Coefficients of univalent harmonic mappings

Abstract

Let ${\mathcal{S}}_H^0$ denote the class of all functions $f(z)=h(z)+\overline{g(z)}=z+{\sum }_{n=2}^{\infty }{a}_n{z}^n+\overline{{\sum }_{n=2}^{\infty }{b}_n{z}^n}$ 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 $f=h+\overline{g}\in {\mathcal{S}}_H^0$ then $|a_n|<5.24\times 10^{-6}{n}^{17}$ and $|b_n|<2.32\times 10^{-7}{n}^{17}$ for all $n\ge 3$. Making use of these coefficient estimates, we also obtain radius of univalence of sections of univalent harmonic mappings.

14 pages; The article is to appear in the journal Monatshefte für Mathematik