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Unbounded convex polygons, Blaschke products and concave schlicht functions. (English) Zbl 1161.30006

Summary: We consider conformal maps \(f\) of the open unit disc \(\mathbb D\) onto a concave domain, i.e., a domain whose complement with respect to \(\mathbb C\) is convex and unbounded. We say that \(f\) is a concave schlicht function if \(f(\mathbb D)\) is a concave domain. We also fix an opening angle for the domain \(f(\mathbb D)\) at \(\infty\) which is less than or equal to \(\pi A\), \(A\in \{1,2\}\), and denote this class of functions by \(CO(A)\). In this paper we prove a representation formula using Blaschke products for those members \(f\) of \(CO(A)\) for which the exterior of \(f(\mathbb D)\) is a convex unbounded polygon. Further, we present some examples supporting our conjecture that these polygonal maps are extreme points of the class \(CO(A)\).

MSC:

30C45 Special classes of univalent and multivalent functions of one complex variable (starlike, convex, bounded rotation, etc.)
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