Growth, distortion and coefficient bounds for Carathéodory families in \(\mathbb C^n\) and complex Banach spaces. (English) Zbl 1295.32009
Summary: Let \(X\) be a complex Banach space with the unit ball \(B\). The family \(\mathcal M\) is a natural generalization to complex Banach spaces of the well-known Carathéodory family of functions with positive real part on the unit disc. We consider subfamilies \(\mathcal M_g\) of \(\mathcal M\) depending on a univalent function \(g\). We obtain growth theorems and coefficient bounds for holomorphic mappings in \(\mathcal M_g\), including some sharp improvements of existing results. When \(g\) is convex, we study the family \(\mathcal R_g\) consisting of holomorphic mappings \(f:B\to X\) which have the property that the mapping \(Df(z)(z)\) belongs to \(\mathcal M_g\). Further, we consider radius problems related to the family \(\mathcal R_g\), when \(X\) is a complex Hilbert space. In particular, if \(X\) is the Euclidean space \(\mathbb C^n\), we obtain some quasiconformal extension results for mappings in \(\mathcal R_g\). We also obtain some sufficient conditions for univalence and starlikeness in complex Banach spaces.
MSC:
| 32A30 | Other generalizations of function theory of one complex variable |
| 30C45 | Special classes of univalent and multivalent functions of one complex variable (starlike, convex, bounded rotation, etc.) |
| 46G20 | Infinite-dimensional holomorphy |
Keywords:
unit ball of a Banach space; Carathéodory family; convex function; Loewner chain; subordinationReferences:
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