Geometric Mappings under the Perturbed Extension Operators in Complex Systems Analysis

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 β and order α, S∗ Ω(β, A, B) as well as almost spiral-like mappings of type β and order α under different conditions on Bergman-Hartogs domains. Sequentially we obtain the conclusions on the unit ballBn and 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.


Introduction
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 [1]) 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.So many scholars devoted themselves to the research of star-like mappings and convex mappings.Recently many subclasses or expansions of starlike and convex mappings are introduced.The properties of biholomorphic mappings with special geometric properties are important research objects in geometric function theories of several complex variables.It is easy to find specific examples of these new subclasses or expansions in C, while it is very difficult in C  .In order to study these subclasses better in several complex variables, we need the specific examples imminently.
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 C  by corresponding functions on the unit disk  of C. That will promote the development of the research of biholomorphic mappings.So the Roper-Suffridge operator plays an important role in several complex variables.In recent years, there are lots of results about the Roper-Suffridge extension operator which was generalized and modified on different domains in different spaces to preserve the geometric characteristics of convex mappings, star-like mappings, and their subclasses.Graham and Kohr gave a survey about the Roper-Suffridge extension operator and the developments in the theory of biholomorphic mappings in several complex variables to which it had led in [8].
Muir and Suffridge [9] introduced the following generalized Roper-Suffridge extension operator:  () = ( ( 1 ) +   ( 1 )  ( 0 ) , √   ( 1 ) 0 )  , (2) where  is a normalized biholomorphic function on the unit disk ,  = ( 1 ,  0 )  ∈   ,  1 ∈ ,  0 = ( 2 , . . .,   )  ∈ C −1 .The branch of the power function is chosen such that √  (0) = 1. : C −1 → C is a homogeneous polynomial of degree 2. The extended operator (2) was proved to preserve star-likeness on ‖‖ ≤ 1/4 and convexity on ‖‖ ≤ 1/2 in [9] and was proved to take the extreme points of normalized convex functions to extreme points of normalized convex mappings of the Euclidean ball in C  under precise conditions by Muir in [10].Also the extended operator (2) was studied by Kohr and Muir in [11,12] with Loewner chains.Later the operator (2) was generalized by Elin and Levenshtein and the generalized operator was proved to preserve the spirallikeness property in [13] and was concluded that it can be embedded in a Loewner chain on the unit ball in C  in [14].Moreover, Elin and Levenshtein presented an extension operator for semigroup generators and concluded that the new one-dimensional covering results established in [13] are crucial.Furthermore, (2) was modified and discussed by Cui et al. in [15].Elin introduced a general construction of the extension operators where (, ) is on the unit ball of the product  *  of two Banach spaces and Γ(ℎ, ) is an operator-valued mapping which satisfies some natural conditions. Γ [ℎ](, ) was proved to preserve star-likeness and spiral-likeness under some conditions in [16].
In the case of  (1) = ⋅ ⋅ ⋅ =  () = 0, (4) leads to the following operator: which can also be seen as the modification of the following generalized Roper-Suffridge extension operator introduced by Muir on the unit ball in complex Banach spaces, where Loewner chain preserving extension operator provided that  satisfies some conditions in [12].
In this paper, we mainly discuss the invariance of several biholomorphic mappings under the generalized Roper-Suffridge extension operators (4) on the Bergman-Hartogs domains Ω    1 ,...,  , which is based on the unit ball   .In Section 2, we give some definitions and lemmas that are used to derive the main results.In Sections 3-5, we detailedly discuss the perturbed Extension Operator (4) preserving the geometric properties of strong and almost spiral-like mappings of type  and order ,  * Ω (, , ), as well as almost spiral-like mappings of type  and order  under different conditions on Bergman-Hartogs domains and thus generalize the conclusions on the unit ball   in C  .At last, we derive that the generalized Roper-Suffridge operators preserve the properties of subclasses of the three kinds of biholomorphic mappings mentioned above.The conclusions include and promote some known results.

Definitions and Lemmas
In the following, let  denote the unit disk in C and   denote the unit ball in C  .Let () denote the Fréchet derivative of  at .
To get the main results, we need the following definitions and lemmas.
Setting  = 0,  = 0, and  =  = 0, Definition 1 reduces to the definition of strong spiral-like mappings of type , strong and almost star-like mappings of order , and strong star-like mappings, respectively.Definition 2 (see [19]).Let Ω be a bounded star-like circular domain in C  .The Minkowski functional () of Ω is  1 except for a lower-dimensional manifold.Let () be a normalized locally biholomorphic mapping on Ω.
Setting  = −1 = − − 2,  = − = −, and  → 1 − in Definition 2, respectively, we obtain the corresponding definitions of spiral-like mappings of type  and order , strong spiral-like mappings of type  and order , and almost spiral-like mappings of type  and order  on Ω.

The Invariance of Strong and Almost Spiral-Like Mappings of Type 𝛽 and Order 𝛼
For simplicity, let Ω denote Ω    1 ,...,  , .In this section we will show that the perturbed Roper-Suffridge extension operator (4) preserves the geometric characteristics of strong and almost spiral-like mappings of type  and order  on Ω, and thus we obtain the conclusion on   ; also we get the invariance of some subclasses.
Proof.Setting  = 0 in (39) and applying Lemma 8 we obtain the following inequality string: where Hence the assertion follows.

The Invariance of Almost Spiral-Like Mappings of Type 𝛽 and Order 𝛼
In the following, we mainly discuss the perturbed Extension Operator (4) preserving the geometric characteristics of almost spiral-like mappings of type  and order  on Ω, and thus we get the conclusion on   as well as the results about some subclasses.
Similar to Theorem 11, the left side of (66) is the real part of a holomorphic mapping and thus is a harmonic function.Due to the minimum principle of harmonic functions, we need only to prove that (66) holds for  ∈ Ω which implies that (, ) = 1.

Remark 18 .
Setting  = −1 = − − 2 and  = − = − in Theorems 15 and 17 and Corollary 16, respectively, we get the corresponding results for spiral-like mappings of type  and order  and strong spiral-like mappings of type  and order .