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Let denote the class of harmonic functions f defined in , and normalized by . In this paper, for , we consider the subclass of , 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 , 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 are close-to-convex in .