PERIODIC POINTS AND NORMALITY CONCERNING MEROMORPHIC FUNCTIONS WITH MULTIPLICITY

【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 nonpolynomial 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 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.更多还原

【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 nonpolynomial 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 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.更多还原