Behavior of domain constants under conformal mappings
Let \(\Omega\) be a plane domain of hyperbolic type and let \(\varphi\) be a universal covering map of the unit disk onto \(\Omega\). Furthermore, let \(\lambda_\Omega(w)= 1/[(1- |z|^2)|\varphi'(z)|\) with \(w= \varphi(z)\) denote the hyperbolic metric. The authors consider six domain constants. The domain constant \[ c(\Omega)= \sup\{\lambda_\Omega(w)\text{ dist}(w, \Omega): w\in \Omega\} \] is \(> 0\) if and only if \(\partial\Omega\) is uniformly perfect. Also, let \[ \eta(\Omega)= \sup\{(1- |z|^2)|\varphi''(z)/[2\varphi'(z)]- \overline z|: |z|< 1\}. \] Now, let \(\Omega\) and \(\Delta\) be two such domains that are conformally equivalent. It is proved that \(|\eta(\Omega)- \eta(\Delta)|\leq 1\). The proof is rather subtle and relies on the fact that the hyperbolic injectivity radius is conformally invariant. Also, it is shown that \[ c(\Delta)/c(\Omega)\leq 2(1- c(\Delta)- c(\Delta)^2)\qquad\text{and even }\leq 2.4336. \] Further domain constants are connected with the Schwarzian deriviate and with maps of hyperbolic disks onto convex domains.
scientific article; zbMATH DE number 820096
scientific article