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

Abstract

LetI be a union of finitely many closed intervals in [−1, 0). Let I^↞ be a single interval of the form [−1, −a] chosen to have the same logarithmic length as I. Let D be the unit disc. Then, Beurling [8] has shown that the harmonic measure of the circle ∂D at the origin in the slit disc D/I is increased if I is replaced by I^↞. We prove a number of cognate results and extensions. For instance, we show that Beurling's result remains true if the intervals in I 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.