Uniform domains, John domains and quasi-isotropic domains

We study for two metrics j and d whether a plane domain D⊊︀C for 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.