Geometric Mappings under the Perturbed Extension Operators in Complex Systems Analysis

Wang, Chaojun; Cui, Yanyan; Liu, Hao

doi:https://doi.org/10.1155/2017/3512326

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Research Article | Open Access

Volume 2017 | Article ID 3512326 | https://doi.org/10.1155/2017/3512326

Geometric Mappings under the Perturbed Extension Operators in Complex Systems Analysis

Chaojun Wang,¹Yanyan Cui,^1,2and Hao Liu³

Academic Editor: Chris Goodrich

Received03 Mar 2017

Accepted08 May 2017

Published15 Jun 2017

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 type and order , as well as almost spiral-like mappings of type and order under different conditions on Bergman-Hartogs domains. Sequentially we obtain the conclusions on the unit ball 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.

1. 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 star-like 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 , while it is very difficult in . In order to study these subclasses better in several complex variables, we need the specific examples imminently.

In 1995, Roper and Suffridge [3] introduced an operator where Roper and Suffridge proved the Roper-Suffridge operator preserves convexity and star-likeness on . Graham et al. generalized the Roper-Suffridge operator and discussed the generalized operators preserving star-likeness and the block property in [4, 5]. In 2002, Graham et al. extended the Roper-Suffridge operator on the unit ball in and proved the extended operator preserves star-likeness and convexity if and only if some conditions are satisfied in [6]. In 2003, Gong and Liu [7] generalized the Roper-Suffridge operator and obtained the generalized operator preserving star-likeness on the Reinhardt domain which leads to star-likeness and convexity in the cases of and , respectively.

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 by corresponding functions on the unit disk of . 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: where is a normalized biholomorphic function on the unit disk . The branch of the power function is chosen such that . is a homogeneous polynomial of degree 2. The extended operator (2) was proved to preserve star-likeness on and convexity on 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 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 spiral-likeness property in [13] and was concluded that it can be embedded in a Loewner chain on the unit ball in 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 is an operator-valued mapping which satisfies some natural conditions. was proved to preserve star-likeness and spiral-likeness under some conditions in [16].

Now, we introduce a new extension operator on the Bergman-Hartogs domain where , is a normalized univalent holomorphic function on , is a holomorphic function in with , and the power functions take the branches such that and . The homogeneous expansion of is , where is a homogeneous polynomial of degree . Equation (4) is the modification of the extension operators discussed in [17].

In the case of , (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 . Muir proved is a 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 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 , , 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 . 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.

2. Definitions and Lemmas

In the following, let denote the unit disk in and denote the unit ball in . Let denote the Fréchet derivative of at .

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

Definition 1 (see [18]). Let be a bounded star-like circular domain in . The Minkowski functional of is except for a lower-dimensional manifold. If is a normalized locally biholomorphic mapping on , let , and Then is called a strong and almost spiral-like mapping of type and order on .

Setting , , and , 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 be a bounded star-like circular domain in . The Minkowski functional of is except for a lower-dimensional manifold. Let be a normalized locally biholomorphic mapping on . If where and , then we call .

Setting , , and 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 .

Definition 3 (see [20]). Let be a bounded star-like circular domain in . The Minkowski functional of is except for a lower-dimensional manifold. Let be a normalized locally biholomorphic mapping on . If where and , then we call an almost spiral-like mapping of type and order .

Setting , in Definition 3, we obtain the definition of spiral-like mappings of type and almost star-like mappings of order on , respectively.

Lemma 4 (see [21]). Let be a bounded star-like circular domain in . The Minkowski functional of is except for a lower-dimensional manifold . Then we have

Lemma 5 (see [12]). Let be a homogeneous polynomial of degree and let be the Fréchet derivative of at . Then

Lemma 6 (see [17]). Let be the Minkowski functional of . Let . Then and where

Lemma 7 (see [1]). If is a normalized biholomorphic function on the unit disk , then

Lemma 8 (see [22]). Let be a strong spiral-like mapping of type on bounded and balanced domain with and . Let the Minkowski functional of be . Then Let be a strong spiral-like function of type on , then

Lemma 9 (see [23]). Let be a bounded star-like circular domain and the Minkowski functional of be except for some submanifolds of lower dimensions. Let be -fold symmetric. Thenor, equivalently,The above estimates are all accurate.

Lemma 10 (see [24]). Let be an almost spiral-like mapping of type and order on the unit ball in complex Banach spaces with and . Then For we get

3. The Invariance of Strong and Almost Spiral-Like Mappings of Type and Order

For simplicity, let denote . 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 , and thus we obtain the conclusion on ; also we get the invariance of some subclasses.

Theorem 11. Let be a strong and almost spiral-like function of type and order on with and . Let be the mapping denoted by (4) with and . Then is a strong and almost spiral-like mapping of type and order on provided that for and where .

