Intrinsic Geometry and Boundary Structure of Plane Domains - Rainio Oona; Sugawa Toshiyuki; Vuorinen Matti (MAIK NAUKA/INTERPERIODICA/SPRINGER, 2022)
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 derivative Omega), respectively. The quantity d(E)/d(E, partial derivative 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 derivative)) 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 H. We show that partial derivative(Omega) is positive if and only if the boundary of Omega is uniformly perfect and partial derivative(Omega) <= kappa(H) for all Omega, with equality holding precisely when Omega is convex.