Periodic points and normal families concerning multiplicity
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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 fk have no fixed points in D, is F normal in D? This problem was solved by Essén 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 fk 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 fk 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 “fk 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 fk 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 “fk having at most one fixed point in D” are the best possible.
Keywordsnormality iteration periodic points
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This work was supported by National Natural Science Foundation of China (Grant Nos. 11371149 and 11231009 ) and the Graduate Student Overseas Study Program from South China Agricultural University (Grant No. 2017LHPY003). The authors thank the referees for valuable suggestions.
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