Proof. By Definition 1, we need to proveIt is obvious that (23) holds for . Otherwise, setting where and , by Lemma 4, we obtain Fix , then is holomorphic with respect to . Due to the maximum modulus principle of holomorphic functions, the left side of (23) gets the maximum value at . So we need only to prove that (23) holds for , which follows .
Let Then and Letting , we have and . In view of , we obtain provided that . By Schwarz Lemma we get which implies that In addition, from (4) we have whereLet , which follows . ThusLemma 5 and a straightforward calculation show thatAs a consequence of Lemma 6 we get where Let Taking into account (25) and (31) we obtain the following equation string:where . We use (26) and to see that Note that implies that lead toThereforeprovided that .
Lemma 7 and (38) eventually lead us to the following inequality string: where for and This completes the proof.

Setting in Theorem 11, the operator (4) reduces to the generalized Roper-Suffridge operator (6) and the Bergman-Hartogs domain reduces to the unit ball ; thus, we draw the following conclusion on in .

Corollary 12. Let be a strong and almost spiral-like function of type and order on with and . Let be the mapping denoted by (6) with and . Then is a strong and almost spiral-like mapping of type and order on provided that for and

Remark 13. Setting and in Theorem 11 and Corollary 12, respectively, we get the corresponding results for strong and almost star-like mappings of order and strong spiral-like mapping of type .

If we have the precise growth result of strong and almost spiral-like mapping of type and order and apply the growth result in Theorem 11, we can get more precise conclusion. But, up to now we only have the growth theorem of strong spiral-like function of type [22]. In the following, we apply it in the process of discussing the invariance of strong spiral-like function of type .

Theorem 14. Let be a strong spiral-like function of type on with and . Let be the mapping denoted by (4) with and . Then is a strong spiral-like mapping of type and order on provided that for and where .

Proof. Setting in (39) and applying Lemma 8 we obtain the following inequality string: where for and Hence the assertion follows.

4. The Invariance of

As above, write . In this section, we mainly study the perturbed Extension Operator (4) preserving the geometric characteristics of on and then generalize the conclusion on and get the invariance of some subclasses.

Theorem 15. Let with and . Let be the mapping denoted by (4) with and . Then provided that for and where .

Proof. By Definition 2, we need to prove Similar to Theorem 11, we need only to prove that (46) holds for which implies .
Since , by Definition 2, we have Let Then and . Letting , we get and . In view of , we obtain where . By Schwarz Lemma, we get which implies that Let Then, (31) and (48) lead to the following equation string: By (49), (38), and Lemma 7 there comes the following inequality string: where for and Therefore (46) holds, which follows .

Setting in Theorem 15, we draw the following invariance of the mappings under the generalized Roper-Suffridge operator (6) on the unit ball in .

Corollary 16. Let with and . Let be the mapping denoted by (6) with and . Then provided that for and

In the process of discussing the invariance of , if we apply the growth theorem of in [23], we can get the following more precise conclusion.

Theorem 17. Let with . Let be the mapping denoted by (4) with and . Let where . Then we have the following conclusions:(1)For , provided that (2)For and , provided that (3)For and , provided that

Proof. Setting in the proof of Theorem 15, by using (52) and Lemma 9 we obtain the following conclusions:(1)For the case that we get where (2)For the case that and we have where (3)For the case that and , similar to (2), we get where From (1) to (3) we get the desired conclusion.

Remark 18. Setting 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 .

5. The Invariance of Almost Spiral-Like Mappings of Type and Order

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 , and thus we get the conclusion on as well as the results about some subclasses.

Theorem 19. Let be an almost spiral-like function of type and order on with and . Let be the mapping denoted by (4) with and . Let and where . Then is an almost spiral-like mapping of type and order on provided that

Proof. By Definition 3, we need to prove which is equivalent to 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 which implies that .
Let Since is an almost spiral-like function of type and order on , then . Therefore . Let Then . Moreover implies ; thus . Applying Schwarz Lemma we obtain ; in other words, which leads to Let We then get the following equation string by (31) and (67): Therefore, by (70), (38), and Lemma 10 there comes the following inequality string: where and This is equal to (66) and hence completes the proof.

Setting in Theorem 19, we draw the following invariance of almost spiral-like mappings of type and order under the generalized Roper-Suffridge operator (6) on the unit ball in .

Corollary 20. Let be an almost spiral-like function of type and order on with and . Let be the mapping denoted by (6) with and . Let for . Then is an almost spiral-like mapping of type and order on provided that

Remark 21. Setting and in Theorem 19 and Corollary 20, respectively, we get the corresponding results for spiral-like mappings of type and almost star-like mappings of order .

Conflicts of Interest

The authors declare that they have no conflicts of interest.

Acknowledgments

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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Copyright © 2017 Chaojun Wang et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

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