  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.
