AAA Abstract and Applied Analysis 1687-0409 1085-3375 Hindawi Publishing Corporation 265718 10.1155/2014/265718 265718 Research Article A Note on Strongly Starlike Mappings in Several Complex Variables Hamada Hidetaka 1 Honda Tatsuhiro 2 Kohr Gabriela 3 Shon Kwang Ho 4 Choi Junesang 1 Faculty of Engineering Kyushu Sangyo University Fukuoka 813-8503 Japan kyusan-u.ac.jp 2 Hiroshima Institute of Technology Hiroshima 731-5193 Japan it-hiroshima.ac.jp 3 Faculty of Mathematics and Computer Science Babeş-Bolyai University 1 M. Kogălniceanu Street 400084 Cluj-Napoca Romania ubbcluj.ro 4 Department of Mathematics College of Natural Sciences Pusan National University Busan 609-735 Republic of Korea pusan.ac.kr 2014 332014 2014 03 12 2013 27 01 2014 3 3 2014 2014 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.

Let f be a normalized biholomorphic mapping on the Euclidean unit ball 𝔹 n in n and let α 0,1 . In this paper, we will show that if f is strongly starlike of order α in the sense of Liczberski and Starkov, then it is also strongly starlike of order α 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 n 2 .

1. Introduction and Preliminaries

Let n denote the space of n complex variables z = ( z 1 , , z n ) with the Euclidean inner product z , w = j = 1 n z j w ¯ j and the norm z = z , z 1 / 2 . The open unit ball { z n : z < 1 } is denoted by 𝔹 n . In the case of one complex variable, 𝔹 1 is denoted by U .

If Ω is a domain in n , let H ( Ω ) be the set of holomorphic mappings from Ω to n . If Ω is a domain in n which contains the origin and f H ( Ω ) , we say that f is normalized if f ( 0 ) = 0 and D f ( 0 ) = I n , where I n is the identity matrix.

A normalized mapping f H ( 𝔹 n ) is said to be starlike if f is biholomorphic on 𝔹 n and t f ( 𝔹 n ) f ( 𝔹 n ) for t [ 0,1 ] , where the last condition says that the image f ( 𝔹 n ) is a starlike domain with respect to the origin. For a normalized locally biholomorphic mapping f on 𝔹 n , f is starlike if and only if (1) [ D f ( z ) ] - 1 f ( z ) , z > 0 , z 𝔹 n { 0 } (see  and the references therein, cf. ).

Let α ( 0,1 ] . A function f H ( U ) , normalized by f ( 0 ) = 0 and f ( 0 ) = 1 , is said to be strongly starlike of order α if (2) | arg z f ( z ) f ( z ) | < α π 2 ,    z U .

If f is strongly starlike of order α , then f is also starlike and thus univalent on U . Stankiewicz  proved that if α ( 0,1 ) , then a domain Ω which contains the origin is α -accessible if and only if Ω = f ( U ) , where U is the unit disc in and f is a strongly starlike function of order 1 - α on U . For strongly starlike functions on U , see also Brannan and Kirwan , Ma and Minda , and Sugawa .

Kohr and Liczberski  introduced the following definition of strongly starlike mappings of order α on 𝔹 n .

Definition 1.

Let 0 < α 1 . A normalized locally biholomorphic mapping f H ( 𝔹 n ) is said to be strongly starlike of order α if (3) | arg [ D f ( z ) ] - 1 f ( z ) , z | < α π 2 , z 𝔹 n { 0 } .

Obviously, if f is strongly starlike of order α , then f is also starlike, and if α = 1 in (3), one obtains the usual notion of starlikeness on the unit ball 𝔹 n .

Using this definition, Hamada and Honda , Hamada and Kohr , Liczberski , and Liu and Li  obtained various results for strongly starlike mappings of order α in several complex variables.

