Computational Methods and Function Theory 4 (2004), No. 1, 1--19
Copyright Heldermann Verlag 2004
Remarks on the Bohr Phenomenon
email@example.com , Department of Mathematics and Computer Science, Seton Hall University, South Orange, New Jersey 07079, U.S.A.
firstname.lastname@example.org , Centre for Mathematical Sciences, Mathematics (Faculty of Science), University of Lund, Box 118, S-221 00, Lund, Sweden.
email@example.com , Department of Mathematics, University of Arkansas, Fayetteville, Arkansas 72701.
H. Bohr's Theorem [Proc. London Math. Soc. (2) 13 (1914) 1--5] states that analytic functions bounded by $1$ in the unit disk have power series $\sum a_n z^n$ such that $\sum |a_n||z|^n < 1$ in the disk of radius $1/3$ (the so-called Bohr radius). On the other hand, it is known that there is no such Bohr phenomenon in Hardy spaces with the usual norm, although it is possible to build equivalent norms for which a Bohr phenomenon does occur. In this paper, we consider Hardy space functions that vanish at the origin and obtain an exact positive Bohr radius. Also, following L. Aizenberg, I. Grossman and Yu. Korobeinik [Izv. Vyssh. Uchebn. Zaved. Mat. 2002, No. 10, 3--10 (in Russian)] and P. Djakov and M. Ramanujan [J. Anal. 8 (2000) 65--77), we discuss the growth and Bohr phenomena for series of the type $\sum |a_n|^p r^n$, $0 < p < 2$, that come from functions $f(z) = \sum a_n z^n$ in the Hardy spaces. We will then consider Bohr phenomena in more general normed spaces of analytic functions and show how renorming a space affects the Bohr radius. Finally, we extend our results to several variables and obtain as a consequence some general Schwarz-Pick type estimates for bounded analytic functions.
Keywords: Taylor series, absolute convergence, Bohr's Theorem, Schwarz-Pick estimates.
MSC 2000: Primary 30H05, Secondary 32A05.
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