涉及正规族与分担值的Hayman问题

Hayman’s question concerning normal families and shared values

设n,k(n≥k+3)是两个正整数,a(≠=0),b是两个有穷复数,F是区域D内的一族亚纯函数,其中族中每个函数的零点都至少是k重.若对于F中的任意两个函数f,g,f(k)afn与g(k)agn在D内分担b,则F在D内正规.两个例子说明函数族中的每个函数的零点都至少是k重以及n≥k+3是最佳的.

Let n, k （n ≥ k ＋ 3） be two positive integers, let a （≠ 0）, b be two finite constants, and let F be a family of meromorphic functions in a domain D, all of whose zeros have multiplicity at least k. If, for each pair of functions f and g in F, f^（k） - af^n and g^（k） - ag^n share b in D, then F is normal in D. Two examples show that all zeros of each function in the family have multiplicity at least k and n ≥ k ＋ 3 are best possible.