Periodic points and normal families concerning multiplicity | |
Deng, Bingmao1; Fang, Mingliang1; Wang, Yuefei2,3 | |
2019-03-01 | |
Source Publication | SCIENCE CHINA-MATHEMATICS |
ISSN | 1674-7283 |
Volume | 62Issue:3Pages:535-552 |
Abstract | 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. |
Keyword | normality iteration periodic points 35D45 |
DOI | 10.1007/s11425-016-9185-4 |
Language | 英语 |
Funding Project | 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 Research Area | Mathematics |
WOS Subject | Mathematics, Applied ; Mathematics |
WOS ID | WOS:000457383300006 |
Publisher | SCIENCE PRESS |
Citation statistics | |
Document Type | 期刊论文 |
Identifier | http://ir.amss.ac.cn/handle/2S8OKBNM/32442 |
Collection | 数学所 |
Corresponding Author | Fang, Mingliang |
Affiliation | 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 |
Recommended Citation 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. |
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