A distortion theorem for conformal mappings with an application to subharmonic functions
Let D be a finitely connected domain in the complex plane such that every component of \({\mathbb{C}}\)-D contains at least two points. If there exist \(\rho >0\) and \(\theta\in (0,1)\) such that for every \(\omega\in \partial D\) a closed circular sector with vertex at \(\omega\) which is congruent to \(S(\rho,\theta)=\{w\in {\mathbb{C}}:\) \(0\leq | w| \leq \rho\), \(0\leq \arg w\leq \pi \theta \}\) lies in \({\mathbb{C}}\)-D, then D is said to satisfy an exterior \(\theta\)-wedge condition. Suppose that \(D_ 0\) is a finitely connected domain whose boundary components are simple analytic curves. The author first proves that if f is a conformal mapping of \(D_ 0\) onto D, and if \(\lambda_{D_ 0}(z)\) is the Poincaré metric for \(D_ 0\), then there exists \(m>0\) such that for all \(z\in D_ 0\), \[ \frac{1}{| f'(z)|}\leq m\lambda_{D_ 0}(z)^{1-\theta}. \] This distortion theorem is then applied to prove the following, which generalizes a theorem of Suzuki. Suppose that D satisfies an exterior \(\theta\)-wedge condition. For \(0<p<\infty\), set \(\beta (p,\vartheta)=2- \min \{1,p\}/(2-\theta)\), and let \(\delta_ D(z)\) be the distance from z in D to \(\partial D\). If a non-negative subharmonic function s on D satisfies \[ \iint_{D}\delta_ D(z)^{-\beta (p,\theta)}s(z)^ p dx dy<\infty, \] then \(s\equiv 0\) on D.
scientific article; zbMATH DE number 4175339