1549055259-Ubiquitous_Quasidisk__The__Gehring_

9.3. PRE-SCHWARZIAN DERIVATIVES AND LOCAL CONNECTIVITY 125

standard distortion theorems applied to h(z)/h'(O) imply that

1 -- I h ( ) YJ I _ < 4 (l _ IY1 IYJ I l)2 and hence I YJ I >^1

6

for j = 1, 2. This, together with (9.3.12), yields inequality (9.3.10). D

LEMMA 9.3.13 (Astala-Gehring [14]). Suppose thatc > 2e^3 7r +l, thatw 0 E R^2 ,

and that 0 < r < oo. If there are points in D \ B(w 0 , er) which cannot be joined in

D \ B(wo, r) , then

(9.3.14) (

1)-l

T(D) ::::; 15 7f log c;

PROOF. Suppose that w 1 , w 2 are points in D\B(w 0 , er) which cannot be joined
in D \ B(wo, r) and let (3 denote the hyperbolic geodesic joining w 1 and w 2 in D.
The hypotheses imply that

(3 n B(w 0 , r) ~ 0 and B(w 0 , r) \ D ~ 0.

Thus we can choose zo E (3 such that

dist(zo, oD) ::::; lzo - wol + dist(wo, oD) < 2r

where
c-1
lw1 - zol 2: lw1 - wol - lzo - wol > -
2

• dist(zo, oD) = d

for j = 1, 2.
Let a denote the component of (3 n B(z 0 , d) which contains zo and let z 1 , z 2
denote the endpoints of a. Then
c-1
(9.3.15) lz - zol ::::; lz1 - zol = lz2 - zol = -
2

• dist(zo, oD)

for z Ea.
If D' denotes the image of D under the similarity mapping
(z - zo)
w = dist(z 0 , oD)'

then it is easy to check that T(D) = T(D'). Hence we may assume without loss of
generality that Zo = 0 and dist(zo, oD) = l. Let g map D conformally onto B so
that g(O) = 0 and 0 < g(z 2 ) = x 2 < l. Then -1 < g(z1) = x1 < 0. Since BCD,
g(B) C B and the Schwarz lemma implies that

( )
l

9 g(z)I min(lzl,l)^2 2 <^4 ()

·^3 ·^16 - z- ::::; lzl ::::; lz l + 1 ::::; dist(z, oD) - PD z

and

(9.3.17)

lg' (z) I

1 - lg(z)I ::::; PD(z)

in D.
Now suppose T(D) > 0 and choose a > 0 so that

(9.3.18) a < min(l, T(~) ),

and for each w E B let

J(z) = zeawG(z), G(z) = t g(() d(.

l o (