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

Boundary limits of subharmonic functions in the disc


Author: M. Stoll
Journal: Proc. Amer. Math. Soc. 93 (1985), 567-568
MSC: Primary 31A20
MathSciNet review: 774024
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Abstract: In this note we prove the following: let $ f\not\equiv - \infty $ be subharmonic in $ \left\vert z \right\vert < 1$ satisfying $ \mathop {\lim }\limits_{r \to 1} \int_0^{2\pi } {f(r{e^{i\theta }})d\theta = 0} $ with $ f(z) \leqslant 0$; then

$\displaystyle \mathop {\lim }\limits_{r \to 1} \sup (1 - r)\mathop {\inf }\limits_{\left\vert z \right\vert = r} f(z) = 0$

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Additional Information

DOI: http://dx.doi.org/10.1090/S0002-9939-1985-0774024-0
PII: S 0002-9939(1985)0774024-0
Article copyright: © Copyright 1985 American Mathematical Society