On a subclass of harmonic close-to-convex mappings

Abstract

Let $\mathcal {H}$ denote the class of harmonic functions f defined in $\mathbb {D}:= \{z\in \mathbb {C}:|z| < 1\}$, and normalized by $f(0) = 0 = f_{z}(0) -1$. In this paper, for $\alpha \ge 0$, we consider the subclass $\mathcal {W}^0_{\mathcal {H}}(\alpha )$ of $\mathcal {H}$, defined by

$$\begin{aligned} \mathcal {W}^0_{\mathcal {H}}(\alpha ):= \left\{ f = h + \overline{g}\in \mathcal {H}: {\mathrm{Re}}\,(h'(z) + \alpha z h''(z)) >|g'(z) + \alpha z g''(z)|, \quad z\in \mathbb {D}\right\} . \end{aligned}$$

For $f\in \mathcal {W}^0_{\mathcal {H}}(\alpha )$, we prove the Clunie–Sheil-Small coefficient conjecture, and give some growth, convolution, and convex combination theorems. We also determine the value of r so that the partial sums of functions in $\mathcal {W}^0_{\mathcal {H}}(\alpha )$ are close-to-convex in $|z|<r$.