1549055259-Ubiquitous_Quasidisk__The__Gehring_

132 9. SECOND SERIES OF IMPLICATIONS

by (9.4.21). Then F(z) = E j(z) - Ej+ 1 (z) is linear in x and y where z = x + iy
and F(B*) is contained in a closed disk of radius

Hence

(9.4.24)

t = a(L - l){rj + rj+1).

2t

IEj(z) - Ej+ 1 (z) I = IF(z)I:::; -lz l = 2a(L -1)(1 + q)lzl

rj

for z E R^2. Next (j E Bj n Bj+l and inequality (9.4.23) with z = (j yields

(9.4.25)

Finally, if n > j, then

l(n - (jl :::; Sn - Sj = (qj - qn) l:::; qj l

and we obtain

lgj((n) - 9j+1((n)I:::; lgj((j) - 9j+1((j)I + IEj((n - (j) - Ej+I((n - (j)I

:::; a(L - 1)(1+q)qJ(ro+2l)

from (9.4.22), (9.4.24), and (9.4.25). This, together with (9.4.19), implies that

IJ((n) - 9o((n)I = lgn((n) - 9o((n)I

n-1

j=O

l+q

:::; a(L-1)--(ro + 2l)

1-q

= a(L - 1) (ro + 2z)2

ro

:::; 9 ac L^2 ( L - 1) l.

Here (n-7Z1 as n-too and we conclude that

lf(z1) - 9o(z1)I:::; 9acL^2 (L - 1) length(l(z1, zo))

and hence with (9.4.17) that

lf(z1) - go(z1)I:::; 9ac2 L2(L -1) lz1; z2I < lz1; z2I

when 1 < L < L(c). Finally,

lz1 - z2I

lf(z2) - 9o(z2)I < 2

by the above argument applied to z 0 , z 2 and f (zi) -I f (z2).

9.5. Rigid domains are linearly locally connected

0

We show next that rigid domains are linearly locally connected. Our proof

makes use of the following technical lemma concerning a special class of bilipschitz

maps.