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An abstract approach to Bohr's phenomenon

Author(s): L. Aizenberg; A. Aytuna; P. Djakov
Journal: Proc. Amer. Math. Soc. 128 (2000), 2611-2619.
MSC (2000): Primary 32A37, 32A05; Secondary 46E10
Posted: March 1, 2000
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Abstract: In 1914 Bohr discovered that there exists $r \in (0,1)$ such that if a power series converges in the unit disk and its sum has modulus less than $1$, then for $ \vert z\vert < r $ the sum of absolute values of its terms is again less than $1$. Recently analogous results were obtained for functions of several variables. Our aim here is to present an abstract approach to the problem and show that Bohr's phenomenon occurs under very general conditions.

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L. Aizenberg, Multidimensional analogues of Bohr's theorem on power series, Proc. Amer. Math. Soc., to appear. CMP 98:16

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L. Aizenberg, A. Aytuna, P. Djakov, Generalization of Bohr's theorem for arbitrary bases in spaces of holomorphic functions of several variables, preprint METU, Mathematics 98/168.

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H. P. Boas, D. Khavinson, Bohr's power series theorem in several variables, Proc. Amer. Math. Soc. 125 (1997), 2975-2979. MR 98i:32002

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H. Bohr, A theorem concerning power series, Proc. London Math. Soc. (2) 13 (1914) 1-5.

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C. Carathéodory, Über den Variabilitätsbereich der Koeffizienten von Potenzreihen, die gegebene werte nicht annemen, Math. Ann. 64 (1907), 95-115.

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Additional Information:

L. Aizenberg
Affiliation: Department of Mathematics and Computer Science, Bar-Ilan University, 52900Ramat-Gan, Israel
Email: aizenbrg@macs.biu.ac.il

A. Aytuna
Affiliation: Department of Mathematics, Middle East Technical University, 06531 Ankara, Turkey
Email: aytuna@rorqual.cc.metu.edu.tr

P. Djakov
Affiliation: Department of Mathematics, Sofia University, 1164 Sofia, Bulgaria
Email: djakov@fmi.uni-sofia.bg

DOI: 10.1090/S0002-9939-00-05270-9
PII: S 0002-9939(00)05270-9
Keywords: Spaces of holomorphic functions, Bohr phenomenon
Received by editor(s): July 17, 1998
Received by editor(s) in revised form: October 15, 1998
Posted: March 1, 2000
Additional Notes: The first author's research was supported by the BSF grant No 94-00113
The second author wishes to thank L. Aizenberg and institute E. Noether for the invitation to Bar-Ilan University and their hospitality during his visit in Israel
The third author's research was supported in part by NRF of Bulgaria, grant no. MM-808/98
Communicated by: Steven R. Bell
Copyright of article: Copyright 2000, American Mathematical Society

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