亚纯函数的例外集

Exceptional Sets for Meromorphic Functions

自从Ｌｅｈｔｏ Ｏ．［１］提出例外集的概念后，这方面已有不少工作，但这些工作基本上是局限于整函数来做的，主要原因是极点给证明带来很大困难．本文证明了任一个满足δ（∞，ｆ）＞３／４的超越亚纯函数ｆ（ｚ），ｆｋＱ［ｆ］在任一不含ｆ和Ｑ［ｆ］极点的可数个圆盘并集之外取任何非零复数无穷次，其中ｋ≥１５／８＋１／２ГＱ，ГＱ是ｆ的微分多项式 Ｑ［ｆ］的权．本文的结果推广了 Ｈａｘｍａｎ等人的结论．

Since Lehto O.[1] presented the concept of exceptional set of entire functions and meromorphic functions, many results have been got. But these results are only on entire functions, the main reasoin is the difficult to prove caused by poles. The paper proves the following result which extended the result of Hayman. Suppose that f is a transcendental meromorphic function such that δ(∞, f) > 3/4, Q[f] is a differential polynomial in f of weight ГQ, and k is an integer not less than 15/8 + 1/2ГQ. Then fkQ[f] assumes ail Values w∈ C, except possibly zero,infinitely often, outside the union of the discs which do not contain the poles of f and Q[f].