1549055259-Ubiquitous_Quasidisk__The__Gehring_

10.3. HYPERBOLIC METRIC AND HYPERBOLIC SEGMENTS 143

This is the first inequality. For the second, if z E 'Y( z 1 , w 1 ), then z E 'Y( (k, (k+l) for

some k, 1 _.:::: k _.:::: m 1 , and

k k

length('Y(z 1 , z)) _.:::: Llength('Y((k,(k+ 1 )) _.:::: b Ldist((k,8D)

j=l j=l

= b(2k - 1) dist(z1, 8D) < b dist( (k+i, 8D)

_.:::: beb/^2 dist(z , 8D)

again by (10.3.13) and (10.3.14).

We show next that if dist(w 1 , 8D) _.:::: dist(w2, 8D), then

(10.3.15)

length('/'( w 1 , w2)) _.:::: 2be^2 b dist( w 1 , 8D),

dist(w2, 8D) _.:::: e^2 b dist(z, 8D)

for all z E 'Y( w 1 , w2). We may assume that w 1 =/:. w 2 since otherwise there is nothing
to prove. By hypothesis

r 2". max(dist(zj, 8D)),

J=l,2

where r is as in (10.3.6), and we have the following two possible subcases:

(10.3.16)

(10.3.17)

r =sup dist(z, 8D),

ZE"f

r=2lz1-z2I·

If (10.3.16) holds, set

t _ _le_n_gt_h--,.('Y_(_w_ 1 _, w--,.--- 2 )_)

• dist(w1, aD).

Then

dist(z,8D) _.:::: r < 2dist(w 1 ,8D)

if z E 'Y(w 1 , w 2 ) by (10.3.11) and we can repeat the proof for the first part of
(10.3.14), with w 1 in place of (k and W2 in place of (k+i, to obtain (10.3.15).
Next if (10.3.17) holds, then

lw1 - w2I _.:::: L length('Y(zj, Wj)) + lz1 - z2I

j=l,2

_.:::: b L dist(wj,8D) +r/2

j=l,2

_.:::: 4b dist( W1, aD)

by (10.3.11) and the first part of (10.3.12). Hence

hD(w1, w2) .:::: a]D(w1, w2) .:::: 2a log (di;(~,~~)) _.:::: 2a log 5b < b.

If z E 'Y(w 1 ,w2), then

e-^2 b dist(w 2 , 8D) .:::: dist(z, 8D) .:::: e^2 b dist(w18D)