CHAPTER 9

Second series of implications

In the last chapter we proved seven ways of characterizing a quasidisk by means

of a circle of implications. Two of the characterizations established there for a

simply connected domain D are as follows.

D is a quasidisk if and only if it is a uniform domain.

D is a quasidisk if and only if it is linearly locally connected.

We now use this information to establish four additional characterizations by prov-
ing the following statements for a simply connected domain D.

1° If D is uniform, then a(D) > 0.

2° If a(D) > 0, then T(D) > 0.

3° If T(D) > 0, then D is linearly locally connected.

4° If D is uniform, then L(D) > 1.

5° If L(D) > 1, then D is linearly locally connected.

6° If D is uniform, then it has the min-max property.

7° If D has the min-max property, then D is linearly locally connected.

Implications 1°, 2°, and 3° concern domains D where a meromorphic function
f is injective whenever its Schwarzian derivative Sf or pre-Schwarzian T1 is not
large compared to the hyperbolic metric PD· Implications 4° and 5° consider the
same problem for functions f locally bilipschitz in D with small lipschitz constant.
Finally, 6° and 7° involve a geometric property of the hyperbolic geodesics in D.

D is a quasidisk

/:C__ L(D) >I ~

D linearly locally D is uniform

connected ~ ~/

~ Min-max prop.

9 .3~ 9.1

T(D) > (^0) 9 .2 a(D) > 0
FIGURE 9.1
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