A logarithmic lower bound for multi-dimensional bohr radii

Bohr’s theorem for holomorphic mappings with values in homogeneous balls

“…Also, in [18], they gave a generalisation of Bohr's theorem to holomorphic mappings of B into itself using homogeneous expansions, where B is one of the four classical domains in C n . For other generalisation of Bohr's theorem to several complex variables, see [2], [3], [4], [5], [9], [10], [11], [14], [19], [20] and [21]. Also, all references on the problem of Bohr's phenomenon can be found in the new research book [17].…”

Remarks on the Bohr and Rogosinski phenomena for power series

“…Generalization and modifications of this result can be found in [1][2][3][4][5][6][7][8][9][10][11][12][13]15,16,[19][20][21][22][23][24][25][26]28,29,31,33]. We especially point out an important, recent result contained in [20].…”

On the Bohr Inequality

“…When p = ∞, the unit ball is to be interpreted as the unit polydisk in , where c > 0 is a constant independent of p, n. The lower bound was then improved to the value (log n/ log log n)/n in [34]. On the other hand, Aizenberg [12] proved that 1 3e 1/3 ≤ K(B ℓ n 1 ) ≤ 1/3, where 1/3 is the best upper bound.…”

“…General lower and upper estimates for the first and the second Bohr radii of bounded complete Reinhardt domains are given in [37]. Results from both papers [34] and [37] were proved using certain theorems from [36], which was the first paper linking multidimensional Bohr study to local Banach space theory. The estimates for K n obtained in [36] were in terms of unconditional basis constants and Banach-Mazur distances.…”