# Coefficients of univalent harmonic mappings

## TL;DR

This paper establishes new bounds on the coefficients of univalent harmonic mappings in the class , providing insights into their structure and the radius of univalence of their sections.

## Contribution

It introduces explicit coefficient bounds for functions in , advancing understanding of their behavior and univalence properties.

## Key findings

- Coefficient bounds: |a_n| < 5.24e-6 n^{17} and |b_n| < 2.32e-7 n^{17} for n 
- Derived radius of univalence for sections of these harmonic mappings
- Enhanced understanding of the coefficient growth in univalent harmonic functions

## Abstract

Let $\mathcal{S}_H^0$ denote the class of all functions $f(z)=h(z)+\overline{g(z)}=z+\sum^\infty_{n=2} a_nz^n +\overline{\sum^\infty_{n=2} b_nz^n}$ that are sense-preserving, harmonic and univalent in the open unit disk $|z|<1$. The coefficient conjecture for $\mathcal{S}_H^0$ is still \emph{open} even for $|a_2|$. The aim of this paper is to show that if $f=h+\overline{g} \in \mathcal{S}^0_H$ then $ |a_n| < 5.24 \times 10^{-6} n^{17}$ and $|b_n| < 2.32 \times 10^{-7}n^{17}$ for all $n \geq 3$. Making use of these coefficient estimates, we also obtain radius of univalence of sections of univalent harmonic mappings.