Quasi-invariant domain constants

Five domain constants are studied in our paper, all related to the hyperbolic geometry in hyperbolic plane regions which are uniformly perfect (in Pommerenke’s terminology). Relations among these domain constants are obtained, from which bounds are derived for the variance ratio of each constant under conformal mappings of the regions, and we also show that each constant may be used to characterize uniformly perfect regions. The authors wish to thank the University of California, San Diego for its hospitality during the 1988–89 academic year when this research was begun. Supported by the Landau Center for Mathematical Research in Analysis. Research partially supported by NSF Grant No. DMS-8801439.