Harmonic mappings onto $R$-convex domains

The plane domain $D$ is called $R$-convex if $D$ contains each compact set bounded by two shortest sub-arcs of the radius $R$ with endpoints $w_1,\,w_2\in D$, $|w_1-w_2|\le 2R$. In this paper, we prove the conditions of $R$-convexity for images of disks under harmonic sense preserving functions. The coefficient bounds for harmonic mappings of the unit disk onto $R$-convex domains are obtained.