Multidimensional analogues of Bohr’s theorem on power series
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- by Lev Aizenberg PDF
- Proc. Amer. Math. Soc. 128 (2000), 1147-1155 Request permission
Abstract:
Generalizing the classical result of Bohr, we show that if an $n$-variable power series converges in $n$-circular bounded complete domain $D$ 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
- L. A. Aĭzenberg, The spaces of functions analytic in $(p,\,q)$-circular regions, Soviet Math. Dokl. 2 (1961), 79–82. MR 0142790
- L. A. Aĭzenberg and B. S. Mitjagin, Spaces of functions analytic in multi-circular domains, Sibirsk. Mat. . 1 (1960), 153–170 (Russian). MR 0124526
- I. A. Aĭzenberg and A. P. Yuzhakov, Integral representations and residues in multidimensional complex analysis, Translations of Mathematical Monographs, vol. 58, American Mathematical Society, Providence, RI, 1983. Translated from the Russian by H. H. McFaden; Translation edited by Lev J. Leifman. MR 735793, DOI 10.1090/mmono/058
- Harold P. Boas and Dmitry Khavinson, Bohr’s power series theorem in several variables, Proc. Amer. Math. Soc. 125 (1997), no. 10, 2975–2979. MR 1443371, DOI 10.1090/S0002-9939-97-04270-6
- H. Bohr, A theorem concerning power series, Proc. London Math. Soc. (2) 13 (1914) 1-5.
- Walter Rudin, Function theory in the unit ball of $\textbf {C}^{n}$, Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 241, Springer-Verlag, New York-Berlin, 1980. MR 601594, DOI 10.1007/978-1-4613-8098-6
- Wolfram Research, Mathematica 3.0, 1996.
Additional Information
- Lev Aizenberg
- Email: aizenbrg@macs.biu.ac.il
- 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
- © Copyright 2000 American Mathematical Society
- Journal: Proc. Amer. Math. Soc. 128 (2000), 1147-1155
- MSC (1991): Primary 32A05
- DOI: https://doi.org/10.1090/S0002-9939-99-05084-4
- MathSciNet review: 1636918