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On the Bohr Inequality

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Book cover Progress in Approximation Theory and Applicable Complex Analysis

Part of the book series: Springer Optimization and Its Applications ((SOIA,volume 117))

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 \(\mathbb{D}\) 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 \(\mathbb{D}\), 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.

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Acknowledgements

This work benefited greatly from the stimulating discussions and careful scrutiny of the manuscript by our student, Zhen Chuan Ng. The work of the second author was supported in parts by a research university grant from Universiti Sains Malaysia. The research of the third author was supported by project RUS/RFBR/P-163 under the Department of Science & Technology, India.

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Authors and Affiliations

  1. Department of Mathematics, American University of Sharjah, Sharjah, UAE

    Yusuf Abu Muhanna

  2. School of Mathematical Sciences, Universiti Sains Malaysia, 11800, Penang, Malaysia

    Rosihan M. Ali

  3. Indian Statistical Institute (ISI), Chennai Centre, SETS (Society for Electronic Transactions and Security), MGR Knowledge City, CIT Campus, Chennai, 600113, Tamil Nadu, India

    Saminathan Ponnusamy

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  1. Yusuf Abu Muhanna
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  2. Rosihan M. Ali
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Correspondence to Rosihan M. Ali .

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Editors and Affiliations

  1. Department of Mathematics and Statistics, Auburn University, Auburn, Alabama, USA

    Narendra Kumar Govil

  2. Department of Mathematics, University of Central Florida, Orlando, Florida, USA

    Ram Mohapatra

  3. Department of Mathematics, Tuskegee University, Tuskegee, Alabama, USA

    Mohammed A. Qazi

  4. Department of Mathematics, University of Erlangen-Nuremberg, Erlangen, Germany

    Gerhard Schmeisser

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Muhanna, Y.A., Ali, R.M., Ponnusamy, S. (2017). On the Bohr Inequality. In: Govil, N., Mohapatra, R., Qazi, M., Schmeisser, G. (eds) Progress in Approximation Theory and Applicable Complex Analysis. Springer Optimization and Its Applications, vol 117. Springer, Cham. https://doi.org/10.1007/978-3-319-49242-1_13

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