Improved Bohr inequality for harmonic mappings. (English) Zbl 1523.31005
Summary: In order to improve the classical Bohr inequality, we explain some refined versions for a quasi-subordination family of functions in this paper, one of which is key to build our results. Using these investigations, we establish an improved Bohr inequality with refined Bohr radius under particular conditions for a family of harmonic mappings defined in the unit disk \(\mathbb{D}\). Along the line of extremal problems concerning the refined Bohr radius, we derive a series of results. Here, the family of harmonic mappings has the form \(f=h+\overline{g}\), where \(g(0)=0\), the analytic part \(h\) is bounded by 1 and that \(|g^{\prime}(z)|\leq k|h^{\prime}(z)|\) in \(\mathbb{D}\) and for some \(k\in [0,1]\).
{© 2022 Wiley-VCH GmbH.}
{© 2022 Wiley-VCH GmbH.}
MSC:
| 31A05 | Harmonic, subharmonic, superharmonic functions in two dimensions |
| 30B10 | Power series (including lacunary series) in one complex variable |
| 30A10 | Inequalities in the complex plane |
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