How round are the complementary components of planar Brownian motion

Consider a Brownian motion W in C started from 0 and run for time 1. Let A(1), A (2), ... denote the bounded connected components of C - W([0, 1]). Let R(i) (resp. r(i)) denote the out-radius (resp. in-radius) of A(i) for i is an element of N. Our main result is that E[Sigma(i) R(i)(2) vertical bar log R(i)vertical bar(theta)] < infinity for any theta < 1. We also prove that Sigma(i) r (i)(2) vertical bar log r(i)vertical bar = infinity almost surely. These results have the interpretation that most of the components A (i) have a rather regular or round shape.