Positive superharmonic functions and the Hölder continuity of conformal mappings

We study the rate at which a positive superharmonic function u can tend to zero at a boundary point z ₀of a plane domain G. In particular, if G is a quasidisk, and a > 0 is given, we show that the condition that lim inf «(z)/dist (z, dG) ^llct> 0 as z -* z ₀in G for any such u is related to the condition that the conformal map/of the unit disk onto G withal) = z ₀is Holder continuous with exponent a at the point 1. This leads us to consider the problem of finding the best exponent a for which /is Holder continuous. The answer depends on how we characterize quasidisks or quasicircles. In this connection we give a negative answer to a question of Nakki and Palka.