Abstract
In this paper, we mainly seek conditions on which the geometric properties of subclasses of biholomorphic mappings remain unchanged under the perturbed Roper-Suffridge extension operators. Firstly we generalize the Roper-Suffridge operator on Bergman-Hartogs domains. Secondly, applying the analytical characteristics and growth results of subclasses of biholomorphic mappings, we conclude that the generalized Roper-Suffridge operators preserve the geometric properties of strong and almost spiral-like mappings of typeβand orderα,SΩ⁎(β,A,B)as well as almost spiral-like mappings of typeβand orderαunder different conditions on Bergman-Hartogs domains. Sequentially we obtain the conclusions on the unit ballBnand for some special cases. The conclusions include and promote some known results and provide new approaches to construct biholomorphic mappings which have special geometric characteristics in several complex variables.
Highlights
The theory of several complex variables derives from the theory of one complex variable
It is natural to think that we can extend these results in several complex variables, while some basic theorems are found not to hold in several complex variables
In 1933, Cartan [2] suggested that we can consider the geometric constraint of biholomorphic mappings, such as star-likeness and convexity
Summary
The theory of several complex variables derives from the theory of one complex variable. We mainly discuss the invariance of several biholomorphic mappings under the generalized RoperSuffridge extension operators (4) on the Bergman-Hartogs domains ΩBp1n,...,ps,q which is based on the unit ball Bn. In Section 2, we give some definitions and lemmas that are used to derive the main results.
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