Bohr’s Inequality for Harmonic Mappings and Beyond

  • Anna Kayumova
  • Ilgiz R. Kayumov
  • Saminathan Ponnusamy
Conference paper
Part of the Communications in Computer and Information Science book series (CCIS, volume 834)


There has been a number of problems closely connected with the classical Bohr inequality for bounded analytic functions defined on the unit disk centered at the origin. Several extensions, generalizations and modifications of it are established by many researchers and they can be found in the literature, for example, in the multidimensional setting and in the case of the Dirichlet series, functional series, function spaces, etc. In this survey article, we mainly focus on the recent developments on this topic and in particular, we discuss new and sharp improvements on the classical Bohr inequality and on the Bohr inequality for harmonic functions.


Bounded analytic functions Univalent functions Bohr radius Rogosinski radius Schwarz-Pick lemma Subordination 

Subject Classifications:

Primary: 30A10 30H05 30C35 Secondary: 30C45 



The research of the first and the second authors were supported by Russian foundation for basic research, Proj. 17-01-00282. The work of the third author is supported by Mathematical Research Impact Centric Support of DST, India (MTR/2017/000367). The third author is currently at Indian Statistical Institute (ISI), Chennai Centre, Chennai, India.


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© Springer Nature Singapore Pte Ltd. 2018

Authors and Affiliations

  1. 1.Kazan Federal UniversityKazanRussia
  2. 2.Department of MathematicsIndian Institute of Technology MadrasChennaiIndia

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