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 Rk(z) has at least k + 1 fixed points in the complex plane for each integer k ≥ 2; (ii) a complete solution to the problem of normality of meromorphic functions with periodic points is given by letting ℱ be a family of meromorphic functions in a domain D, and letting k ≥ 2 be a positive integer. If, for each f ∈ ℱ, all zeros and poles of f(z) − z are multiple, and its iteration fk has at most k distinct fixed points in D, then ℱ is normal in D. Examples show that all of the conditions are the best possible.
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The first author was supported by the NNSF of China (11901119, 11701188); The third author was supported by the NNSF of China (11688101).
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Deng, B., Fang, M. & Wang, Y. Periodic Points and Normality Concerning Meromorphic Functions with Multiplicity. Acta Math Sci 40, 1429–1444 (2020). https://doi.org/10.1007/s10473-020-0515-9
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DOI: https://doi.org/10.1007/s10473-020-0515-9