Affine and linear invariant families of harmonic mappings

TL;DR

This paper investigates the properties of affine and linear invariant families of harmonic mappings in the unit disk, linking their order to the Schwarzian norm and confirming conjectures about the harmonic Koebe function.

Contribution

It determines the order of these families and relates it to the Schwarzian norm, providing new insights into the structure of harmonic univalent mappings.

Findings

Order of affine and linear invariant families characterized

Connection established between Schwarzian norm and family order

Consistency with conjectured order of harmonic Koebe function

Abstract

We study the order of affine and linear invariant families of planar harmonic mappings in the unit disk and determine the order of the family of mappings with bounded Schwarzian norm. The result shows that finding the order of the class $\mathcal{S}_H$ of univalent harmonic mappings can be formulated as a question about Schwarzian norm and, in particular, our result shows consistency between the conjectured order of $\mathcal{S}_H$ and the Schwarzian norm of the harmonic Koebe function.