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Proceedings of the American Mathematical Society

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A note on Hall's lemma


Author: Dieter Gaier
Journal: Proc. Amer. Math. Soc. 37 (1973), 97-99
MSC: Primary 30A44
DOI: https://doi.org/10.1090/S0002-9939-1973-0310231-0
MathSciNet review: 0310231
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Abstract: Let H be the half plane $ \{ z:\operatorname{Re} z > 0\} $. Let y be a Jordan arc joining $ z = 0$ and $ z = {e^{i\alpha }}\;(0 \leqq \alpha < \pi /2)$ in $ H \cap \{ \vert z\vert \leqq 1\} $. Let $ {\gamma ^\ast}$ be the segment $ z = iy\;(0 \leqq y \leqq 1)$ of the imaginary axis. If $ \omega (z,\gamma )$ is the harmonic measure of $ \gamma $ with respect to $ H\backslash \gamma $ and $ \omega (z,{\gamma ^\ast})$ the harmonic measure of $ {\gamma ^\ast}$ with respect to H, then $ \omega (x + iy,\gamma ) > \omega (x - i\vert y\vert,{\gamma ^\ast})$.


References [Enhancements On Off] (What's this?)

  • [1] Peter L. Duren, Theory of 𝐻^{𝑝} spaces, Pure and Applied Mathematics, Vol. 38, Academic Press, New York-London, 1970. MR 0268655
  • [2] W. H. J. Fuchs, Topics in the theory of functions of one complex variable, Manuscript prepared with the collaboration of Alan Schumitsky. Van Nostrand Mathematical Studies, No. 12, D. Van Nostrand Co., Inc., Princeton, N.J.-Toronto, Ont.-London, 1967. MR 0220902

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Additional Information

DOI: https://doi.org/10.1090/S0002-9939-1973-0310231-0
Keywords: Harmonic measure, Hall lemma
Article copyright: © Copyright 1973 American Mathematical Society