Multidimensional analogues of Bohr’s theorem on power series

Abstract

Abstract. Generalizing the classical result of Bohr, we show that if an nvariable 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 ·D, where r = 1− n 2/3. This constant is near to the best one for the domain D = {z : |z 1 | + . . . + |zn| < 1}.

Some generalizations of Bohr's theorem

“…Then, we have for z ∈ (1 / 3) G . Moreover, the constant 1 / 3 is best possible. Remark Aizenberg obtained the aforementioned corollary when G is a bounded balanced domain in in ,Theorem 8. However, in , he assumed that G is convex to deduce that the constant 1 / 3 is best possible.…”

“…Moreover, the constant 1 / 3 is best possible. Remark Aizenberg obtained the aforementioned corollary when G is a bounded balanced domain in in ,Theorem 8. However, in , he assumed that G is convex to deduce that the constant 1 / 3 is best possible. In the aforementioned corollary, we do not need the convexity of G . Proof of Theorem The estimation is proved in [, Theorem 3.1].…”

“…Many mathematicians have studied generalizations of Bohr's theorem in. They obtained some generalizations using the power series of a holomorphic function defined in bounded complete Reinhardt domains in .…”

“…On the other hand, using homogeneous expansions of holomorphic functions, Aizenberg obtained another generalization of Bohr's theorem in ,Theorem 8; Liu and Wang gave a generalization of Bohr's theorem to holomorphic mappings of B into itself, where B is one of the four classical domains in in . Hamada, Honda and Kohr generalized to infinite dimensional spaces in .…”

A Bohr phenomenon on the punctured unit disk

“…This result became known as the Bohr theorem. The radius 6 1 was later improved independently to the sharp constant 3 1 by Wiener, Riesz and Schur (see [2][3][4]). Analogous results to the Bohr for several complex variables have been established by replacing U with a complete Reinhardt domain [ 5], a unit ball or hypercone in higher dimensions [ 6].…”