24 
MATHEMATICS: T. H. GRONWALL 
Since S w | fl^n|^ converges, the right hand member in (2) may be made less 
than an arbitrarily small 5 for p < sin e by taking N sufficiently large, and 
having fixed N, the right hand member in (l) may be made less than 8 by tak- 
ing p sufficiently small. Thus, for p sufficiently small, \w{z) — w{z') | < 2'5 
independently of d and 6', which proves the theorem. 
Let us now assume in particular that w = w{z) maps the circle | z | < 1 
conformally on a simple (i.e. simply connected and nowhere overlapping) 
region D in the w-plane, and that all points of D are within a circle of 
radius R (this latter condition can always be brought about by a linear trans- 
formation on w and the extraction of a square root^; then 
converges^ (and is less than i?^). Finally, suppose that 1^(2') approaches 
a limit Wq when approaches unity on the real axis; our theorem then 
shows that 'w{z) approaches the same point Wq on the boundary of D 
when z tends toward unity along any curve interior to both the unit circle 
and an angle less than tt formed by two straight lines through z = 1 and 
symmetrical in respect to the real axis. This proposition is usually derived,, 
somewhat less directly, from the distortion theorem.^ 
iKoebe, /. Math., Berlin, 145, 1915. 
^Fe}6T, Festschrift . . . H. A. Schwarz, Berlin, 1914:, 
'Koebe, 1. c, and Study, Konforme Abbildung einfach zusammenhangender Bereiche^ 
Leipzig, 1913. 
