Uniform domains, John domains and quasi-isotropic domains

We study for two metrics j and d whether a plane domain D Cfor which there exists a constant c >0 with j (z,w) cd(z,w) for all z,w ∈ D is a uniform domain. In particular, we study the case when d is the λ-Apollonian metric a . We also study for a simply connected domain that whether quasi-isotropic domains are John disks and conversely. © 2008 Elsevier Inc. All rights reserved