  For $\alpha > -1$ and $\beta >0, $ let $\mathcal{B}_{\mathcal{H}}^0(\alpha,
\beta)$ denote the class of sense preserving harmonic mappings
$f=h+\overline{g}$ in the open unit disk $\mathbb{D}$ satisfying
$|zh''(z)+\alpha(h'(z)-1)|\leq \beta-|zg''(z)+\alpha g'(z)|.$ First, we
establish that each function belonging to this class is close-to-convex in the
open unit disk if $\beta \in (0, 1+\alpha]$. Next, we obtain coefficient
bounds, growth estimates and convolution properties. We end the paper with
applications and will construct harmonic univalent polynomials belonging to
this class.
