关于亚纯星形映射的弧长问题

On the Arc Length Problem of Meromorphic Starlike Mapping

若f属于亚纯星形函数族∑~*(p,w_0),以Lr(f)表示圆周|z|=r在w=f(z)映射下像的弧长。Miller J曾得到渐近结果Lr(f)=O(log(1/|r-p|(1-r)).当r→p时这一结果中的数量级有误。本文指出,正确的结论应当是Lr(f)=O(1/|r-p|)log(1/(1-r)),而且对渐近式中的数量阶进行了分析。

Suppose f belongs to the class∑~*(p,w_0)meromorphic starlike functions, let L_r(f) denote the arc length of the map of the circle |z|=r under the mapping w=f(z). In this note we correct the asymptotic estimation of J. for L_r(f).