1549055259-Ubiquitous_Quasidisk__The__Gehring_

106 8. FIRST SERIES OF IMPLICATIONS

FIGURE 8.8

before, uniform. Thus conditions (3.5.2) and (3.5.3) in Definition 3.5.1 for a uniform
domain can be replaced by the substantially weaker hypotheses (8.3.2) and (8.3.3)
when D is simply connected.

8.4. Linear local connectivity and the three-point condition
We show next that a simply connected domain which is linearly locally con-
nected satisfies the three-point condition.

THEOREM 8.4.1 (Gehring [49]). Suppose that Dis a simply connected domain
in R^2 and that there exists a constant c ~ 1 such that for each z 0 E R^2 and r > 0

(8.4.2)

(8.4.3)

D n B(zo, 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).

Then D is a Jordan domain and for each pair of points z 1 , z 2 E 8D \ { oo}

(8.4.4) min diam(lj) :S: c^2 lz1 - z2 I

J=l,2

where ')' 1 , ')' 2 are the components of 8D \ {z1, z2}.

PROOF. The above hypotheses imply that Dis locally connected at each point
of its boundary and hence a Jordan domain by the converse of the Jordan curve
theorem (Newman [140]).
Next suppose that (8.4.4) does not hold for two points z 1 , z 2 E 8D \ { oo} and
set
1
r = - 2 lz1 - z2I.

Then there exist t with r < t < oo and finite points w 1 , w 2 such that

Wj E /'j \ B(z 0 , c^2 t)

for j = 1,2.