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 $\phi$, and the purpose is to look into two cases: when $\phi$ is convex, or a general univalent function in $\ID$. 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 $$\sum_{n=1}^{\infty}(|a_n|+|b_n|)r^n\leq \dist (φ(0),\partialφ(\ID)) ~\mbox{ for $r\leq r^*$} $$ and give estimates for the largest possible $r^{\ast }$ depending only on the geometric property of $φ(\ID)$ 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 $K\to \infty$.

15 pages; To appear in Bulletin of the Malaysian Mathematical Sciences Society

References in corpus (1)

• On the Bohr inequality