A generalization of Study’s theorem on convex maps

Abstract:

Study proved that disks of radius r, $0 < r \leqslant 1$, get mapped onto convex (starlike) sets under all convex (starlike) maps of the unit disk U. Pommerenke and Heins gave a geometric characterization of the subsets of U that get mapped onto convex sets under all convex maps. In this note we give an analytical characterization of the subsets of U that get mapped onto $\alpha$-starlike domains under all $\alpha$-starlike maps of U for all $\alpha \geqslant 0$ noting that 0-starlikeness equals starlikeness and 1-starlikeness equals convexity.