Periodic Points and Normality Concerning Meromorphic Functions with Multiplicity

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 ≥ 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 f k 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.