Intrinsic geometry and boundary structure of plane domains

Given a nonempty compact set $E$ in a proper subdomain $\Omega$ of the complex plane, we denote the diameter of $E$ and the distance from $E$ to the boundary of $\Omega$ by $d(E)$ and $d(E,\partial\Omega)$, respectively. The quantity $d(E)/d(E,\partial\Omega)$ is invariant under similarities and plays an important role in geometric function theory. In case $\Omega$ has the hyperbolic distance $h_\Omega(z,w)$, we consider the infimum $\kappa(\Omega)$ of the quantity $h_\Omega(E)/\log(1+d(E)/d(E,\partial\Omega))$ over compact subsets $E$ of $\Omega$ with at least two points, where $h_\Omega(E)$ stands for the hyperbolic diameter of $E$. Let the upper half-plane be $\Bbb{H}$. We show that $\kappa(\Omega)$ is positive if and only if the boundary of $\Omega$ is uniformly perfect and $\kappa(\Omega)\le \kappa(\Bbb{H})$ for all $\Omega$, with equality holding precisely when $\Omega$ is convex.