Domains with growth conditions for the quasihyperbolic metric

For a given growth functionH, we say that a domainD ?R ⁿ is anH-domain if δ_D x≤δ_D(x ₀)H(k _D(x,x ₀)),x ∈D, where δ_D(x)=d(x?D) andk _D denotes the quasihyperbolic distance. We show that ifH satisfiesH(0)=1, |H'|≤H, andH"≤H, then there exists an extremalH-domain. Using this fact, we investigate some fundamental properties ofH-domains.