Bohr's power series theorem in several variables

Authors:
Harold P. Boas and Dmitry Khavinson

Journal:
Proc. Amer. Math. Soc. **125** (1997), 2975-2979

MSC (1991):
Primary 32A05

DOI:
https://doi.org/10.1090/S0002-9939-97-04270-6

MathSciNet review:
1443371

Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: Generalizing a classical one-variable theorem of Bohr, we show that if an -variable power series has modulus less than in the unit polydisc, then the sum of the moduli of the terms is less than in the polydisc of radius .

**1.**H. F. Bohnenblust and Einar Hille,*On the absolute convergence of Dirichlet series*, Ann. of Math. (2)**32**(1931), 600-622.**2.**Harald Bohr,*A theorem concerning power series*, Proc. London Math. Soc. (2)**13**(1914), 1-5.**3.**Seán Dineen and Richard M. Timoney,*Absolute bases, tensor products and a theorem of Bohr*, Studia Math.**94**(1989), 227-234. MR**91e:46006****4.**-,*On a problem of H. Bohr*, Bull. Soc. Roy. Sci. Liège**60**(1991), no. 6, 401-404. MR**93e:46050****5.**Jean-Pierre Kahane,*Some random series of functions*, second ed., Cambridge University Press, 1985. MR**87m:60119****6.**Steven G. Krantz,*Function theory of several complex variables*, second ed., Wadsworth & Brooks/Cole, Pacific Grove, CA, 1992. MR**93c:32001****7.**Anna Maria Mantero and Andrew Tonge,*The Schur multiplication in tensor algebras*, Studia Math.**68**(1980), no. 1, 1-24. MR**81k:46074****8.**Walter Rudin,*Function theory in polydiscs*, Benjamin, New York, 1969. MR**41:501****9.**S. Sidon,*Über einen Satz von Herrn Bohr*, Math. Z.**26**(1927), 731-732.**10.**M. Tomic,*Sur un théorème de H. Bohr*, Math. Scand.**11**(1962), 103-106. MR**31:316****11.**N. Th. Varopoulos,*On an inequality of von Neumann and an application of the metric theory of tensor products to operators theory*, J. Funct. Anal.**16**(1974), 83-100. MR**50:8116**

Retrieve articles in *Proceedings of the American Mathematical Society*
with MSC (1991):
32A05

Retrieve articles in all journals with MSC (1991): 32A05

Additional Information

**Harold P. Boas**

Affiliation:
Department of Mathematics, Texas A&M University, College Station, Texas 77843–3368

Email:
boas@math.tamu.edu

**Dmitry Khavinson**

Affiliation:
Department of Mathematical Sciences, University of Arkansas, Fayetteville, Arkansas 72701

Email:
dmitry@comp.uark.edu

DOI:
https://doi.org/10.1090/S0002-9939-97-04270-6

Received by editor(s):
May 8, 1996

Additional Notes:
The first author’s research was supported in part by NSF grant number DMS 9500916 and in part at the Mathematical Sciences Research Institute by NSF grant number DMS 9022140.

Communicated by:
Theodore W. Gamelin

Article copyright:
© Copyright 1997
American Mathematical Society