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

Dariusz Partyka¹ ; Józef Zając²

¹ The John Paul II Catholic University of Lublin, Department of Mathematical Analysis, and The University College of Applied Sciences in Chełm, Institute of Mathematics and Information Technology
² The University College of Applied Sciences in Chełm, Institute of Mathematics and Information Technology

Annales Fennici Mathematici, Tome 49 (2024) no. 1, p. 49–63.

Résumé

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.  

Détail

DOI : 10.54330/afm.143007

Keywords: Boundary normalization, harmonic mappings, Poisson integral, Schwarz Lemma