Annular functions in probability

Author:
Russell W. Howell

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

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Abstract: A function holomorphic in the open unit disk is said to be strongly annular if there exists a sequence of concentric circles converging outward to the boundary of such that the minimum of on tends to infinity as increases. We show here that such functions with Maclaurin coefficients form a residual set in the space of functions with coefficients . We also show that the set of in for which is strongly annular ( is the th Rademacher function) is residual, and measurable with measure either 0 or .

**[1]**D. D. Bonar and F. W. Carroll,*Annular functions form a residual set*, J. Reine Angew. Math.**272**(1975), 23–24. MR**0417428****[2]**-,*Some examples and counterexamples in annular functions*(unpublished manuscript).**[3]**F. W. Carroll, D. J. Eustice and T. Figiel,*On the minimum modulus of a polynomial*(unpublished manuscript).**[4]**Anri Kartan,*\cyrÈlementarnaya teoriya analiticheskikh funktsiĭodnogo i neskol′kikh kompleksnykh peremennykh*, Izdat. Inostran. Lit., Moscow, 1963 (Russian). MR**0220963****[5]**Jean-Pierre Kahane,*Some random series of functions*, D. C. Heath and Co. Raytheon Education Co., Lexington, Mass., 1968. MR**0254888**

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Additional Information

DOI:
https://doi.org/10.1090/S0002-9939-1975-0374398-2

Keywords:
Annular function,
Rademacher function,
residual set,
zero-one law

Article copyright:
© Copyright 1975
American Mathematical Society