1549055259-Ubiquitous_Quasidisk__The__Gehring_

9.3. PRE-SCHWARZIAN DERIVATIVES AND LOCAL CONNECTIVITY 123

9.3. Pre-Schwarzian derivatives and local connectivity
We saw in the previous section that the pre-Schwarzian radius of injectivity
r(D) of a simply connected domain D is positive whenever its Schwarzian radius
of injectivity a-(D) is positive. We show here that the domain D is linearly locally
connected whenever r(D) > 0.
The proof of this fact depends on the following four geometric lemmas.
LEMMA 9.3.1 (Gehring-Martio [64]). Suppose that c > 1 and that there exist
two points in DnB(zo, r) which cannot be joined in DnB(z 0 , er). Then there exist
points z 1 , z 2 ED and wo E 8B(zo, er)\ D such that

. 2

(9.3.2) Jh(zi) - h(z2) - 2-rriJ ~ -,

c-1

where h(z) = log(z - wo).

PROOF. Let z~, z~ be two points in D n B ( z 0 , r) which cannot be joined in
D n B(z 0 , er), and let a.' be the segment and /3' a rectifiable arc which join z~, z~
in R^2 and D , respectively. We may choose /3' so that it intersects a.' in a finite
set of points E. Then there exist two adjacent points z 1 , z2 E E which cannot
be joined in D n B ( zo, er). Let a. and /3 denote the parts of a.' and /3' between
z 1 and z 2. Then/ = a U /3 is a Jordan curve and we denote by Do the bounded

component of R

2
\ f. The fact that z 1 , z2 cannot be joined in D n B(zo, er) and a
simple topological argument based on Kerekjarto's theorem (Newman [140]) imply
the existence of a point w 0 such that
-2
w 0 E (R \ D) n 8B(zo, er) n Do.

For the details see Gehring [49]. Since D is simply connected, we can choose an
analytic branch of h(z) = log(z - wo) in D, and

h(z1)-h(z2) = r h'(z)dz = 2-rrin(!,wo)-1 __!!!_'

} f3 0 Z - Wo

where n(!, w 0 ) is the winding number of/ with respect to wo. Now

n(!, wo) = n = ±1,

and hence

(9.3.3)

Since a C B(z 0 , r) and wo E 8B(zo, er),

1

Jdzl < length(a) < _2_

0 Jz-wol- (c-l)r -c-1

and (9.3.2) follows from (9.3.3) if n = l. If n = -1, the result follows by inter-
changing z 1 and z2. 0

LEMMA 9.3.4 (Astala-Gehring [14]). Suppose that c > 2, that zo E R^2 , and
that O < r < oo. If there are points in D n B(z 0 , r) which cannot be joined in
DnB(z 0 ,cr), then

(9.3.5) r(D) <

2

• -rr(c - 1) - 1