  We consider proper subdomains $G$ of $\mathbb{R}^n$ and their images
$G'=f(G)$ under quasiconformal mappings $f$ of $\mathbb{R}^n$. We compare the
distance ratio metrics of $G$ and $G'$; as an application we show that
$\varphi$-uniform domains are preserved under quasiconformal mappings of
$\mathbb{R}^n$. A sufficient condition for $\varphi$-uniformity is obtained in
terms of the quasi-symmetry condition. We give a geometric condition for
uniformity: If $G\subset\mathbb{R}^n$ is $\phi$-uniform and satisfies the
twisted cone condition, then it is uniform. We also construct a planar
$\phi$-uniform domain whose complement is not $\psi$-uniform for any $\psi$.
