Extreme points and support points of families of harmonic Bloch mappings

TL;DR

This paper investigates the extreme and support points of families of harmonic Bloch mappings, providing necessary conditions and characterizations based on Bloch unit-valued sets within the unit disk.

Contribution

It introduces new criteria for identifying extreme and support points of harmonic Bloch mappings using Bloch unit-valued sets, advancing the understanding of their geometric structure.

Findings

Necessary condition for extreme points in harmonic Bloch spaces

Support points characterized by non-empty Bloch unit-valued sets

Complete characterization of support points in harmonic Bloch spaces

Abstract

In this paper, the main aim is to discuss the existence of the extreme points and support points of families of harmonic Bloch mappings and little harmonic Bloch mappings. First, in terms of the Bloch unit-valued set, we prove a necessary condition for a harmonic Bloch mapping (resp. a little harmonic Bloch mapping) to be an extreme point of the unit ball of the normalized harmonic Bloch spaces (resp. the normalized little harmonic Bloch spaces) in the unit disk $\mathbb{D}$. Then we show that a harmonic Bloch mapping $f$ is a support point of the unit ball of the normalized harmonic Bloch spaces in $\mathbb{D}$ if and only if the Bloch unit-valued set of $f$ is not empty. We also give a characterization for the support points of the unit ball of the harmonic Bloch spaces in $\mathbb{D}$.