About the Gradient of the Conformal Radius

Abstract

Let D be a simply connected domain in $\overline{\mathbb{C}}$ and R(D, z) the conformal radius of D at the point $z \in D/ \{\infty\}$We discuss the function $\nabla R(D,z)$ In particular, we prove that $\nabla R(D,z)$ is a quasi-conformal mapping of D for different types of domains.

Mathematics Subject Classification (2000). Primary 30C35; Secondary 30C62.