Abstract
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.
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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.
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Pruss, A.R. Radial rearrangement, harmonic measures and extensions of Beurling's shove theorem. Ark. Mat. 37, 183–210 (1999). https://doi.org/10.1007/BF02384833
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DOI: https://doi.org/10.1007/BF02384833