1549055259-Ubiquitous_Quasidisk__The__Gehring_

102 8. FIRST SERIES OF IMPLICATIONS

f B

FIGURE 8.3

for z E 1, we see that it is sufficient to establish (8.1.6) and (8.1.7) for the case

where I is a hyperbolic line in D and D is bounded.

Assume now that this is the case and let D' and 1' be the images of D and 1

under

Z - Z 1

w=g(z) = --.

Z - Z2

Let f map H conformally onto D'. Again f extends to a homeomorphism of H

onto D' and we may assume that f(O) = 0 and f(oo) = oo.

Now D' is a K-quasidisk and 1' is the image of the positive imaginary axis

under f. Hence if w E 1', then w = f(iy) for some 0 < y < oo and

(8.1.8) s = 1y If' (it) I dt ::::; C1 dist(!( i y ), aD') = C1 dist( w, aD')

by Lemma 8.1.2 where s denotes the arclength of 1' between 0 and w. Let h = g-^1.

Then

length(/)= f lh'(w)l ldwl = lz1 - z2 1 J 1 ldwl

"I' "I' w - 1 12

(^00 ds

= lz1 - z2I lo lw(s) - 112 ,

where w = w( s) is the arclength representation for 1'. If we let

s ---C1

0 - C1+1 '

then for 0 < s ::::; so,

1

lw(s) - ll ::::>: 1-lw(s)I ::::>: 1 - s ::::>: cl+ 1 ,

while for s 0 ::::; s < oo,

lw(s) - ll ::::>: dist(w(s), aD') ::::>: ~

C1

by (8.1.8) since D c R^2 implies that

1 = g(oo) ej_ D'.

Thus

r= ds ro 1= c2

lo lw(s) - 112 ::::; lo (c1+1)2 ds + so s; ds = 2c1(c1+1) = c2