On the Boundary Behavior of a Conformal Map

The object of this paper is to indicate the immediate usefulness of Caratheodory's theory of the conformal mapping of variable regions in the study of boundary behavior of a fixed but arbitrary conformal map. We study especially the mapping of an infinite strip and its behavior at infinity. Studies of an infinite strip by other methods have previously been made, especially by Ahlfors, Ferrand, Ostrowski, and Warschawski; for a resume see Gattegno and Ostrowski [6]. The present results are more directly geometric than these previous ones in both method and conclusion; broadly speaking, they are in some respects more and in other respects less general than the previous ones. The theory of conformal mapping of variable regions introduced by Caratheodory in 1912 was employed by Montel in 1917 to study the properties of prime ends under conformal mapping. That theory has more recently been used in the study of boundary behavior of conformal maps by Ferrand [7; 8], emphasized by Walsh [2], and used for the study of strips by Madame Lelong-Ferrand [3]. The essential difference between the latter and the present paper is that here we consistently use both translation and stretching of the original region to obtain a sequence of variable regions possessing a kernel, whereas Madame Lelong-Ferrand uses primarily translation. The present results are thus more general, both where the width of the strip (the variable 2<b(u), in the notation of §2 below) has an infinite number of limit values, and more especially where that limit is zero or infinite. We introduce a new condition (property B, below) on the boundary of an infinite strip, which is useful in the study of conformal mapping of the strip. The intrinsic properties of this condition are studied in §1, and applications to conformal mapping in successively more general situations are considered in §§2-5. The extension of property B and its implications for finite boundary points are studied in §6, and the relation of property B to various other conditions on the boundary of an infinite strip is investigated in §7. Property B is exhibited as a sufficient (but not necessary) condition for the fundamental asymptotic relations, such as (2.1), a less restrictive condition than that for an 7,-strip. 1. Property B. If <p(u) is a real function of the real variable u defined for