势位论原理在共形映射上的新应用

利用势位论原理，借助有界单连城关于Ｏ的内映射半径和外映射半径，研究复平面上的有界 环域．建立由含原点的有界单连域Ｄ２挖去含原点的单连域Ｄ１的闭包所形成的环域关于它们的内 外映射半径的不等式，从而得到有界环域成为以原点为中心的圆环的一些充要条件．建立环域模 数的一个不等式，从而得到环域成为圆环的相应充要条件。给出套环的模数定理的一个简捷证 明。

Some New Applications of Potential Theory to Conformal Mappings

In this paper, using the principle of potential theory, we study a bounded ring domain on the complex plane by means of the inner mapping radius with respect to origin and the outer mapping radius of bounded simply-connected domains. First, we establish an inequality for the inner mapping radius with respect to origin and the outer mapping radius of the bounded simply-connected domain D 1 and D2, containing the origin, which consists of a bounded ring domain D2 omitted the clocure of D1. And then we obtain some necessary and sufficient conditions for a ring domain becoming a circular ring with the origin as its center. Second. we establish an inequality for the modulus of a ring domain and then obtain corresponding necessary and sufficient condition for a ring domain becoming a circular ring. As an application of the preceding results, we give finally a simp1er proof of the theorem on the modulus of a ring domain overlaping one another.