Koebe invariant functions and extremal problems for holomorphic mappings in the unit ball of \(\mathbb C^n\)
Let \(B\) be the Euclidean ball in \(\mathbb{C}^{n},n\geq2\). Consider \(\sigma\) a holomorphic automorphism of the ball \(B\). A holomorphic mapping \(g:B\rightarrow\mathbb{C}^{n}\) that is invertible in a neighborhood of \(g(\sigma(0))\) is said to be invariant if \(\Lambda_{\sigma}(g)=g\), where \[ \Lambda_{\sigma}(g)(z)=D\sigma(0)^{-1}Dg(\sigma(0))^{-1} \left(g(\sigma(z))-g(\sigma(0))\right). \] In the paper, the authors give a complete characterization of all invariant functions and show how to construct invariant functions for arbitrary \(n\geq2\). A connection between the invariance concept and the solution of extremal problems in certain linear invariant families is also given.
scientific article; zbMATH DE number 5281676