New subclasses of biholomorphic mappings and the modified Roper-Suffridge operator. (English) Zbl 1352.32003
Summary: The authors propose a new approach to construct subclasses of biholomorphic mappings with special geometric properties in several complex variables. The Roper-Suffridge operator on the unit ball \(B^n\) in \(\mathbb C^n\) is modified. By the analytical characteristics and the growth theorems of subclasses of spirallike mappings, it is proved that the modified Roper-Suffridge operator \([\Phi_{G,\gamma} (f)](z)\) preserves the properties of \(S_{\Omega}^{*}(A,B)\), as well as strong and almost spirallikeness of type \(\beta\) and order \(\alpha\) on \(B^n\). Thus, the mappings in \(S_{\Omega}^{*}(A,B)\), as well as strong and almost spirallike mappings, can be constructed through the corresponding functions in one complex variable. The conclusions follow some special cases and contain the elementary results.
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.) |
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