On the Bohr inequality with a fixed zero coefficient. (English) Zbl 1428.30001
Summary: 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.
MSC:
| 30A10 | Inequalities in the complex plane |
| 30B10 | Power series (including lacunary series) in one complex variable |
| 30C62 | Quasiconformal mappings in the complex plane |
| 30C80 | Maximum principle, Schwarz’s lemma, Lindelöf principle, analogues and generalizations; subordination |
| 31A05 | Harmonic, subharmonic, superharmonic functions in two dimensions |
Keywords:
Bohr inequality; harmonic mappings; quasiconformal mappings; subordination; quasisubordinationReferences:
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