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 ∑ ∞ n=0 |a n |r n ≤ 1 holds whenever | ∑ ∞ n=0 a n z n | ≤ 1 in the unit disk D of the complex plane. The exact value of this largest radius, known as the 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 D, as well as for analytic functions from 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 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.

Bohr radius for locally univalent harmonic mappings

“…is a sense-preserving harmonic mapping of the disk , where ℎ is a bounded function in . Then 1) and the number 1∕5 is sharp. Moreover, either 0 = 0 or | 0 | in (1.1) is replaced by | 0 | 2 , then the constant 1∕5 could be replaced by 1∕3 which is also sharp.…”

“…Many mathematicians have contributed toward the understanding of this problem in several settings. We refer to the recent survey on this topic by Abu-Muhanna et al [1] for the importance, background, and several other recent results and extensions. For certain recent results, see [2,11,12].…”

“…We refer to the recent survey on this topic by Abu‐Muhanna et al. for the importance, background, and several other recent results and extensions. For certain recent results, see .…”

On the Bohr inequality for the Cesáro operator

“…In the recent years, there has been a great deal of research activity including refinements, ramifications and extensions of Bohr-type theorems in different settings. See [2,[10][11][12] and the survey chapters from [1,8], and the references therein. Interestingly, Dixon [6] used Bohr's phenomena in operator theory in connection with the long-standing problem of characterization of Banach algebras that satisfy the von Neumann inequality [14].…”

Bohr theorems for slice regular functions over octonions

“…The Bohr type inequality has also been formulated for holomorphic functions valued in Banach spaces [8,23,25]. The reader may refer to the survey [1] and references therein for various generalizations and variants of the Bohr theorem, such as in the setting of complex polynomials, harmonic (or poly-harmonic) mappings, holomorphic functions in several complex variables and more abstract settings. Historically, Bohr in [10] discovered a weak version of theorem 1.1 for |z| 1/6 by the Carathéodory inequality…”