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 \(\Omega\subset\hat{\mathbb{C}}\), the authors introduce the range set \(RS_{\mathcal F}(A)\) of a set \(A\subset\Omega\) under the class in question, i.e. the set of all \(w\in\Bbb C\) 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 \(\Bbb T\) of the same length \(2\pi/3\) and covering \(\Bbb T\). The paper deals with the range set \(RS_{\mathcal F}(\{0\})\), where \(\mathcal F\) is the class of all complex-valued harmonic functions \(F\) of the unit disk \(\Bbb D\) 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 \(F^*(z)\) 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 \(F\in\mathcal F\),

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