On a subclass of close-to-convex harmonic mappings

Abstract

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 $|z{h}^{\prime \prime }(z)+\alpha (h^{\prime }(z)-1)|\le \beta -|z{g}^{\prime \prime }(z)+\alpha g^{\prime }(z)|.$ 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.