Complex Disk Products and Cartesian Ovals

Abstract

Let $$D_R$$, $$D_r$$, $$D_S$$, $$D_s$$ be complex disks with common center 1 and radii R, r, S, s, respectively. We consider the Minkowski products $$A := D_R D_r$$ and $$B := D_S D_s$$ and give necessary and sufficient conditions for A being a subset or superset of B. Partially, this extends to n-fold disk products $$D_1\ldots D_n$$, $$n>2$$. It is well-known that the boundaries of A and B are outer loops of Cartesian ovals. Therefore, our results translate to necessary and sufficient conditions under which such loops encircle each other.