1549055259-Ubiquitous_Quasidisk__The__Gehring_

8.7. REFLECTIONS AND QUASIDISKS 113

The following result yields a Euclidean analo gue of Theorem 2.1.11.

-2 -2

LEMMA 8.7.3. Suppose that h: R -tR is K-quasiconformal with h(oo) = oo

and that

(8.7.4)

for z 1 , z2 E H where D = h(H) and D* = h(H*). Then

~ < lh(zi) - h(z2)I < M

(8 .7.5) M - I h ( z1) - h(z2)I -

for z1, z2 EH where M = M(K, L).

PROOF. Suppose that z E H. Then for each z 0 E 8H,

lz - zol = lz - zol

and

(8. 7.6)

1 lh(z) - h( zo)I

• < <c
  c - lh(z) - h(zo)I -
  by Theorem 1.3.4 where c = e^8 K. In particular if we choose z 0 in (8.7.6) first so
  that

and then so that

we obtain

Thus

(8.7.7)

lh(z) - h(zo)I = dist(h(z), 8D)

lh(z) - h(zo)I = dist(h(z), 8D*),

1 dist(h(z ) , 8D)

• < < c
  c - dist(h(z), 8D*) - ·

1 PD(h(z))

• < < 4 c
  4c - PD· (h(z)) -
  by (3 .2.1) where PD and PD• denote the hyperbolic densities in D and D*. Since

PD (h( zo )) = zr .:,rr;o hD(h(lh(z) z-), h(h( zzo)I o))

and

_. hD· (h(z),h(z 0 ))

PD· (h(zo)) = }_:,n;o lh(z) - h(zo)I '

we conclude from (8.7.4) and (8.7.7) that

1.. lh(z) - h(zo)I. lh(z) - h(zo)I

(8.7.8) M - < hmmf z-+zo lh(-) Z - h(-Z o )I :::; hmz-+zo sup lh(-) Z - h(-Z o )I < M

for each z 0 E H where M > 4cL. Inequality (8 .7.6) shows that (8.7.8) also holds
for z 0 EH. Finally, since (8.7.8) is symmetric, it also holds for zo EH* and hence
for zo E R^2.
Suppose that/ is any arc in R^2 with endpoints z 1 , z2. By (8.7.8) we can choose
consecutive points w1,w2, ... ,wn+l E /such that W1 = z1, Wn+l = z2, and