The range set of zero for harmonic mappings of the unit disk with sectorial boundary normalization
Given a family \(F\) of all complex-valued functions in a domain \(ΩC \), the authors introduce the range set \(RS_F(A)\) of a set \(AΩ\) under the class in question, i.e. the set of all \(wC\) such that \(wF(A)\) for a certain \(FF\). Let \(T_1,T_2,T_3\) be closed arcs contained in the unit circle \(T\) of the same length \(2π/3\) and covering \(T\). The paper deals with the range set \(RS_F({0})\), where \(F\) is the class of all complex-valued harmonic functions \(F\) of the unit disk \(D\) into itself satisfying the following sectorial condition: For each \(k{1,2,3}\) and for almost every \(zT_k\) 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 \(FF\), \(|F(z)|43-2π(3 1+2|z|), zD\), by which \(|F(0)|2/3\). This implies that \(RS_F({0})\) is a subset of the closed disk of radius 2/3 and centred at the origin. In the paper the range set \(RS_F({0})\) is precisely determined. , number=1, journal=Annales Fennici Mathematici, author=Partyka, Dariusz and Zając, Józef, year=2024, month=Feb., pages=49–63