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Properties of the modified Roper-Suffridge extension operators on Reinhardt domains. (English) Zbl 1374.30065

Summary: In this paper, we mainly discuss the properties of the modified Roper-Suffridge operators on Reinhardt domains. By the analytical characteristics and distortion results of subclasses of biholomorphic mappings, we conclude that the modified Roper-Suffridge operators preserve the properties of \(S_\Omega^*(\beta, A, B)\), almost starlike mapping of complex order \(\lambda\) on \(\Omega_{n, p_2, \dots, p_n}\). Sequentially, we get that the modified Roper-Suffridge operators preserve spirallikeness of type \(\beta\) and order \(\alpha\), strongly spirallikeness of type \(\beta\) and order \(\alpha\), almost starlikeness of order \(\alpha\) on \(\Omega_{n, p_2,\dots, p_n}\). The conclusions provide a new approach to construct these biholomorphic mappings which have special geometric properties in several complex variables.

MSC:

30C45 Special classes of univalent and multivalent functions of one complex variable (starlike, convex, bounded rotation, etc.)
32A07 Special domains in \({\mathbb C}^n\) (Reinhardt, Hartogs, circular, tube) (MSC2010)
32A30 Other generalizations of function theory of one complex variable
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