Concave functions, Blaschke products, and polygonal mappings

We consider the class $\mathrm{Co}(p)$ of all conformal maps of the unit disk onto the exterior of a bounded convex set. We prove that the triangle mappings, i.e., the functions that map the unit disk onto the exterior of a triangle, are among the extreme points of the closed convex hull of $\mathrm{Co}(p)$. Moreover, we prove a conjecture on the closed convex hull of $\mathrm{Co}(p)$ for all $p\in(0,1)$ which had partially been proved by the authors for some values of $p\in(0,1)$.