International Journal of Mathematics and Mathematical Sciences
Volume 16 (1993), Issue 4, Pages 679-686
doi:10.1155/S0161171293000857

Diameter problems for univalent functions with quasiconformal extension

Abstract

This paper utilizes the method of extremal length to study several diameter problems for functions conformal outside of a disc centered at the origin, with a standard normalization, which possess a quasiconformal extension to a ring subdomain of this disc. Known results on the diameter of a complementary component of the image domain of a univalent function are extended. Applications to the transfinite diameters of families of non-overlapping functions and an extension of the Koebe one-quarter theorem are included.