On the Bohr inequality

The Bohr inequality, first introduced by Harald Bohr in 1914, deals with finding the largest radius $r$, $0<r<1$, such that $\sum_{n=0}^\infty |a_n|r^n \leq 1$ holds whenever $|\sum_{n=0}^\infty a_nz^n|\leq 1$ in the unit disk $\mathbb{D}$ of the complex plane. The exact value of this largest radius, known as the \emph{Bohr radius}, has been established to be $1/3.$ This paper surveys recent advances and generalizations on the Bohr inequality. It discusses the Bohr radius for certain power series in $\mathbb{D},$ as well as for analytic functions from $\mathbb{D}$ into particular domains. These domains include the punctured unit disk, the exterior of the closed unit disk, and concave wedge-domains. The analogous Bohr radius is also studied for harmonic and starlike logharmonic mappings in $\mathbb{D}.$ The Bohr phenomenon which is described in terms of the Euclidean distance is further investigated using the spherical chordal metric and the hyperbolic metric. The exposition concludes with a discussion on the $n$-dimensional Bohr radius

Multidimensional Bohr radii for vector-valued holomorphic functions

In this paper, we study a new multidimensional Bohr radius, namely the arithmetic Bohr radius, which was first introduced by Defant, Maestre and Prengel [Q. J. Math. 59, (2008), pp. 189--205], in more general settings. We consider the well-known Bohr inequality for the class of holomorphic mappings defined on complete Reinhardt domains in $\mathbb{C}^n$ into complex Banach spaces (possibly infinite dimensional). Further, we give answer to some open questions related to exact values of multidimensional Bohr radii using the concept of arithmetic Bohr radius.Comment: 15 p

Monomial convergence for holomorphic functions on ℓ_r\ell\_r

Let $\mathcal F$ be either the set of all bounded holomorphic functions or the set of all $m$-homogeneous polynomials on the unit ball of $\ell\_r$. We give a systematic study of the sets of all $u\in\ell\_r$ for which the monomial expansion $\sum\_{\alpha}\frac{\partial^\alpha f(0)}{\alpha !}u^\alpha$ of every $f\in\mathcal F$ converges. Inspired by recent results from the general theory of Dirichlet series, we establish as our main tool, independently interesting, upper estimates for the unconditional basis constants of spaces of polynomials on $\ell\_r$ spanned by finite sets of monomials

Bohr Phenomena for Laplace-Beltrami Operators

Cataloged from PDF version of article.We investigate a Bohr phenomenon on the spaces of solutions of weighted Laplace-Beltrami operators associated with the hyperbolic metric of the unit ball in ℂN. These solutions do not satisfy the usual maximum principle, and the spaces have natural bases none of whose members is a constant function. We show that these bases exhibit a Bohr phenomenon, define a Bohr radius for them that extends the classical Bohr radius, and compute it exactly. We also compute the classical Bohr radius of the invariant harmonic functions on the real hyperbolic space. © 2006 Royal Netherlands Academy of Arts and Sciences