The local structure of surface of interior conformal radius for a plane domain (Q1398506)

Let \({\mathcal D}\subset \mathbb{C}\) denote a simply connected domain in the complex plane \(z\), \(b\in{\mathcal D}\), \(R({\mathcal D},b)\) -- the interior conformal radius of the domain \({\mathcal D}\) at the point \(b\). Let \(\zeta={\mathcal F}(z)\) be a conformal mapping of \({\mathcal D}\) onto the unit disc \(E=\{\zeta\in\mathbb{C}: |\zeta |<1\}\) with the usual norming \({\mathcal F}(b)=0\), \(\arg{\mathcal F}'(b)= \beta\). Then we have \(R({\mathcal D},b)=1/ |{\mathcal F}'(b)|\). We consider the inverse function \(z={\mathcal F}^{-1}(\zeta)=f(T(\zeta))\), \(T(\zeta)= (\zeta+a)/(1+ \overline a\zeta)\), which maps the disc \(E\) onto \({\mathcal D}\) so that \(f(a)=b\). We obtain \(R({\mathcal D},b)=|f'(a)|(1-|a|^2)= R(f(E),f(a))\). In this article the author studies the local behavior of the surface of the interior conformal radius over a disk and over a domain \({\mathcal D}\). In particular he obtained: Theorem 1. If a regular function \(f(\zeta)\) satisfies in \(E\) the conditions \(|\{f(\zeta), \zeta\}|\leq 2/(1-|\zeta|^2)\), \(\zeta\in E\), with the Schwarz derivative \(\{f(\zeta),\zeta\}\), then the surface \(\omega =\lambda(f(E),f(\zeta))\), constructed over \(E\), is convex downwards \((\lambda( {\mathcal D},b)=1/R({\mathcal D},b))\). Theorem 2. The surface for the density of the hyperbolic metric \(\lambda_1(\zeta)=1/R_1(\zeta)\) with the equation \[ \omega =\lambda_1\bigl(T (\zeta)\bigr)= \Bigl[\biggl|f'\bigl(T(\zeta) \bigr) \biggr|\biggl(1-\bigl|T^2(\zeta) \bigr|\biggr) \Bigr]^{-1}, \] over a neighborhood of the point \(a\in E\) is convex downwards under the condition \[ \biggl|\bigl\{f(\zeta),\zeta \bigr\}\biggr|\leq 2/\bigl(1-|\zeta |^2\bigr)^2,\;\zeta\in E\biggl(R_1(\zeta): =R\bigl(f(E), f(\zeta) \bigr) \biggr). \]