Bohr’s power series theorem in several variables

Abstract

Abstract. 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 ).

Note on Schwarz-Pick estimates for bounded and positive real part analytic functions

“…The proof of Theorem 1.1 is based on the following Lemma, which should be well-known and for which we include a simple proof (see [4]).…”

Bohr-type inequalities of analytic functions

“…The problem was considered by Bohr when he was working on the absolute convergence problem for Dirichlet series of the form , but now it has become a very interesting problem. Bohr’s idea naturally extends to functions of several complex variables [1, 2, 5, 11] and a variety of results on Bohr’s theorem in higher dimensions appeared recently.…”

Bohr radius for the punctured disk

“…The generalization of Bohr theorem to higher dimensions was pioneered by Dineen and Timoney in the setting of the unit polydisk, which led to a partial solution of the problem. Several years later, Boas and Khavinson provided the estimate for the n ‐dimensional Bohr radius. Recently, Defant et al.…”