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Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)



Annular functions in probability

Author: Russell W. Howell
Journal: Proc. Amer. Math. Soc. 52 (1975), 217-221
MSC: Primary 30A10; Secondary 60G50
MathSciNet review: 0374398
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Abstract: A function $ f$ holomorphic in the open unit disk $ U$ is said to be strongly annular if there exists a sequence $ \{ {C_n}\} $ of concentric circles converging outward to the boundary of $ U$ such that the minimum of $ \vert f\vert$ on $ {C_n}$ tends to infinity as $ n$ increases. We show here that such functions with Maclaurin coefficients $ \pm 1$ form a residual set in the space of functions with coefficients $ \pm 1$. We also show that the set of $ t$ in $ [0,1]$ for which $ \sum {{r_n}(t){z^n}} $ is strongly annular ($ {r_n}$ is the $ n$th Rademacher function) is residual, and measurable with measure either 0 or $ 1$.

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Keywords: Annular function, Rademacher function, residual set, zero-one law
Article copyright: © Copyright 1975 American Mathematical Society

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