Abstract and Applied Analysis

Volume 2014 (2014), Article ID 265718, 4 pages

http://dx.doi.org/10.1155/2014/265718

Research Article

A Note on Strongly Starlike Mappings in Several Complex Variables

Hidetaka Hamada,¹ Tatsuhiro Honda,² Gabriela Kohr,³ and Kwang Ho Shon⁴

¹Faculty of Engineering, Kyushu Sangyo University, Fukuoka 813-8503, Japan

²Hiroshima Institute of Technology, Hiroshima 731-5193, Japan

³Faculty of Mathematics and Computer Science, Babeş-Bolyai University, 1 M. Kogălniceanu Street, 400084 Cluj-Napoca, Romania

⁴Department of Mathematics, College of Natural Sciences, Pusan National University, Busan 609-735, Republic of Korea

Received 3 December 2013; Accepted 27 January 2014; Published 3 March 2014

Academic Editor: Junesang Choi

Copyright © 2014 Hidetaka Hamada 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.

Let be a normalized biholomorphic mapping on the Euclidean unit ball in and let . In this paper, we will show that if is strongly starlike of order in the sense of Liczberski and Starkov, then it is also strongly starlike of order in the sense of Kohr and Liczberski. We also give an example which shows that the converse of the above result does not hold in dimension .

1. Introduction and Preliminaries

Let denote the space of complex variables with the Euclidean inner product and the norm . The open unit ball is denoted by . In the case of one complex variable, is denoted by .

If is a domain in , let be the set of holomorphic mappings from to . If is a domain in which contains the origin and , we say that is normalized if and , where is the identity matrix.

A normalized mapping is said to be starlike if is biholomorphic on and for , where the last condition says that the image is a starlike domain with respect to the origin. For a normalized locally biholomorphic mapping on , is starlike if and only if

(see [1–4] and the references therein, cf. [5]).

Let . A function , normalized by and , is said to be strongly starlike of order if

If is strongly starlike of order , then is also starlike and thus univalent on . Stankiewicz [6] proved that if , then a domain which contains the origin is -accessible if and only if , where is the unit disc in and is a strongly starlike function of order on . For strongly starlike functions on , see also Brannan and Kirwan [7], Ma and Minda [8], and Sugawa [9].

Kohr and Liczberski [10] introduced the following definition of strongly starlike mappings of order on .

Definition 1. Let . A normalized locally biholomorphic mapping is said to be strongly starlike of order if

Obviously, if is strongly starlike of order , then is also starlike, and if in (3), one obtains the usual notion of starlikeness on the unit ball .

Using this definition, Hamada and Honda [11], Hamada and Kohr [12], Liczberski [13], and Liu and Li [14] obtained various results for strongly starlike mappings of order in several complex variables.

Recently, Liczberski and Starkov [15] gave another definition of strongly starlike mappings of order on the Euclidean unit ball in , where , and proved that a normalized biholomorphic mapping on is strongly starlike of order if and only if is an -accessible domain in for . Their definition is as follows.

Definition 2. Let . A normalized locally biholomorphic mapping is said to be strongly starlike of order in the sense of Liczberski and Starkov if

In the case , it is obvious that both notions of strong starlikeness of order are equivalent. Thus, the following natural question arises in dimension .

Question 1. Let . Is there any relation between the above two definitions of strong starlikeness of order ?

Let be a normalized biholomorphic mapping on the Euclidean unit ball in and let . In this paper, we will show that if is strongly starlike of order in the sense of Definition 2, then it is also strongly starlike of order in the sense of Definition 1. As a corollary, the results obtained in [11–14] for strongly starlike mappings of order in the sense of Definition 1 also hold for strongly starlike mappings of order in the sense of Definition 2. We also give an example which shows that the converse of the above result does not hold in dimension .

2. Main Results

Let denote the angle between regarding , as real vectors in .

Lemma 3. Let be such that . If and , then

Proof. Let , . Then we have for some and

Since and , we have

Therefore, we have , as desired.

Theorem 4. Let be a normalized biholomorphic mapping on the Euclidean unit ball in and let . If is strongly starlike of order in the sense of Definition 2, then it is also strongly starlike of order in the sense of Definition 1.

Proof. Assume that is strongly starlike of order in the sense of Definition 2. Then by (4), we have and

Using Lemma 3, we have

For fixed , let and

Then is a holomorphic function on with for . Since is a harmonic function on and , by applying the maximum and minimum principles for harmonic functions, we obtain for . Thus, we have

Hence is strongly starlike of order in the sense of Definition 1, as desired.

The following example shows that the converse of the above theorem does not hold in dimension .

Example 5. For , let

where

Then

Therefore,

Since , for , we obtain that for . This implies that lies in the disc of center and radius for each and thus

Therefore, is strongly starlike of order in the sense of Definition 1.

On the other hand,

So, for , we have

where

Then, we obtain

Since

is increasing on and positive for , we have

for .

On the other hand, for , we have

Then, we obtain

Since is positive for , we have

for .

Thus, is not strongly starlike of order in the sense of Definition 2 for .

Conflict of Interests

The authors declare that there is no conflict of interests regarding the publication of this paper.

Acknowledgments

Hidetaka Hamada is supported by JSPS KAKENHI Grant no. 25400151. Tatsuhiro Honda is partially supported by Brain Korea Project, 2013. The work of Gabriela Kohr was supported by a grant of the Romanian National Authority for Scientific Research, CNCS-UEFISCDI, Project no. PN-II-ID-PCE-2011-3-0899. Kwang Ho Shon was supported by a 2-year research grant of Pusan National University.

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