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Multidimensional analogues of Bohr's theorem on power series

Author(s): Lev Aizenberg
Journal: Proc. Amer. Math. Soc. 128 (2000), 1147-1155.
MSC (1991): Primary 32A05
Posted: August 5, 1999
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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 $r \cdot D$ , where $r = 1- \sqrt[n]{2/3}$ . This constant is near to the best one for the domain $D = \{z: |z_1 |+ \ldots + |z_n |$ $< 1 \} .$

References:

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 $\mathbf{ C}^n$ , Springer-Verlag, 1980, 436 pp. MR 82i:32002
7.: Wolfram Research, Mathematica 3.0, 1996.

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Additional Information:

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

DOI: 10.1090/S0002-9939-99-05084-4
PII: S 0002-9939(99)05084-4
Received by editor(s): April 28, 1998
Received by editor(s) in revised form: June 8, 1998
Posted: August 5, 1999
Additional Notes: This work was supported by the BSF, grant No 94-00113.
Communicated by: Steven R. Bell
Copyright of article: Copyright 2000, American Mathematical Society


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