The range set of zero for harmonic mappings of the unit disk with sectorial boundary normalization

Abstract

Given a family $\mathcal{F}$ of all complex-valued functions in a domain $\mathrm{\Omega }\subset \hat{\mathbb{C}}$, the authors introduce the range set $R{S}_{\mathcal{F}}(A)$ of a set $A\subset \mathrm{\Omega }$ under the class in question, i.e. the set of all $w\in \mathbb{C}$ such that $w\in F(A)$ for a certain $F\in \mathcal{F}$. Let $T_1,T_2,T_3$ be closed arcs contained in the unit circle $\mathbb{T}$ of the same length $2\pi /3$ and covering $\mathbb{T}$. The paper deals with the range set $R{S}_{\mathcal{F}}(\{0\})$, where $\mathcal{F}$ is the class of all complex-valued harmonic functions $F$ of the unit disk $\mathbb{D}$ into itself satisfying the following sectorial condition: For each $k\in \{1,2,3\}$ and for almost every $z\in T_k$ the radial limit $F^{\ast }(z)$ of the function $F$ at the point $z$ belongs to the angular sector determined by the convex hull spanned by the origin and arc $T_k$. In 2014 the authors proved that for any $F\in \mathcal{F}$,

$|F(z)|\le \frac{4}{3}-\frac{2}{\pi }\arctan \left(\frac{\sqrt{3}}{1+2|z|}\right),z\in \mathbb{D}$,   by which $|F(0)|\le 2/3$. This implies that $R{S}_{\mathcal{F}}(\{0\})$ is a subset of the closed disk of radius 2/3 and centred at the origin. In the paper the range set $R{S}_{\mathcal{F}}(\{0\})$ is precisely determined.