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International Journal of Mathematics and Mathematical Sciences
Volume 16, Issue 4, Pages 679-686
http://dx.doi.org/10.1155/S0161171293000857

Diameter problems for univalent functions with quasiconformal extension

Department of Mathematics, Louisiana State University in Shreveport, Shreveport, Louisiana 71115, USA

Received 23 April 1992; Revised 4 September 1992

Copyright © 1993 Hindawi Publishing Corporation. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

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.