Research on Geometric Mappings in Complex Systems Analysis

Abstract

We mainly discuss the properties of a new subclass of starlike functions, namely, almost starlike functions of complex order , in one and several complex variables. We get the growth and distortion results for almost starlike functions of complex order . By the properties of functions with positive real parts and considering the zero of order , we obtain the coefficient estimates for almost starlike functions of complex order on . We also discuss the invariance of almost starlike mappings of complex order on Reinhardt domains and on the unit ball in complex Banach spaces. The conclusions contain and generalize some known results.

1. Introduction

The growth, distortion theorems, and coefficient estimates for univalent functions are important research contents in geometric function theory of one and several complex variables. In one complex variable we have the following known growth and distortion theorems.

Theorem 1 (see [1]). Let be a normalized biholomorphic function on the unit disk in . Then The distortion theorem for univalent functions on was introduced by Koebe. In the process of generalizing the distortion theorem for univalent functions on to the unit ball in , Cartan et al. [2, 3] found the distortion theorem did not hold for general biholomorphic mappings. Cartan et al. suggested restricting the mappings by geometric properties, such as starlikeness and convexity. So many people began to discuss the growth and distortion theorems for biholomorphic mappings with special geometric properties. Starlike mappings and convex mappings are discussed most [4–7]. While they discussed starlike mappings and convex mappings, they introduced some subclasses of those mappings. Furthermore, they always discussed these new subclasses in one complex variable firstly. The growth and distortion results [8] for starlike functions are the same as Theorem 1. We have the following results with respect to convex functions.

Theorem 2 (see [9]). Let be a convex function on with and . Then In 1936, Robertson [10] obtained the growth, covering theorems and coefficient estimates for starlike functions of order .

Theorem 3 (see [10]). Let be a starlike function of order on . Then In 1966, Boyd [11] obtained the coefficient estimates for the starlike function of order , where and is the zero of order of . In 1975, Silverman [12] discussed the distortion theorems and coefficient estimates for univalent analytic functions with negative coefficients. In 1991, Srivastava and Owa [13] obtained the distortion theorems and coefficient estimates for a subclass of starlike functions.

Recently, there are many nice results about the growth, distortion theorems, and coefficient estimates for subclasses of starlike functions and convex functions. Obviously we hope there exist similar results in several complex variables. In the process of discussing the properties of subclasses of starlike mappings and convex mappings, we need the specific examples of the mappings. It is easy to find specific examples of these new subclasses in , while it is very difficult in .

In 1995, Roper and Suffridge [14] introduced an operator where , , , , and . They proved the operator preserves convexity and starlikeness on . Graham and Kohr [15] proved the Roper-Suffridge operator preserves the properties of Bloch mappings on . Thus, we can construct lots of convex mappings and starlike mappings on by corresponding functions on by the Roper-Suffridge operator. So the Roper-Suffridge operator plays an important role in several complex variables. Later, many people generalized the Roper-Suffridge operator on different domains and different spaces so as to construct biholomorphic mappings with specific geometric properties in several complex variables. Therefore we can discuss the properties of biholomorphic mappings in several variables preferably. In recent years, there are many results about the generalized Roper-Suffridge operators (such as [16–18]).

In this paper, we mainly discuss almost starlike functions of complex order in one and several complex variables. In Sections 2 and 3, we discuss the growth, distortion theorems, and coefficient estimates for almost starlike functions of complex order on , respectively. In Section 4, we discuss the invariance of almost starlike mappings of complex order on Reinhardt domains and on the unit ball in complex Banach spaces. Thereby, we can construct lots of almost starlike mappings of complex order in several complex variables through almost starlike functions of complex order on . The conclusions generalize some known results.

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

Definition 4 (see [19]). Let be a bounded starlike and circular domain whose Minkowski functional is except for a lower-dimensional manifold. Let be a normalized locally biholomorphic mapping on . Let where and . Then is called an almost starlike mapping of complex order on .
For , the condition in Definition 4 reduces to

Definition 5 (see [19]). Let be a normalized locally biholomorphic mapping on the unit ball in complex Banach spaces. Letwhere and . Then is called an almost starlike mapping of complex order on .

Setting , in Definitions 4 and 5, we obtain the definition of almost starlike mappings of order .

