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On the Bohr inequality with a fixed zero coefficient


Authors: Seraj A. Alkhaleefah, Ilgiz R. Kayumov and Saminathan Ponnusamy
Journal: Proc. Amer. Math. Soc. 147 (2019), 5263-5274
MSC (2010): Primary 30A10, 30B10, 30C62, 30H05, 31A05, 41A58; Secondary 30C75, 40A30
DOI: https://doi.org/10.1090/proc/14634
Published electronically: July 30, 2019
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Abstract: In this paper, we introduce the study of the Bohr phenomenon for a quasisubordination family of functions, and establish the classical Bohr's inequality for the class of quasisubordinate functions. As a consequence, we improve and obtain the exact version of the classical Bohr's inequality for bounded analytic functions and also for $ K$-quasiconformal harmonic mappings by replacing the constant term by the absolute value of the analytic part of the given function. We also obtain the Bohr radius for the subordination family of odd analytic functions.


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Additional Information

Seraj A. Alkhaleefah
Affiliation: Institute of Mathematics and Mechanics, Kazan Federal University, 420 008 Kazan, Russia
Email: s.alkhaleefah@gmail.com

Ilgiz R. Kayumov
Affiliation: Institute of Mathematics and Mechanics, Kazan Federal University, 420 008 Kazan, Russia
Email: ikayumov@gmail.com

Saminathan Ponnusamy
Affiliation: Department of Mathematics, Indian Institute of Technology Madras, Chennai-600 036, India
Email: samy@iitm.ac.in

DOI: https://doi.org/10.1090/proc/14634
Keywords: Bohr inequality, harmonic mappings, sense-preserving $K$-quasiconformal mappings, locally univalent functions, analytic functions, odd functions, $p$-symmetric functions, subordination and quasisubordination
Received by editor(s): October 6, 2018
Received by editor(s) in revised form: March 22, 2019
Published electronically: July 30, 2019
Additional Notes: The research of the second author was funded by the subsidy allocated to Kazan Federal University for the state assignment in the sphere of scientific activities, project 1.13556.2019/13.1. The work of the third author was supported by Mathematical Research Impact Centric Support of DST, India (MTR/2017/000367)
Communicated by: Stephan Ramon Garcia
Article copyright: © Copyright 2019 American Mathematical Society