Coefficients of univalent harmonic mappings

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.