Recently, Liczberski and Starkov  gave another definition of strongly starlike mappings of order α on the Euclidean unit ball 𝔹 n in n , where α ( 0,1 ] , and proved that a normalized biholomorphic mapping f on 𝔹 n is strongly starlike of order 1 - α if and only if f ( 𝔹 n ) is an α -accessible domain in n for α ( 0,1 ) . Their definition is as follows.

Definition 2.

Let 0 < α 1 . A normalized locally biholomorphic mapping f H ( 𝔹 n ) is said to be strongly starlike of order α ( in the sense of Liczberski and Starkov ) if (4) [ D f ( z ) ] - 1 f ( z ) , z ( [ D f ( z ) ] - 1 ) * z · f ( z ) sin ( ( 1 - α ) π 2 ) , z 𝔹 n { 0 } .

In the case n = 1 , it is obvious that both notions of strong starlikeness of order α are equivalent. Thus, the following natural question arises in dimension n 2 .

Question 1.

Let α ( 0,1 ) . Is there any relation between the above two definitions of strong starlikeness of order α ?

Let f be a normalized biholomorphic mapping on the Euclidean unit ball 𝔹 n in n and let α ( 0,1 ) . In this paper, we will show that if f is strongly starlike of order α in the sense of Definition 2, then it is also strongly starlike of order α in the sense of Definition 1. As a corollary, the results obtained in  for strongly starlike mappings of order α in the sense of Definition 1 also hold for strongly starlike mappings of order α 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 n 2 .

2. Main Results

Let ( a , b ) denote the angle between a , b n { 0 } regarding a , b as real vectors in 2 n .

Lemma 3.

Let a , b n { 0 } be such that a , b 0 . If | arg a , b | π and 0 ( a , b ) < π / 2 , then (5) | arg a , b | ( a , b ) .

Proof.

Let θ = arg a , b , φ = ( a , b ) . Then we have a , b = r e i θ for some r 0 and (6) a , b = a · b cos φ = r cos θ . Since cos φ > 0 and r = | a , b | a · b , we have (7) cos φ cos θ . Therefore, we have | θ | φ , as desired.

Theorem 4.

Let f be a normalized biholomorphic mapping on the Euclidean unit ball 𝔹 n in n and let α ( 0,1 ) . If f is strongly starlike of order α in the sense of Definition 2, then it is also strongly starlike of order α in the sense of Definition 1.

Proof.

Assume that f is strongly starlike of order α in the sense of Definition 2. Then by (4), we have [ D f ( z ) ] - 1 f ( z ) , z 0 and (8) ( ( [ D f ( z ) ] - 1 ) * z , f ( z ) )    α π 2 , z 𝔹 n { 0 } . Using Lemma 3, we have (9) | arg [ D f ( z ) ] - 1 f ( z ) , z | = | arg f ( z ) , ( [ D f ( z ) ] - 1 ) * z | ( ( [ D f ( z ) ] - 1 ) * z , f ( z ) ) α π 2 , z 𝔹 n { 0 } . For fixed z 𝔹 n { 0 } , let w = z / z and (10) p ( ζ ) = { 1 ζ [ D f ( ζ w ) ] - 1 f ( ζ w ) , w , for ζ U { 0 } , 1 , for ζ = 0 . Then p is a holomorphic function on U with | arg p ( ζ ) | π α / 2 for ζ U . Since arg p is a harmonic function on U and arg p ( 0 ) = 0 , by applying the maximum and minimum principles for harmonic functions, we obtain | arg p ( ζ ) | < π α / 2 for ζ U . Thus, we have (11) | arg [ D f ( z ) ] - 1 f ( z ) , z | < α π 2 , z 𝔹 n { 0 } . Hence f is strongly starlike of order α in the sense of Definition 1, as desired.

The following example shows that the converse of the above theorem does not hold in dimension n 2 .

Example 5.

