Bohr Radius for Subordination and K-quasiconformal Harmonic Mappings

Abstract

The present article concerns the Bohr radius for K-quasiconformal sense-preserving harmonic mappings $$f=h+\overline{g}$$ in the unit disk $$\mathbb {D}$$ for which the analytic part h is subordinated to some analytic function $$\varphi $$ , and the purpose is to look into two cases: when $$\varphi $$ is convex, or a general univalent function in $$\mathbb {D}$$ . The results state that if $$h(z) =\sum _{n=0}^{\infty }a_n z^n$$ and $$g(z)=\sum _{n=1}^{\infty }b_n z^n$$ , then $$\begin{aligned} \sum _{n=1}^{\infty }(|a_n|+|b_n|)r^n\le {{\text {dist}}}(\varphi (0),\partial \varphi (\mathbb {D})) \quad \text{ for } r\le r^* \end{aligned}$$ and give estimates for the largest possible $$r^*$$ depending only on the geometric property of $$\varphi (\mathbb {D})$$ and the parameter K. Improved versions of the theorems are given for the case when $$b_1 = 0$$ and corollaries are drawn for the case when $$K\rightarrow \infty $$ .