92 7. MISCELLANEOUS PROPERTIES
FIGURE 7.1Hjelle shows that D is the union of an increasing sequence of simply connected
domains which lie in {z = x + iy: !xi < 2}. Hence Dis itself a simply connected
domain. See, for example, page 81 in Newman [140].
Next Do is a Jordan domain which contains E. A theorem due to Teichmi.iller
implies that for each p air of points z 1 , z 2 E E there exists a K-quasiconformal
m apping f : R^2 ---+ R^2 such that
1° f(z1) = z2,
2° f(z ) = z for z E R^2 \Do,where K depends only on hDo (z 1 , z 2 ). See Teichmi.iller [158]. Thus D is homoge-
neous with respect to the family
F =Go {!} o G c QC(K).
However D is not locally connected a t oo E 8D. Hence D is not a Jordan doma in
and, in p articular, not a quasidisk.
On the other hand, the hypothesis that D be a Jordan domain in Theorem 7.4.2
is not needed when K = 1.
THEOREM 7.4.4 (Erka ma [38], Kimel'fel'd [103]). A simply connected domain
D is a disk or half-plane if and only if it is homogeneous with respect to the family
QC(l).