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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 type

The theory of several complex variables derives from the theory of one complex variable. There are many excellent results in geometric function theories of one complex variable. It is natural to think that we can extend these results in several complex variables, while some basic theorems (such as the models of the coefficients of the homogeneous expansion for biholomorphic functions being bounded on the unit disk [

In 1995, Roper and Suffridge [

All of the above illustrate that the Roper-Suffridge operator has good properties. Through the Roper-Suffridge operator or its generalizations we can construct lots of convex mappings and star-like mappings in

Muir and Suffridge [

Now, we introduce a new extension operator

In the case of

In this paper, we mainly discuss the invariance of several biholomorphic mappings under the generalized Roper-Suffridge extension operators (

In the following, let

To get the main results, we need the following definitions and lemmas.

Let

Setting

Let

Setting

Let

Setting

Let

Let

Let

If

Let

Let

Let

For simplicity, let

Let

By Definition

Let

In addition, from (

Let

Note that

Lemma

Setting

Let

Setting

If we have the precise growth result of strong and almost spiral-like mapping of type

Let

Setting

As above, write

Let

By Definition

Since

Let

Setting

Let

In the process of discussing the invariance of

Let

For

For

For

Setting

For the case that

For the case that

For the case that

Setting

In the following, we mainly discuss the perturbed Extension Operator (

Let

By Definition

Let

Let

Setting

Let

Setting

The authors declare that they have no conflicts of interest.

This work was supported by NSF of China (no. 11271359) and Science and Technology Research Projects of Henan Provincial Education Department (no. 17A110041).

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