Bohr radius for locally univalent harmonic mappings

Abstract

We consider the class of all sense-preserving harmonic mappings $f=h+\overline{g}$ of the unit disk $\ID$, where $h$ and $g$ are analytic with $g(0)=0$, and determine the Bohr radius if any one of the following conditions holds: \bee $h$ is bounded in $\ID$. $h$ satisfies the condition $\mathrm{Re}h(z)\le 1$ in $\mathbb{D}$ with $h(0)>0$. both $h$ and $g$ are bounded in $\ID$. $h$ is bounded and $g^{\prime }(0)=0$. \eee We also consider the problem of determining the Bohr radius when the supremum of the modulus of the dilatation of $f$ in $\ID$ is strictly less than $1$. In addition, we determine the Bohr radius for the space $\mathcal{B}$ of analytic Bloch functions and the space ${\mathcal{B}}_H$ of harmonic Bloch functions. The paper concludes with two conjectures.

arXiv comment13 pages; The article will appear in Mathematische Nachrichten