Equivalence problem for annuli and Bell representations in the plane (Q854003)

\textit{S. Bell} studied in [Houston J. Math. 26, No. 2, 277--297 (2000; Zbl 0981.30007)] Bergman kernels and introduced some classes of multiply-connected plane domains. One example for the connectivity 2 case is the domain \[ A(r) =\{ z \in \mathbb{C} : | z+ 1/z| < r \}, \quad r >2 . \] The present authors study the problem of finding an annulus \(\Omega(\rho^2)=\) \(\{ \rho<| z| <1 \}\) conformally equivalent to \(A(r).\) They prove that \(\Omega(\rho^2)\) is conformally equivalent to \(A(r)\) if and only if \(r= 2/c(\rho)\) and establish some infinite series expansions for \(c(\rho).\) \{Reviewer's remark. Alternatively, \[ c(\rho)^2 = \mu^{-1} \left( 2\log \frac{1}{\rho} \right) \] where \(\mu(r)= (\pi K(\sqrt{1-r^2}))/(2 K(r))\) and \(K(r)\) is the complete elliptic integral.\}