Periodic points and normal families concerning multiplicity

doi:10.1007/s11425-016-9185-4

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	Periodic points and normal families concerning multiplicity
	Deng, Bingmao 1; Fang, Mingliang 1; Wang, Yuefei2,3
刊名	SCIENCE CHINA-MATHEMATICS
	2019-03-01
卷号	62 期号:3 页码:535-552
关键词	normality iteration periodic points 35D45
ISSN号	1674-7283
DOI	10.1007/s11425-016-9185-4
英文摘要	In 1992, Yang Lo posed the following problem: let F be a family of entire functions, let D be a domain in C, and let k 2 be a positive integer. If, for every f F, both f and its iteration f(k) have no fixed points in D, is F normal in D? This problem was solved by Essen and Wu in 1998, and then solved for meromorphic functions by Chang and Fang in 2005. In this paper, we study the problem in which f and f(k) have fixed points. We give positive answers for holomorphic and meromorphic functions.Let F be a family of holomorphic functions in a domain D and let k 2 be a positive integer. If, for each f F, all zeros of f(z) - z are multiple and f(k) has at most k distinct fixed points in D, then F is normal in D. Examples show that the conditions all zeros of f(z) - z are multiple and f(k) having at most k distinct fixed points in D are the best possible.Let F be a family of meromorphic functions in a domain D, and let k 2; l be two positive integers satisfying l 4 for k = 2 and l 3 for k 3. If, for each f F, all zeros of f(z) - z have a multiplicity at least l and f(k) has at most one fixed point in D, then F is normal in D. Examples show that the conditions l 3 for k 3 and f(k) having at most one fixed point in D are the best possible.
资助项目	National Natural Science Foundation of China[11371149] ; National Natural Science Foundation of China[11231009] ; Graduate Student Overseas Study Program from South China Agricultural University[2017LHPY003]
WOS研究方向	Mathematics
语种	英语
出版者	SCIENCE PRESS
WOS记录号	WOS:000457383300006
内容类型	期刊论文
源URL	[http://ir.amss.ac.cn/handle/2S8OKBNM/32441]
专题	数学所
通讯作者	Fang, Mingliang
作者单位	1.South China Agr Univ, Inst Appl Math, Guangzhou 510642, Guangdong, Peoples R China 2.Chinese Acad Sci, Acad Math & Syst Sci, Beijing 100190, Peoples R China 3.Univ Chinese Acad Sci, Sch Math Sci, Beijing 100049, Peoples R China
推荐引用方式 GB/T 7714	Deng, Bingmao,Fang, Mingliang,Wang, Yuefei. Periodic points and normal families concerning multiplicity[J]. SCIENCE CHINA-MATHEMATICS,2019,62(3):535-552.
APA	Deng, Bingmao,Fang, Mingliang,&Wang, Yuefei.(2019).Periodic points and normal families concerning multiplicity.SCIENCE CHINA-MATHEMATICS,62(3),535-552.
MLA	Deng, Bingmao,et al."Periodic points and normal families concerning multiplicity".SCIENCE CHINA-MATHEMATICS 62.3(2019):535-552.