A conformal invariant for nonvanishing analytic functions and its applications
The authors of this paper study a new conformal invariant defined for nonvanishing analytic functions. More precisely, let \(\varphi:D \rightarrow\mathbb{C}^*=\mathbb{C}\backslash \{0\}\) be an analytic function defined on a hyperbolic domain \(D\) with hyperbolic density \(\rho_D(z)\), and let \[ V_D(\varphi) = \sup_{z\in D}\rho_D(z)^{-1}\left| \frac{\varphi'(z)}{\varphi(z)}\right| \, . \] This quantity is conformally invariant in the sense that \(V_D(\varphi)=V_{D_0}(\varphi\circ f)\) for a conformal (univalent, onto) map \(f:D_0\rightarrow D\). In general, \(V_D\) decreases under composition \(\varphi\circ g\) with analytic maps \(g\). The authors study a number of properties of the invariant \(V_D\) and its connection with other relevant analytic quantities related to the geometry of domains and with the behavior of analytic functions defined on them. In particular, they relate \(V_D\) to the circular width (with respect to the origin) \[ W(\Omega)= \left(\text{ inf}_{w\in \Omega}| w| \rho_{\Omega}(w)\right)^{-1} \] of a hyperbolic domain \(\Omega\) not containing the origin. The circular width is not easy to compute, but the authors establish several important properties of \(W(\Omega)\) as well as a geometric characterization of when it is infinite. They also provide a series of examples when the circular width can be determined explicitly. Finally, the authors study the connection of \(V_D\) with the concept of uniform perfectness, relevant in the study of univalent functions and quasiconformal extensions.
scientific article; zbMATH DE number 5140640
scientific article