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Proceedings of the American Mathematical Society

Published by the American Mathematical Society, the Proceedings of the American Mathematical Society (PROC) is devoted to research articles of the highest quality in all areas of pure and applied mathematics.

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

The 2020 MCQ for Proceedings of the American Mathematical Society is 0.85.

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Annular functions in probability
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by Russell W. Howell PDF
Proc. Amer. Math. Soc. 52 (1975), 217-221 Request permission

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 $|f|$ 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 $1$.
References
  • D. D. Bonar and F. W. Carroll, Annular functions form a residual set, J. Reine Angew. Math. 272 (1975), 23–24. MR 417428
  • —, Some examples and counterexamples in annular functions (unpublished manuscript). F. W. Carroll, D. J. Eustice and T. Figiel, On the minimum modulus of a polynomial (unpublished manuscript).
  • Anri Kartan, Èlementarnaya teoriya analiticheskikh funktsiĭodnogo i neskol′kikh kompleksnykh peremennykh, Izdat. Inostr. Lit., Moscow, 1963 (Russian). MR 0220963
  • Jean-Pierre Kahane, Some random series of functions, D. C. Heath and Company Raytheon Education Company, Lexington, Mass., 1968. MR 0254888
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Additional Information
  • © Copyright 1975 American Mathematical Society
  • Journal: Proc. Amer. Math. Soc. 52 (1975), 217-221
  • MSC: Primary 30A10; Secondary 60G50
  • DOI: https://doi.org/10.1090/S0002-9939-1975-0374398-2
  • MathSciNet review: 0374398