正规族与高阶导数分担值

A Normal Family and Shared Values of Higher Order Derivatives

设f(z)和g(z)在复平面的一个区域G内亚纯,α∈C=C∪│∞│,若f(z)-α和g(z)-α在G内具有相同的零点,则α称为函数f(z)和g(z)在G内的分担值,当零点计重数或不计重数时,则α分别称为函数f(z)和g(z)在G内的CM分担值或IM分担值.研究在函数与其高阶导数具有分担值的条件下函数族的正规性定则,证明了一个区域G上的全纯函数族F是正规的,如果两个不同的有穷复数为族F中每个函数及其k阶导数在G中的CM分担值,且族F中每个函数的零点重级≥k(k为自然数).例子表明本文定理中对函数零点重级的限制至少在k=2时是精确的.

Let f(z) and g (z) be meromorphic functions in a domain G of the fcomplex plane, α ∈ C = C ∪│∞│ . If /(z) - α and g(z) - α have same zeros in G,α is said to be a shared value in G of f(z) and g(z); when zeros are counted in multiplicities or not respectively, a is said to be a CM or IM shared value respectively.The first approach to normality of a family of meromorphic functions concerning shared values was done by [1]. He proved that a family of meromorphic functions in a domain G was a normal family, if three distinct finite complex numbers α1 ,α2,α3 were IM shared values in G of/and its derivative f' for each function f in the family.Recently, Chen and Hua[2], Xu[3,4] have studied this kind of problem, and have obtained many normality criteria of families of holomorphic functions under the condition that each function in families and its lower order derivatives (mainly one order derivative)share the same values.This paper studies the problem under the condition that each function in families and its higher order derivative share the same values, and shows that a family F of holomorphic functions in a domain G is a normal family, if f and its k-th order derivative f(k) share two distinct finite values CM in G for each function f in the family F and each function in F has only zeros of order≥k, where k is a positive integer.The counterexample shows that the constriction to the multiplicities of zeros of functions in our theorem is precise at least when k = 2.