On the squeezing function for finitely connected planar domains (Q2171001)

The authors disprove the conjecture that a corresponding formula for the squeezing functions for an annulus is valid for planar domains of any finite connectivity. Let \(\Omega\subset\mathbb C^d\), \(d\geq1\), be a domain such that the class \(\mathcal U(\Omega)\) of all injective holomorphic mappings \(f:\Omega\to\mathbb B:=\{z\in\mathbb C^d:|z_1|^2+\dots+|z_d|^2\}<1\) is not empty. The squeezing function \(S_{\Omega}:\Omega\to\mathbb R\) of \(\Omega\) is defined by \[S_{\Omega}(z):=\sup\big\{\mathrm{dist}(0,\partial f(\Omega)):f\in\mathcal U(\Omega),f(z)=0\big\},\;\;z\in\Omega.\] A circularly slit disk is a subdomain \(D\) of the unit disk \(\mathbb D=\{z\in\mathbb C:|z|<1\}\), \(0\in D\), such that \(\mathbb D\setminus D\) consists of \(\partial\mathbb D\) and closed arcs lying on concentric circles centered at the origin. Let \(\Omega\) be a domain in \(\mathbb C\) with at least one non-degenerate boundary component. If \(\Omega\) is finitely connected with non-degenerate boundary components \(\Gamma_0,\Gamma_1,\dots,\Gamma_n\), then for each \(z\in\Omega\) and each \(j=0,1,\dots,n\) there is a unique conformal map \(f_{z,j}\) of \(\Omega\) onto a circularly slit disk normalized by \(f_{z,j}(z)=0\), \(f'_{z,j}(z)>0\), and \(f_{z,j}(\Gamma_j)=\partial\mathbb D\). The main result is given in Theorem 1 which states that for each \(m\geq3\), there exists an \(m\)-connected domain \(\Omega\subset\mathbb C\) without degenerate boundary components and a point \(z\in\Omega\) such that \(S_{\Omega}(z)\neq\max_{j=0,\dots,m-1}\mathrm{dist}(0,\partial f_{z,j}(\Omega)).\) Theorem 1 is not true in the doubly connected case. Theorem 2 gives more precise information by identifying all extremal functions. Theorem 2. Formula \[S_{\Omega}(z)=\max_{j=0,\dots,m-1}\mathrm{dist}(0,\partial f_{z,j}(\Omega))\] holds for any doubly connected domain \(\Omega\) with at least one non-degenerate boundary component and for any \(z\in\Omega\). Each extremal function is a conformal map onto a circularly slit disk. In particular, for any \(r\in(0,1)\) and any \(z\in\mathbb A_r:=\{z\in\mathbb C:r<|z|<1\}\), \[S_{\mathbb A_r}(z)=\max\{|z|,r/|z|\}.\]