: 田宏根 E-mail:tianhg@xjnu.cn
作者简介:王琼,E-mail:wq1298592600@163.com
基金资助:
国家自然科学基金（11461070，11271090）、广东省自然科学基金（S2012010010121，2015A030313346）和新疆科研创新项目（XJGRI2015106）

Normality Criteria of Meromorphic Functions with Zero Numbers

Wang Qiong¹, Chen Wei², Yuan Wenjun³, Tian Honggen¹

1. School of Mathematics Science, Xinjiang Normal University, Urumqi 830054;
2. Department of Mathematics, Shandong University, Jinan 250100;
3. School of Mathematics and Information Sciences, Guangzhou University, Guangzhou 510006

Received:2016-06-22 Revised:2016-12-02 Online:2017-02-26 Published:2017-02-26
Supported by:
Supported by the NSFC (11461070, 11271090), the Nature Science Foundation of Guangdong Province (S2012010010121, 2015A030313346) and Innovation Projects (XJGRI2015106) of Xinjiang Province

摘要/Abstract

摘要：

文章证明了涉及零点个数的亚纯函数族的正规定则：设F为区域D内的一族亚纯函数，a（≠0），b为两个有穷复数，m，n，k为正整数，其中n≥m+2，设任意函数f∈F且f零点重级至少是k和极点重级至少是k+1，当f^（k）-afⁿ-b至多有m个不同零点时，则F在区域D内正规.这一结果提高了邓炳茂等人^[18]的定理1，并推广了Ye等^[16]，张庆彩等^[22]及陈玮等^[19]的相关结果.此外，我们举例说明了结论的精确性.

关键词: 亚纯函数, 零点个数, 正规定则

Abstract:

This paper considers normality criteria for a family of meromorphic functions concerning zero numbers. Let F be a family of meromorphic functions defined in a domain D, all of whose zeros and poles have multiplicity at least k and k+1 respectively, let m, n and k be three positive integers satisfying n≥m+2, and a(≠0), b be two finite constants. If for each function f∈F, f^(k)-afⁿ-b has at most m distinct zeros in D, then F is normal in D. Our results improve theorem 1 by Deng et al.^[18] and generalize the related theorems of Ye et al.^[16], Zhang et al.^[22] and Chen et al.^[19]. Meanwhile, some examples are given to show the sharpness of our results.

Key words: Meromorphic function, Zero numbers, Normal criterion

中图分类号:

O174.52

王琼, 陈玮, 袁文俊, 田宏根. 与零点个数有关的亚纯函数正规定则[J]. 数学物理学报, 2017, 37(1): 7-17.

Wang Qiong, Chen Wei, Yuan Wenjun, Tian Honggen. Normality Criteria of Meromorphic Functions with Zero Numbers[J]. Acta mathematica scientia,Series A, 2017, 37(1): 7-17.

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