On Some Subclass of Harmonic Close-to-convex Mappings

Abstract

Let $\mathcal{H}$ denote the class of harmonic functions $f$ in $\mathbb{D}:=\{z\in \mathbb{C}:|z|<1\}$ normalized by $f(0)=0=f_z(0)-1$. For $\alpha \ge 0$, we consider the following class $${\mathcal{W}}_{\mathcal{H}}^0(\alpha ):=\{f=h+\overline{g}\in \mathcal{H}:\mathrm{Re}(h^{\prime }(z)+\alpha z{h}^{\prime \prime }(z))>|g^{\prime }(z)+\alpha z{g}^{\prime \prime }(z)|,z\in \mathbb{D}\}.$$ In this paper, we first prove the coefficient conjecture of Clunie and Sheil-Small for functions in the class ${\mathcal{W}}_{\mathcal{H}}^0(\alpha )$. We also prove growth theorem, convolution, convex combination properties for functions in the class ${\mathcal{W}}_{\mathcal{H}}^0(\alpha )$. Finally, we determine the value of $r$ so that the partial sums of functions in the class ${\mathcal{W}}_{\mathcal{H}}^0(\alpha )$ are close-to-convex in $|z|<r$.

18 Pages