126 9. SECOND SERIES OF IMPLICATIONS
Then f is analytic in D with f' f:. 0 and
I
(^1) " I I g( z) g' ( z) I
f =a lwl -z-+ 1 +aw g(z)
:5. a (lg(z) I+ Jg'(z)I ) :5. 5apD(z)
z 1 - lg(z)I
in D. Since 5a < T(D), f is injective and
l f:. j (z2) = Z2 eaw I = eill+aw I
f (z1) Z1
for all w EB where IBI :5. 7r,
-1z2 g(z) -1x2 h'(x)
I - dz - h( ) x dx,
Z1 Z X1 X
and h = g-^1. Thus -iB/aI (j. B , whence
a :5. l;I < 37r (log c; 1 )-1
by Lemma 9.3.8 with b = (c - 1)/2. Taking the supremum over all a which satisfy
(9.3.18) yields
min(l, -T(D) ( c -1)-l
5
-) :5. 37r log -
2
by our choice of c, and we obtain (9.3.14). 0
THEOREM 9.3.19. Suppose D C R^2 is simply connected and T(D) > 0. Then
D is c-linearly locally connected where c = c(T(D)).PROOF. By hypothesis, a function f is injective in D if f is analytic and locally
injective in D with
l
f"(z)I -1
~~g f'(z) PD(z) < T(D).We must show that this hypothesis implies there exists a constant c = c(T(D)) > 1
such that D is c-locally connected, that is, for each z 0 E D and each r > 0
(9.3.20)
(9.3.21)D n B(z 0 , r) lies in a component of D n B(z 0 , er),
D \ B(zo, r) lies in a component of D \ B(z 0 , r /c).Lemma 9.3.4 implies that points in DnB(z 0 ,r) can be joined in DnB(z 0 ,cr)
and hence that (9.3.20) holds provided that
T(D) + 2
c > H(D) + l.Lemma 9.3.13 implies that points in D \ B(z 0 ,r) can be joined in D \ B(z 0 ,r/c)
and hence that (9.3.21) holds provided that
c > 2 exp ( 7
1
(~)) + l.This completes the proof. 0