2. Growth and Distortion Results

Lemma 6 (see [20]). indicates a circle in whose center is and whose radius is , where

Theorem 7. Let be an almost starlike function of complex order on with , . Then

Proof. Since is an almost starlike function of complex order on , then which follows Let Therefore . Let . We have and . Let Then and . By Schwarz lemma we obtain which follows For then, by (14), we obtain which follows By Lemma 6 we get that is in a circle whose center is and whose radius is , where So we obtain By direct computation we have Similarly, we can get where Let Immediately, we have where which follows . Therefore is monotone increasing. So where , . By (19) we have . Therefore Integrating both ends of (26) on , we obtain ThereforeSincesetting in (28), we have Therefore Obviously for . Thus . So we obtain Similarly, by (19) we can get

Theorem 8. Let be an almost starlike function of complex order on with . Then

Proof. By (14) we have which follows . Therefore For we have Similar to Theorem 7, we get Let ; we obtain . Hence .

Setting in Theorems 7 and 8, we get the following corollary.

Corollary 9. Let be an almost starlike function of order . Then

Theorem 10. Let be an almost starlike function of complex order on with , . Then

Proof. By (17) we have Since then we have which follows On the other hand, we also have which leads to Hence Then, by Theorem 7 we obtain the desired conclusion (39).

Setting in Theorem 10, we get the following result.

Corollary 11. Let be an almost starlike function of order with . Then

3. Coefficient Estimates of Almost Starlike Functions of Complex Order

Lemma 12 (see [21]). Let be a holomorphic function on with . Then .

Lemma 13 (see [22]). Let be holomorphic on . Then if and only if

Theorem 14. Let be an almost starlike function of complex order with , on . Then

Proof. Since is an almost starlike function of complex order , then Let . Then we have and and . Let . Then by Lemma 12. For , (51) follows Therefore So By (52) we get By mathematical induction we obtain the desired conclusion.

Theorem 15. Let be an almost starlike function of complex order with , on , and let be the zero of order of . Let . Then

Proof. Since , similar to Theorem 14, by (51) we obtain which follows Therefore By (52) we get the desired conclusion.

Theorem 16. is an almost starlike function of complex order with and on if and only if

Proof. Let . Since is an almost starlike function of complex order , then is holomorphic on and and . By Lemma 13 we get which follows For , we have which leads to the desired conclusion.

Remark 17. Setting in Theorems 14–16, we get the corresponding results for almost starlike functions of order .

4. The Invariance of Almost Starlike Mappings of Complex Order

In this section, we mainly discuss the invariance of almost starlike mappings of complex order on Reinhardt domains and on the unit ball in complex Banach spaces under some generalized Roper-Suffridge operators.

Lemma 18 (see [23]). Let be a bounded and completely Reinhardt domain whose Minkowski functional is except for a lower-dimensional manifold . Then for , we have

Theorem 19. Let be an almost starlike function of complex order on with and . Let be a bounded and completely Reinhardt domain whose Minkowski functional is on . Let where , , and . Then is an almost starlike mapping of complex order on .

Proof. For immediately we haveTherefore Then By Lemma 18 we get Since is an almost starlike function of complex order , then Therefore, by Lemma 18 we obtain Hence is an almost starlike mapping of complex order on by Definition 4.

Theorem 20. Let be almost starlike functions of complex order on with and . Let be a bounded starlike and circular domain whose Minkowski functional is on . Let be the radius of the disk whose center is zero. Let where , , and . Then is an almost starlike mapping of complex order on .

Proof. Let which follows . Then . By direct computation we get Since are almost starlike functions of complex order on , then Therefore Then . It is obvious that and , so is a normalized locally biholomorphic mapping on and Let (77) follows that is, . For , by (76), (79), and Lemma 18 we obtain which leads to Hence is an almost starlike mapping of complex order on by Definition 4.

Similar to Theorem 20, we can get the following conclusion.

Theorem 21. Let be almost starlike functions of complex order on with and . Let where , , , and is the unit vector of the complex Banach space . Then is an almost starlike mapping of complex order on the unit ball in complex Banach space .

Theorem 22. Let be an almost starlike function of complex order on with and . Let where , , , , and are linearly independent and such that , , and for . Then is an almost starlike mapping of complex order on the unit ball in complex Banach space .

Proof. From [24] we get is a normalized biholomorphic mapping on andSince is an almost starlike function of complex order , then Therefore which lead to the desired conclusion by Definition 5.

Similar to Theorem 22, we can get the following conclusion.

Theorem 23. Let be an almost starlike function of complex order on with , . Let where , , , , and for . Then is an almost starlike mapping of complex order on .

Remark 24. Setting in Theorems 19–23, we get the corresponding results with respect to almost starlike mappings of order .

Competing Interests

The authors declare that they have no conflict of interests.

Acknowledgments

This work is supported by NSF of China (no. U1204618) and Science and Technology Research Projects of Henan Provincial Education Department (nos. 14B110015 and 14B110016).