On the Bohr inequality

Abstract

The Bohr inequality, first introduced by Harald Bohr in 1914, deals with finding the largest radius $r$, $0<r<1$, such that ${\sum }_{n=0}^{\infty }|a_n|r^n\le 1$ holds whenever $|{\sum }_{n=0}^{\infty }{a}_n{z}^n|\le 1$ 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.

References in corpus (3)

• The Bohnenblust--Hille inequality for homogeneous polynomials is hypercontractive
• Bohr property of bases in the space of entire functions and its generalizations
• H. Bohr's theorem for bounded symmetric domains

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• A generalization of the Bohr inequality for bounded analytic functions on simply connected domains and its applications
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• Bohr operator on analytic functions
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• Modifications of Bohr's inequality in various settings
• On Certain Generalizations of ${\mathcal{S}}^{\ast }(\psi )$
• Bohr inequalities for certain integral operators
• The Bohr's Phenomenon for the class of K-quasiconformal harmonic mappings
• Improved Bohr-type Inequalities for the Cesaro Operator
• A generalization of the Bohr-Rogosinski sum
• Note on Improved Bohr Inequality for harmonic mappings
• Bohr radius for subordination and $K$-quasiconformal harmonic mappings
• Generalized Bohr inequalities for K-quasiconformal harmonic mappings and their applications