1549055259-Ubiquitous_Quasidisk__The__Gehring_

w" J

11.3. HOMOGENEITY AND QUASIDISKS

/

w'J.

FIGURE 11. 6

157

then the points wj, w 1 , z 21 form triples which reduce the situation to case 1, since
by (11.3.2)

min(diam~~~~~~~~ ,81,diam,Bj) > ~~min{lw1 ~~~~~~~~ - z 21 I , lw1 - z11I}

lwj - w1I - lwj -w1I

lw J - z2 J ·I - lw J · - z1 J ·I -700

• 2r1 lw1 - z21I

where ,8 1 and ,Bj denote the components of oD \ { w 1 , wj}.
The situation r 1 = r > 0 for all j in (11.3.4) can also be treated much the same
way as in case 1. Fix a ED and let fj E QC(K) satisfy f1(D) = D and fj(a) = c1.

• 2 - 2
  Next let Lj : R -7 R be an affine map so that L 1 (B 1 ) =Band L(w1) = 1. Again
  the composed mappings 9jlD omit 1 and oo. Moreover, since 91(a) = 0 E 91(D)
  for all j, we conclude as before that, by passing to subsequences if necessary, the
  sequences (g 1 ) and (gj^1 ) converge uniformly to K-quasiconformal mappings g and
  g-^1 , respectively.
  Now let w'j be the last point on the ray { w 1 + t( Wj - z 21 ) : t 2: 0} which meets
  oD. We may suppose that

Here

_lim f 1 -^1 (z11) = Z1 E oD, _lim f1-^1 (z21) = Z2 E oD,

J-+ 00 J-+ 00

. 1 lm. f j-^1 ( w1 ) = w E u !:ID ,. l" lm 1-j^1 ( wj ") = W II E u !:ID.
J-+ 00 J-+ 00

1

1gjuj-^1 (z1j)) - 9juj-^1 (z2j))I = ILj(Z1j) - Lj(Z2j)I = - lz1j - Z2j1,

r

which tends to 0 by (11.3.3). Therefore z 1 = z 2. Furthermore,

91Uj -1 (z2j)) = 1 + - ,^1 gj(fj-^1 (wj)) = 1, gj(fj-^1 (w'j)) = -s

r