Bohr’s Power Series Theorem in Several Variables

Generalizing a classical one-variable theorem of Bohr, we show that if an n-variable power series has modulus less than 1 in the unit polydisc, then the sum of the moduli of the terms is less than 1 in the polydisc of radius 1/(3 √ n ). How large can the sum of the moduli of the terms of a convergent power series be? Harald Bohr addressed this question in 1914 with the following remarkable result on power series in one complex variable. Theorem 1 (Bohr). Suppose that the power series ∑∞ k=0 ckz k converges for z in the unit disk, and |∑∞k=0 ckz| < 1 when |z| < 1. Then ∑∞ k=0 |ckz| < 1 when |z| < 1/3. Moreover, the radius 1/3 is the best possible. Bohr’s paper [2], compiled by G. H. Hardy from correspondence, indicates that Bohr initially obtained the radius 1/6, but this was quickly improved to the sharp result by M. Riesz, I. Schur, and N. Wiener, independently. Bohr’s paper presents both his own proof and Wiener’s. Some years later, S. Sidon gave a different proof [9], which was subsequently rediscovered by M. Tomić [10]. In this note, we formulate a version of Bohr’s theorem in higher dimensions. We write an n-variable power series ∑ α cαz α using the standard multi-index notation: α denotes an n-tuple (α1, α2, . . . , αn) of nonnegative integers, |α| denotes the sum α1 + · · ·+ αn of its components, α! denotes the product α1!α2! . . . αn! of the factorials of its components, z denotes an n-tuple (z1, . . . , zn) of complex numbers, and z denotes the product z1 1 z α2 2 . . . z αn n . Let Kn denote the n-dimensional Bohr radius: the largest number such that if ∑ α cαz α converges in the unit polydisc {(z1, . . . , zn) : max1≤j≤n |zj | < 1}, and if | ∑ α cαz α| < 1 in the unit polydisc, then ∑ α |cαz| < 1 when max1≤j≤n |zj| < Kn. 1991 Mathematics Subject Classification. Primary 32A05. 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.