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Publisher's Note Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations
- Publisher
- Springer Journals
- Copyright
- Copyright © 2019 by Malaysian Mathematical Sciences Society and Penerbit Universiti Sains Malaysia
- Subject
- Mathematics; Mathematics, general; Applications of Mathematics
- ISSN
- 0126-6705
- eISSN
- 2180-4206
- DOI
- 10.1007/s40840-019-00795-9
- Publisher site
- See Article on Publisher Site
Abstract
The present article concerns the Bohr radius for K-quasiconformal sense-preserving harmonic mappings $$f=h+\overline{g}$$ f = h + g ¯ in the unit disk $$\mathbb {D}$$ D for which the analytic part h is subordinated to some analytic function $$\varphi $$ φ , and the purpose is to look into two cases: when $$\varphi $$ φ is convex, or a general univalent function in $$\mathbb {D}$$ D . The results state that if $$h(z) =\sum _{n=0}^{\infty }a_n z^n$$ h ( z ) = ∑ n = 0 ∞ a n z n and $$g(z)=\sum _{n=1}^{\infty }b_n z^n$$ g ( z ) = ∑ n = 1 ∞ b n z n , then $$\begin{aligned} \sum _{n=1}^{\infty }(|a_n|+|b_n|)r^n\le {{\text {dist}}}(\varphi (0),\partial \varphi (\mathbb {D})) \quad \text{ for } r\le r^* \end{aligned}$$ ∑ n = 1 ∞ ( | a n | + | b n | ) r n ≤ dist ( φ ( 0 ) , ∂ φ ( D ) ) for r ≤ r ∗ and give estimates for the largest possible $$r^*$$ r ∗ depending only on the geometric property of $$\varphi (\mathbb {D})$$ φ ( D ) and the parameter K. Improved versions of the theorems are given for the case when $$b_1 = 0$$ b 1 = 0 and corollaries are drawn for the case when $$K\rightarrow \infty $$ K → ∞ .
Journal
Bulletin of the Malaysian Mathematical Sciences Society – Springer Journals
Published: Jul 1, 2019