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Proceedings of the American Mathematical Society
Proceedings of the American Mathematical Society
ISSN 1088-6826(online) ISSN 0002-9939(print)

 

Distortion of sets by inner functions


Author: D. H. Hamilton
Journal: Proc. Amer. Math. Soc. 117 (1993), 771-774
MSC: Primary 30A10; Secondary 30D50
MathSciNet review: 1139469
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Abstract: For any inner function $ f$ with $ f(0) = 0$ and any Borel set $ E \subset {\mathbf{D}}$

$\displaystyle {M_\alpha }(z \in {\mathbf{D}}:f(z) \in E) \geqslant {M_\alpha }(E),\qquad 0 < \alpha \leqslant 1,$

where $ {M_\alpha }$ denotes $ \alpha $-dimensional Hausdorff measure. In the case that $ 0 < {M_\alpha }(E) < \infty $ we have equality only for rotations of the identity.

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Additional Information

DOI: http://dx.doi.org/10.1090/S0002-9939-1993-1139469-2
PII: S 0002-9939(1993)1139469-2
Article copyright: © Copyright 1993 American Mathematical Society