  In this note, we prove a version of the Phragmen-Lindelof principle using
probabilistic techniques. In particular, we will show that if the p-th 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 infinity along some sequence more rapidly than
$e^{|z|^{2p}}$. We also present a method for constructing domains whose exit
time has finite p-th moment, thereby giving a general Phragmen-Lindelof
principle for spiral-like and star-like domains, and in the process giving a
probabilistic proof of a theorem of Hansen. A number of auxiliary results are
presented as well.
