78 6. TWO-SIDED CRITERIA
FIGURE 6.1The corresponding result for property 2° does not follow for merely simply
connected domains since a John disk need not be a Jordan domain. A disk minus
a radius provides a standard counterexample.THEOREM 6.1.2 (Nakki-Vaisala [136]). A Jordan domain D is a quasidisk if
and only if D and D* are John disks.6.2. Hardy-Littlewood property
Suppose that D is a domain in R^2 and that f is a real-or complex-valued
function defined in D. We say that f is in the Lipschitz class Lip°'(D), 0 <a:::; 1,
if there exists a constant m such that
(6.2.1)for all z1 and z2 in D, and we let llflla denote the infimum of the numbers m
so that (6.2.1) holds. If there exists a constant m such that (6.2.1) holds in all
disks BCD, we say that flies in the local Lipschitz class locLip°'(D), and llfll~c
denotes the corresponding infimum over all such m.
For an analytic function satisfying (6.2.1) it follows from the Cauchy integral
formula for the derivative that
(6.2.2) lf'(z)I:::; m dist(z,8D)°'-^1.Conversely, the following well-known result of Hardy and Littlewood shows that
condition (6.2.2) is also sufficient to guarantee that flies in Lip°'(D) when Dis a
disk. See, for example, Theorem 5.1 in [34].
THEOREM 6.2.3. If f is analytic in a disk B = B(zo, r) with
lf'(z )I:::; m dist(z,8B)°'-^1 ,