About this Journal Submit a Manuscript Table of Contents
Abstract and Applied Analysis
Volume 2014 (2014), Article ID 719652, 15 pages
http://dx.doi.org/10.1155/2014/719652
Research Article

Construction of Biholomorphic Convex Mappings of Order on

1School of Mathematical Sciences, South China Normal University, Guangzhou 510631, China
2Department of Mathematics, Faculty of Science and Technology, University of Macau, Taipa, Macao, China
3School of Information Technology in Education, South China Normal University, Guangzhou 510631, China

Received 1 October 2013; Accepted 20 December 2013; Published 8 April 2014

Academic Editor: Tatsuhiro Honda

Copyright © 2014 Ming-Sheng Liu 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.

Abstract

Some sufficient conditions for biholomorphic convex mappings of order on the Reinhardt domain in are given; from that, criteria for biholomorphic convex mappings of order with particular form become direct. As applications of these sufficient conditions, some concrete biholomorphic convex mappings of order   on are provided.

1. Introduction and Preliminaries

The analytic functions of one complex variable, which map the unit disk onto starlike domains or convex domains, have been extensively studied. These functions are easily characterized by simple analytic or geometric conditions. In the case of one complex variable, the following notions are well known.

Let  =  : be analytic in with . A function is said to be convex if is convex, that is, given , for all . We let denote the class of univalent convex functions in . Suppose . If satisfies for all and the following inequality: then we call a convex function of order in . We let denote the class of convex functions of order in . It is evident that .

In higher dimensions, demanding that a mapping takes the unit ball to a convex domain turned out to be a very restrictive condition. It is rather hard to construct concrete biholomorphic convex mappings on some domains in , even on the Euclidean unit ball.

Suppose is a fixed positive integer, . Let be the space of complex variables with the usual inner product , where . We introduce the -norm of   , and let ; it is evident that is a Reinhardt domain. For simplicity, let .

Let be the class of holomorphic mappings in the Reinhardt domain , where . A mapping is said to be locally biholomorphic in if has a local inverse at each point or, equivalently, if the first Fréchet derivative is nonsingular at each point in .

The second Fréchet derivative of a mapping is a symmetric bilinear operator on , and is the linear operator obtained by restricting to . The matrix representation of is where .

Let denote the class of all locally biholomorphic mappings such that , where is the unit matrix of . If is a biholomorphic mapping on and is a convex domain in , then we call a biholomorphic convex mapping on . The class of all biholomorphic convex mappings on is denoted by . Obviously, . The biholomorphic convex mapping of order on was introduced and investigated in [15]; the starlike and quasi-convex mappings were investigated in [4, 6].

Definition 1 (see [13, 5]). Suppose , and . Assume that for any and with , we have where . Then, is called a biholomorphic convex mapping of order on . The class of all biholomorphic convex mappings of order on is denoted by . It is evident that and .

In 1995, Roper and Suffridge [7] proved that if and , where , then . is popularly referred to as the Roper-Suffridge operator. Using this operator, we may construct a lot of concrete biholomorphic convex mappings on . Roper and suffridge [8] also obtained some sufficient conditions for biholomorphic convex mappings on the Euclidean unit ball. Liu and Zhu [9] had given some sufficient conditions and concrete examples of biholomorphic convex mappings on the Reinhardt domain . Liu [3] also gave some sufficient conditions for biholomorphic convex mappings of order on . A problem is naturally posed: can we give several direct criteria for biholomorphic convex mapping of order on ? For example, can we get some sufficient conditions such that the mapping of the form is a biholomorphic convex mapping of order on ?

The aim of this paper is to give an answer to the above problem. From these, we may construct some concrete biholomorphic convex mappings of order on .

2. Main Results

Theorem 2. Suppose that , , . Let where and is holomorphic with . If satisfies the following conditions: for all , then .

Proof. By direct computation of the Fréchet derivatives of , we obtain where Taking such that , by the hypothesis of Theorem 2, we have By Hölder’s inequality, we have Hence, we conclude from the above inequalities and the hypothesis of Theorem 2 that for all such that . Thus, it follows from Definition 1 that . The proof is complete.

Remark 3. Setting , in Theorem 2, we get Theorem 1 of [9].

Let us give two examples to illustrate the application of Theorem 2 in the following.

Example 4. Suppose that and is a positive integer such that . Let
If and then .

Proof. Let Then, So it follows from that
Since , we have By straightforward calculations, we obtain Set . Then, where .
When , we have .
When , we have and so Hence, when we have By Theorem 2, we obtain that . The proof is complete.

Example 5. Suppose that and is a positive integer such that . Let If and then .

By applying the same method of the proof for Theorem 2, we may prove the following result.

Theorem 6. Suppose that and are analytic on , , . Let , when .
If for any , we have then .

Remark 7. Setting in Theorem 6, we get Theorem 1 in [9].

Example 8. Suppose that and is a positive integer such that . Let where . If and then .
Now, we give another sufficient condition for , which gives an answer to the problem mentioned in the introduction.

Theorem 9. Suppose that and is a positive integer such that . Let where is holomorphic with and is holomorphic with for . If satisfies the following conditions: then .

Proof. By calculating the Fréchet derivatives of straightforwardly, we obtain where
Taking such that , by Definition 1 and the hypothesis of Theorem 9, we have for all such that . Thus, it follows from Definition 1 that . The proof is complete.

Corollary 10. Suppose that , and is a positive integer such that . Let where is holomorphic with and is holomorphic with . If satisfies the following conditions: