Coefficient bounds for biholomorphic mappings which have a parametric representation
Let \(\mathbb{C}^{n}\) be the space of \(n\) complex variables with respect to an arbitrary norm \(\|.\|\) and let \(B=\{z\in\mathbb{C}^{n}: \|z\|<1\}\) be the unit ball. \(H(B)\) denotes the set of all holomorphic mappings from \(B\) into \(\mathbb{C}^{n}\). Let \(S_{g}^{0}(B)\) be the class of mappings \(f\in H(B)\) which have \(g\)-parametric representation on \(B\). Denote by \(S_{g,k+1}^{0}(B)\) the subset of \(S_{g}^{0}(B)\) consisting of mappings \(f\) for which there exists a \(g\)-Loewner chain \(f(z,t)\) such that \(\{e^{-t}f(z,t)\}_{t\geq0}\) is a normal family on \(B\), \(f=f(.,0)\) and \(z=0\) is a zero of order \(k+1\) of \(e^{-t}f(z,t)-z\) for each \(t\geq0\). In the paper, the authors give coefficient bounds of mappings in \(S_{g,k+1}^{0}(B)\). These results generalize the related works of Hamada, Honda and Kohr.
scientific article; zbMATH DE number 5555240