# MathSciDoc: An Archive for Mathematician ∫

#### TBDmathscidoc:1701.332911

Arkiv for Matematik, 37, (1), 183-210, 1996.6
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.
```@inproceedings{alexander1996radial,
title={Radial rearrangement, harmonic measures and extensions of Beurling's shove theorem},
author={Alexander R. Pruss},
url={http://archive.ymsc.tsinghua.edu.cn/pacm_paperurl/20170108203555452690720},
booktitle={Arkiv for Matematik},
volume={37},
number={1},
pages={183-210},
year={1996},
}
```
Alexander R. Pruss. Radial rearrangement, harmonic measures and extensions of Beurling's shove theorem. 1996. Vol. 37. In Arkiv for Matematik. pp.183-210. http://archive.ymsc.tsinghua.edu.cn/pacm_paperurl/20170108203555452690720.