1549055259-Ubiquitous_Quasidisk__The__Gehring_

142 10. THIRD SERIES OF IMPLICATIONS

We shall show first that

(10.3.12)

length('Y(zj, Wj)) ::::; b dist( Wj, oD),

length(t(zj, z)) ::::; be^612 dist(z, 8D)

for j = 1, 2 and z E 'f'(Zj,wj) where b = 64a^2. We need only consider the case

where j = 1 and m 1 ~ 1.

Choose points ( 1 ,( 2 ,.. .,(m 1 +i E 1'(z 1 ,wi) so that ( 1 = z1 and (k is the first

point of 1'(z1, w1) with

(10.3.13)

as we traverse 1' from z 1 towards w 1. Then (m, + 1 = w 1. Fix k , 1 ::::; k ::::; m 1 , and

let

t = length('Y((k,(k+1)) > l(k - (k+1I.

dist((k, oD) - dist((k, oD)

If z E "r'((k, (k+1), then

dist(z, 8D) ::::; dist((k+i, 8D) = 2 dist((k, 8D)

and

t::::; 21 k dist(~~8D)::::; 4hD((k,(k+i),

where "r'k = "r'((k, (k+1). Next

JD. ( (k,(k+i ) ::::;2log ( dist((kl(k - (k+i ,oD) I +1 ) ::::;2logt+l ( ) ::::;2vt ri

since the function

f(x) = Vx -log(x + 1)

is increasing for x > 0 with f (0) = 0. Hence

t::::; 4hD((k,(k+i)::::; 4afo((k,(k+1)::::; 8aVt

by (10.3.4), whence t::::; 64a^2 =band

hD((k, (k+l) ::::; 2a Vt::::; b/4.

Next if z E "r'((k> (k+i), then

(

dist((k+ 1 , 8D))

0 < log dist(z, oD) ::::; 2 hD(z, (k+1)::::; 2 hD((k, (k+^1 )::::; b/2

by Lemma 10.3.1. We conclude that

length('Y((k, (k+i))::::; bdist((k, 8D),

(10.3.14)

dist((k+ 1 , 8D)::::; e^612 dist(z, 8D)

fork= 1, 2,... , m1 and z E "r'((k, (k+1).

We now complete the proof for the inequalities in (10.3.12) as follows. By

(10.3.13) and (10.3.14)

m1 m1

length('Y(z1, w1)) = L length('Y((k, (k+i)) ::::; b L dist((k, 8D)

k=l k=l

= b (2m^1 - 1) dist(z 1 , 8D) < bdist(w 1 , 8D).