PERIODIC POINTS AND NORMALITY CONCERNING MEROMORPHIC FUNCTIONS WITH MULTIPLICITY

doi:10.1007/s10473-020-0515-9

数学物理学报(英文版) ›› 2020, Vol. 40 ›› Issue (5): 1429-1444.doi: 10.1007/s10473-020-0515-9

PERIODIC POINTS AND NORMALITY CONCERNING MEROMORPHIC FUNCTIONS WITH MULTIPLICITY

邓炳茂¹, 方明亮², 王跃飞^3,4

1. School of Financial Mathematics and Statistics, Guangdong University of Finance, Guangzhou 510521, China;
2. Department of Mathematics, Hangzhou Dianzi University, Hangzhou 310012, China;
3. School of Mathematics and Statistics, Shenzhen University, Shenzhen 518060, China;
4. Academy of Mathematics and Systems Science, Chinese Academy of Sciences, Beijing 100190, China

收稿日期:2018-12-04 修回日期:2020-01-04 出版日期:2020-10-25 发布日期:2020-11-04
通讯作者: Mingliang FANG E-mail:mlfang@hdu.edu.cn
作者简介:Bingmao DENG,E-mail:dbmao2012@163.com;Yuefei WANG,E-mail:wangyf@math.ac.cn
基金资助:
The first author was supported by the NNSF of China (11901119, 11701188); The third author was supported by the NNSF of China (11688101).

PERIODIC POINTS AND NORMALITY CONCERNING MEROMORPHIC FUNCTIONS WITH MULTIPLICITY

Bingmao DENG¹, Mingliang FANG², Yuefei WANG^3,4

1. School of Financial Mathematics and Statistics, Guangdong University of Finance, Guangzhou 510521, China;
2. Department of Mathematics, Hangzhou Dianzi University, Hangzhou 310012, China;
3. School of Mathematics and Statistics, Shenzhen University, Shenzhen 518060, China;
4. Academy of Mathematics and Systems Science, Chinese Academy of Sciences, Beijing 100190, China

Received:2018-12-04 Revised:2020-01-04 Online:2020-10-25 Published:2020-11-04
Contact: Mingliang FANG E-mail:mlfang@hdu.edu.cn
Supported by:
The first author was supported by the NNSF of China (11901119, 11701188); The third author was supported by the NNSF of China (11688101).

摘要/Abstract

摘要： In this article, two results concerning the periodic points and normality of meromorphic functions are obtained: (i) the exact lower bound for the numbers of periodic points of rational functions with multiple fixed points and zeros is proven by letting $R(z)$ be a non-polynomial rational function, and if all zeros and poles of $R(z)-z$ are multiple, then $R^k(z)$ has at least $k+1$ fixed points in the complex plane for each integer $k\ge 2$; (ii) a complete solution to the problem of normality of meromorphic functions with periodic points is given by letting $\mathcal{F}$ be a family of meromorphic functions in a domain $D$, and letting $k\ge 2$ be a positive integer. If, for each $f\in \mathcal{F}$, all zeros and poles of $f(z)-z$ are multiple, and its iteration $f^k$ has at most $k$ distinct fixed points in $D$, then $\mathcal{F}$ is normal in $D$. Examples show that all of the conditions are the best possible.

关键词: normality, iteration, periodic points

Abstract: In this article, two results concerning the periodic points and normality of meromorphic functions are obtained: (i) the exact lower bound for the numbers of periodic points of rational functions with multiple fixed points and zeros is proven by letting $R(z)$ be a non-polynomial rational function, and if all zeros and poles of $R(z)-z$ are multiple, then $R^k(z)$ has at least $k+1$ fixed points in the complex plane for each integer $k\ge 2$; (ii) a complete solution to the problem of normality of meromorphic functions with periodic points is given by letting $\mathcal{F}$ be a family of meromorphic functions in a domain $D$, and letting $k\ge 2$ be a positive integer. If, for each $f\in \mathcal{F}$, all zeros and poles of $f(z)-z$ are multiple, and its iteration $f^k$ has at most $k$ distinct fixed points in $D$, then $\mathcal{F}$ is normal in $D$. Examples show that all of the conditions are the best possible.

Key words: normality, iteration, periodic points

中图分类号:

30D45

邓炳茂, 方明亮, 王跃飞. PERIODIC POINTS AND NORMALITY CONCERNING MEROMORPHIC FUNCTIONS WITH MULTIPLICITY[J]. 数学物理学报(英文版), 2020, 40(5): 1429-1444.

Bingmao DENG, Mingliang FANG, Yuefei WANG. PERIODIC POINTS AND NORMALITY CONCERNING MEROMORPHIC FUNCTIONS WITH MULTIPLICITY[J]. Acta mathematica scientia,Series B, 2020, 40(5): 1429-1444.

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PERIODIC POINTS AND NORMALITY CONCERNING MEROMORPHIC FUNCTIONS WITH MULTIPLICITY

PERIODIC POINTS AND NORMALITY CONCERNING MEROMORPHIC FUNCTIONS WITH MULTIPLICITY

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