Asymptotic Bohr radius for the polynomials in one complex variable

“…Subsequently, Fournier [45] computed and obtained an explicit formula for R n by applying the notion of bounded-preserving operators. The following result concerning the asymptotic behaviour of R n was proved only recently in [33]:…”

Section: Theorem 24mentioning

confidence: 99%

On the Bohr Inequality

Ali

Ponnusamy

2017

Springer Optimization and Its Applications

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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.

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“…For the last twenty years, this result has been generalized in many ways: to polynomials in one complex variable by Guadarrama [13], Fournier [11] and Chu [6]. To several complex variables by Boas-Khavinson [4], to the polydisk by Defant-Ortega-Cerdà-Ounaïes-Seip, [8].…”

Section: Introductionmentioning

confidence: 99%