# Sections of univalent harmonic mappings

## TL;DR

This paper determines the univalence radius of sections of normalized univalent harmonic mappings with convex, starlike, or close-to-convex ranges, providing sharp results especially for convex ranges, and compares with classical conformal mapping results.

## Contribution

It establishes the sharp radius of univalence for sections of harmonic mappings with various geometric range conditions, extending classical conformal mapping results.

## Key findings

- The radius of univalence for sections with convex range is sharp.
- Each section $s_{n,n}(f)$ is univalent in the disk of radius 1/4 for all $n extgreater{}2$.
- Results extend classical univalence radius results to harmonic mappings.

## Abstract

In this article, we determine the radius of univalence of sections of normalized univalent harmonic mappings for which the range is convex (resp. starlike, close-to-convex, convex in one direction). Our result on the radius of univalence of section $s_{n,n}(f)$ is sharp especially when the corresponding mappings have convex range. In this case, each section $s_{n,n}(f)$ is univalent in the disk of radius $1/4$ for all $n\geq2$, which may be compared with classical result of Szeg\"{o} on conformal mappings.