Bounds of all terms of homogeneous expansions for a subclass of \(g\)-parametric biholomorphic mappings in \(\mathbb{C}^n\). (English) Zbl 1517.32050
Summary: Let \(\mathbb{B_X}\) be the unit ball in a complex Banach space \(\mathbb{X}\) and \(\mathbb{D}^n\) be the unit polydisc in the space of \(n\)-dimensional complex variables. We obtain the bounds of all terms of homogeneous expansions for a subclass of \(g\)-parametric biholomorphic mappings on \(\mathbb{B_X}\) (resp. \(\mathbb{D}^n\)), where \(g\) is a convex function. Our results generalize some known works and can be regarded as an extended example to hold the weak version of the Bieberbach conjecture in several complex variables.
MSC:
| 32K12 | Holomorphic maps with infinite-dimensional arguments or values |
| 46G20 | Infinite-dimensional holomorphy |
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