Convergence regions with bounded convex complements for continued fractions K(1/bn)

In this paper we resurrect some twin convergence region results of Thron from the 1940s for continued fractions K(1/bn) and derive from them best simple convergence region results for these continued fractions. Our principal contribution is an elementary proof of what we call the Uniform Circle Theorem. This theorem says that a certain one-parameter family of regions which are complements of open disks containing the origin is a family of best uniform convergence regions for continued fractions K(1/bn) and, moreover, it contains a sharp useful estimate for the speed of convergence. We apply this theorem to obtain a new convergence result for variable element continued fractions of Stieltjes type, which we call the Limacon Theorem.