关于亚纯函够点、极点的比较定理

AComparative Theorm of the Zeros and Poles about Meromorphic Function

1982年M.H.Shih证明:在一包含原点的有界区域内解析、在边界上连续的函数f(z),若对边界上每一点Z,Re[Zf(z)]>0,则f(z)在D内恰有一个零点;1987年钟玉泉将f(z)在D内解析改为在D内除可能有极点外解析,其余条件不变,证明了:N(f,?D)-P(f,?D)=1。本文继此推广得到:

In this paper we mainly obtain the following Theorem: Suppose (i) Q is a inside or outside region, that is surrounded by a simple closed curve or finite simple closed curves, or an half plane. (ii) f(z), h(z) are meromorphic in Q, they are continuous and haven't Zero point on the boundary of Q. If ∝∈Q, it is not the zero point of f(z) and h(z). (iii) If the real part (or the imaginary part) of h(z)、f(z) is always positive or negative on the boundary of Q. Then the equality: N(f,?Q)-P(f,?Q)=N(h,?Q-P(h,?Q) (1) holds. The signs N(f,?Q) and P(f,?Q) are respectively expressed the total number in- cluding zero and poles of f(z) in Q (counted with their multiplicities).