The exit time of planar Brownian motion and the Phragmén-Lindelöf principle

In this paper a version of the Phragmén-Lindelöf principle is proved using probabilistic techniques. In particular, we will show that if the pth moment of the exit time of Brownian motion from a planar domain is finite, then an analytic function on that domain is either bounded by its supremum on the boundary or else goes to ∞ along some sequence more rapidly than e|z|2p. We also provide a method of constructing domains whose exit time has finite pth moment. This allows us to give a general Phragmén-Lindelöf principle for spiral-like and star-like domains, as well as a new proof of a theorem of Hansen. A number of auxiliary results are presented as well.