1549055259-Ubiquitous_Quasidisk__The__Gehring_

160 11. FOURTH SERIES OF IMPLICATIONS

Now let r 1 be the family of curves in r D which lie in B ( Z2, s) where

(11.4.7) s = r (47fC)

4

exp b; = r b1,

and let f2 = fv \ f1. If each curve 'YE f1 has length Z('Y) ?'.: l, then

{

z-^1 if iz - z2I:::; s,

p(z) = 0

otherwise

is in adm(f 1 ) and hence

(11.4.8)

Next each curve 'YE f2 joins the circles lz - z2I = r/4 and lz - z2I = s. Thus

p(z) = {(log(4s/r) iz - z2i)-^1 if r/4 ~ lz - z2I:::; s,

0 otherwise

is in adm(f2) and

(11.4.9)

27f

mod(f2) :::; log(4s/r)

Inequalities (11.4.6), (11.4.8), and(ll.4.9) with (11.4.7) then imply that

~ 7f~ ~

-Z:::; mod(fv):::; mod(f1) + mod(f2):::; ~ +

2

c

and we obtain

where b 1 = 87f /bo = 48.
Set b2 = exp(b1c). Then there is a rectifiable curve 'Yo E fv with
length('Yo):::; b2r = b2lz1 - z2I

and endpoints w 1 and w 1 in C such that

Next set ri = lz1 - w1 I and let C1 and C2 denote the components of C n
B(z1, ri/4) and 'Yon B(wi, ri/4) which contain z 1 and w 1 , respectively. Then

min diam( Ci) ?'.: ri ?'.: ~dist( C1, C2)

J=l,2 4 4

and arguing as above we obtain a curve 'Yi in D with

lz1 - z2I

length('Y1) :::; b2 42

such that 'Yi joins 'Yo to a point w 2 E C with

1

lz1 - w2I :::; 42 lz1 - z2I·

An analogous procedure yields a subcurve ii in D with

_ lz1 - z2I

length( 'Y1) :::; b2 42