Radial rearrangement, harmonic measures and extensions of Beurling's shove theorem

LetI be a union of finitely many closed intervals in −1, 0). LetI ^↞ be a single interval of the form −1, −a] chosen to have the same logarithmic length asI. LetD be the unit disc. Then, Beurling 8] has shown that the harmonic measure of the circle ∂D at the origin in the slit discD/I is increased ifI is replaced byI ^↞. We prove a number of cognate results and extensions. For instance, we show that Beurling's result remains true if the intervals inI are not just one-dimensional, but if they in fact constitute polar rectangles centred on the negative real axis and having some fixed constant angular width. In doing this, we obtain a new proof of Beurling's result. We also discuss a conjecture of Matheson and Pruss 25] and some other open problems. Much of the present paper has been adapted from Chapter IV of the author's doctoral dissertation. The research was partially supported by Professor J. J. F. Fournier's NSERC Grant #4822.