The spherical metric and univalent harmonic mappings

TL;DR

This paper improves bounds on the second coefficient of harmonic univalent maps in the unit disk and relates the spherical areas of their covering surfaces, advancing understanding of harmonic mappings' geometric properties.

Contribution

It provides the first qualitative improvement on coefficient estimates for harmonic univalent maps since the 1980s and relates spherical areas of covering surfaces under dilatation constraints.

Findings

Improved estimate for the second coefficient of $h$.

Established dominance of spherical area of $h$'s surface over that of $f$ when dilatation is bounded.

First qualitative improvement since foundational papers in 1984 and 1990.

Abstract

Let $f=h+\overline{g}$ be a harmonic univalent map in the unit disk $\mathbb{D}$, where $h $ and $g$ are analytic. We obtain an improved estimate for the second coefficient of $h$. This indeed is the first qualitative improvement after the appearance of the papers by Clunie and Sheil-Small in 1984, and by Sheil-Small in 1990. Also, when the sup-norm of the dilatation is less than $1$, it is shown that the spherical area of the covering surface of $h$ is dominated by the spherical area of the covering surface of $f.$