Multidimensional analogues of Bohr's theorem

on power series

Author:
Lev Aizenberg

Journal:
Proc. Amer. Math. Soc. **128** (2000), 1147-1155

MSC (1991):
Primary 32A05

DOI:
https://doi.org/10.1090/S0002-9939-99-05084-4

Published electronically:
August 5, 1999

MathSciNet review:
1636918

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Abstract | References | Similar Articles | Additional Information

Abstract: Generalizing the classical result of Bohr, we show that if an -variable power series converges in -circular bounded complete domain and its sum has modulus less than 1, then the sum of the maximum of the modulii of the terms is less than 1 in the homothetic domain , where . This constant is near to the best one for the domain

**1.**L.A. Aizenberg, The spaces of functions analytic in -circular regions, Soviet Math. Dokl.**2**(1960), 79-82. MR**26:359****2.**L.A. Aizenberg, B.S. Mityagin, The spaces of functions analytic in multicircular domains, Sibirsk. Math. Zh.**1**(1960), 153-170 (Russian). MR**23:A1838****3.**L.A. Aizenberg, A.P. Yuzhakov, Integral Representations and Residues in Multidimensional Complex Analysis, AMS, 1983, 283 pp. MR**85a:32006****4.**H.P. Boas, D. Khavinson, Bohr's power series theorem in several variables, Proc. Amer. Math. Soc.**125**(1997), 2975-2979. MR**98i:32002****5.**H. Bohr, A theorem concerning power series, Proc. London Math. Soc. (2)**13**(1914) 1-5.**6.**W.Rudin, Function theory in the unit ball of , Springer-Verlag, 1980, 436 pp. MR**82i:32002****7.**Wolfram Research, Mathematica 3.0, 1996.

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Additional Information

**Lev Aizenberg**

Email:
aizenbrg@macs.biu.ac.il

DOI:
https://doi.org/10.1090/S0002-9939-99-05084-4

Received by editor(s):
April 28, 1998

Received by editor(s) in revised form:
June 8, 1998

Published electronically:
August 5, 1999

Additional Notes:
This work was supported by the BSF, grant No 94-00113.

Communicated by:
Steven R. Bell

Article copyright:
© Copyright 2000
American Mathematical Society