MathSciDoc: An Archive for Mathematician ∫

Let$I$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.

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