1549055259-Ubiquitous_Quasidisk__The__Gehring_

124 9. SECOND SERIES OF IMPLICATIONS

PROOF. By Lemma 9.3.l there exist points z 1 , z 2 ED and wo E R^2 \ D such
that

(9.3.6)

. 2
lh(z1) - h(z2) - 27ril:::; --,
c-1

where his any continuous branch of log(z - w 0 ) in D. Set

f(z ) = eah(z),

Then f is analytic with f' =/= 0 in D and

(9.3.7)

I

f" I 1a - 11 2

f = lz-wol ::;^2 ja-llpD::;7r(c-1)-1PD

by the Koebe distortion theorem and (9.3.6). Since

f (zi) = ea(h(zi)-h(z 2 )) = 1

f (z2) '

f is not injective in D and (9.3.5) follows from the definition of T(D). 0

LEMMA 9.3.8 (Astala-Gehring [14]). Suppose that h: B -+D is conformal with
h(O) = 0 and that
dist(h(O), 8D) = 1.

Suppose also that -1 < x 1 < 0 < xz < 1 and that

(9.3.9) ih(x)I:::; ih(xi)I = lh (x2)I = b > 0

for x1 :::; x :::; x2. Then

(9.3.10)

PROOF. We may assume that b > 1 since otherwise (9.3.10) follows trivially.
Choose Y1 and Y2 so that x1 < Y1 < 0 < Y2 < xz and

(9.3.11) lh(x)I:::; lh(y1)I = lh(yz)I = 1

for YI :::; x :::; Y2. Then z h' ( z ) / h( z) is analytic in B and

d~ log lh(x)I =Re ( ~(~;)

for -1 < x < 1, x =/= 0. Thus

I

r

2

h'(x ) I r

2

(h'(x))

lx, h(x ) xdx 2: lx, Re h(x ) xdx

r 2 1

(9.3.12) =(x2- xI)logb+ lx, loglh(x)I dx

2: (y^2 - YI) log b + (1~' + i~

2

) log lhtx)I dx

2: (y2 - YI) log b

by integration by parts, (9.3.9) and (9.3.11). Since

lh'(O)I:::; 4dist(h(O), 8D) = 4,