1549055259-Ubiquitous_Quasidisk__The__Gehring_

156 11. FOURTH SERIES OF IMPLICATIONS

-2 - 2

let L j : R --+ R be a n affine mapping so that

and let gj = Lj o fj for j = 1, 2,. ... Then

gj(a) = Lj(mj) = i/2

for all j and from the three-point condit ion we infer that ILj(wj)I and ILj(wj)I
tend to infinity, since

lw· J - z2·J I - IL(J w ·) J -11 --+oo

lz 1 j-Z2jl - 2 '

Because z 1 j, z 2 j E 8D, t he sequ ence of m aps gj omits the points -1, 1 and is
normal in D. Hence {gj} has a subsequence which converges uniformly on compact
subsets to g 0 : D --+ D, where g 0 is either a K -quasiconformal mapping or a
constant. See Lehto-Virtanen [117]. Since H C gj(D) for every j, the constant
must lie in g 0 (D) \ H. But we know that gj(a) = i/2 so this is impossible. Hence
g 0 is a K-quasiconformal mapping.
By passing to a subsequence once more, we conclude that the sequence {gj}
converges uniformly to a K-quasiconformal mapping g: R
2
--+ R
2
where gjD = g 0.
Moreover, g is injective and since the inverse mappings gj^1 converge uniformly to
g-^1 , we may assume that

Hm fj-^1 (z1j) = z1 E 8D,

J-'>00

_lim 1j-^1 (z2j) = Z2 E 8D,

J-'>00

_lim fj-^1 (wj) = w E 8D,

J-'>00

Hm j-:--^1 (w'·) = w' E 8D.

J-'>00 J J

By the construction of g, g(zi) = -1, g(z 2 ) = 1, and g(w) = g(w') = oo. But this
implies that w = w' and that w is different from the distinct points z 1 , z 2. This is
impossible.

Proof for case 2

We may assume that

lwi - Z2jl =sup {lw - z2jl : w E /'j} > lz1j - Z2jl

for all j. Then since /'j C B(z 2 j, lwj - z 2 jl) while dist(wj,/'j) > 0 , there is an
rj > 0 such that

(11.3.4)

with Cj = Wj + rj(Wj - Z2j)-
Next by passing to a subsequence we assume that rj = r > 0 for all j. To see
t his, let rj b e maximal in (11.3.4) and choose

If

lim r = 0

j-too J '