Convolution of a harmonic mapping with $n$-starlike mappings and its partial sums

We investigate the univalency and the directional convexity of the convolution $\varphi \tilde{\ast }f=\varphi \ast h+\overline{\varphi \ast g}$ of the harmonic mapping $f=h+\bar{g}$ with a mapping $\varphi$ whose convolution with the mapping $z+{\sum }_{k=2}^{\mathrm{\infty }}{k}^n{z}^k$ is starlike (and such a mapping $\varphi$ is called $n$-starlike). In addition, we investigate the directional convexity of (i) the convolution of an analytic convex mapping with the slanted half-plane mapping, and (ii) the partial sums of the convolution of a $6$-starlike mapping with the harmonic Koebe mapping and the harmonic half-plane mapping.