Periodic points and normal families concerning multiplicity

Autor: Fang Mingliang, Wang Yuefei, Deng Bingmao
Rok vydání: 2018
Předmět:
Zdroj: Science China Mathematics. 62:535-552
ISSN: 1869-1862
1674-7283
DOI: 10.1007/s11425-016-9185-4
Popis: In 1992, Yang Lo posed the following problem: let $\mathcal{F}$ be a family of entire functions, let $D$ be a domain in $\mathbb{C}$, and let $k\ge~2$ be a positive integer.If, for every $f\in~\mathcal{F}$, both $f$ and its iteration $f^k$ have no fixed points in $D$, is $\mathcal{F}$ normal in $D$?This problem was solved by Ess${\rm\acute{e}}$n 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. (I) Let $\mathcal{F}$ be a family of holomorphic functions in a domain $D$ and let $k\ge~2$ be a positive integer.If, for each $f\in~\mathcal{F}$, all zeros of $f(z)-z$ are multiple and $f^k$ has at most $k$ distinct fixed points in $D$, then $\mathcal{F}$ 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. (II) Let $\mathcal{F}$ be a family of meromorphic functions in a domain $D$, and let $k\ge~2$ and $l$ be two positive integers satisfying $l\ge~4$ for $k=2$ and $l\ge~3$ for $k\ge~3$.If, for each $f\in~\mathcal{F}$, all zeros of $f(z)-z$ have a multiplicity at least $l$ and $f^k$ has at most one fixed point in $D$, then $\mathcal{F}$ is normal in $D$. Examples show that the conditions “$l\ge~3$ for $k\ge~3$" and “$f^k$ having at most one fixed point in $D$" are the best possible.
Databáze: OpenAIRE
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