For α ( 0,1 ) , let (12) f ( z ) = f α ( z ) = ( z 1 + b z 2 2 , z 2 ) , z = ( z 1 , z 2 ) 𝔹 2 , where (13) b = 3 3 2 sin ( α π 2 ) . Then (14) D f ( z ) = [ 1 2 b z 2 0 1 ] , [ D f ( z ) ] - 1 = [ 1 - 2 b z 2 0 1 ] . Therefore, (15) [ D f ( z ) ] - 1 f ( z ) , z    = ( z 1 + b z 2 2 - 2 b z 2 2 ) z 1 ¯ + | z 2 | 2 = | z 1 | 2 + | z 2 | 2 - b z 1 ¯ z 2 2 . Since | z 1 z 2 2 | 2 / ( 3 3 ) , for z 𝔹 2 , we obtain that | b z 1 z 2 2 | sin ( α π / 2 ) z 3 for z 𝔹 2 . This implies that [ D f ( z ) ] - 1 f ( z ) , z lies in the disc of center z 2 and radius sin ( α π / 2 ) z 2 for each z 𝔹 2 { 0 } and thus (16) | arg [ D f ( z ) ] - 1 f ( z ) , z | < α π 2 , z 𝔹 2 { 0 } . Therefore, f = f α is strongly starlike of order α in the sense of Definition 1.

On the other hand, (17) ( [ D f ( z ) ] - 1 ) * z = ( z 1 , z 2 - 2 b z ¯ 2 z 1 ) . So, for z 0 = ( 1 / 3 , 2 / 3 ) , we have (18) [ D f ( z 0 ) ] - 1 f ( z 0 ) , z 0    = 1 - m , ( [ D f ( z 0 ) ] - 1 ) * z 0 2 = 1 3 + 2 3 ( 1 - 3 m ) 2 , f ( z 0 ) 2 = 1 3 ( 1 + 3 m ) 2 + 2 3 , sin ( ( 1 - α ) π 2 ) = 1 - m 2 , where (19) m = sin ( α π 2 ) . Then, we obtain (20) ( [ D f ( z 0 ) ] - 1 ) * z 0 2 f ( z 0 ) 2 sin 2 ( ( 1 - α ) π 2 ) - ( [ D f ( z 0 ) ] - 1 f ( z 0 ) , z 0 ) 2 = ( 1 - m ) { [ 1 3 + 2 3 ( 1 - 3 m ) 2 ] [ 1 3 ( 1 + 3 m ) 2 + 2 3 ] × ( 1 + m ) - ( 1 - m ) 1 3 } . Since (21) [ 1 3 + 2 3 ( 1 - 3 m ) 2 ] [ 1 3 ( 1 + 3 m ) 2 + 2 3 ] ( 1 + m ) - ( 1 - m ) is increasing on [ 1 / 3,1 ] and positive for m = 1 / 3 , we have (22) [ D f ( z 0 ) ] - 1 f ( z 0 ) , z 0 < ( [ D f ( z 0 ) ] - 1 ) * z 0 × f ( z 0 ) sin ( ( 1 - α ) π 2 ) for m [ 1 / 3,1 ) .

On the other hand, for z ~ 0 = ( i / 3 , 2 / 3 ) , we have (23) [ D f ( z ~ 0 ) ] - 1 f ( z ~ 0 ) , z ~ 0 = 1 + m i , ( [ D f ( z ~ 0 ) ] - 1 ) * z ~ 0 2 = 1 3 + 2 3 | 1 - 3 m i | 2 = 6 m 2 + 1 , f ( z ~ 0 ) 2 = 1 3 | i + 3 m | 2 + 2 3 = 3 m 2 + 1 . Then, we obtain (24) ( [ D f ( z ~ 0 ) ] - 1 ) * z ~ 0 2 f ( z ~ 0 ) 2 sin 2 ( ( 1 - α ) π 2 ) - ( [ D f ( z ~ 0 ) ] - 1 f ( z ~ 0 ) , z ~ 0 ) 2 = ( 6 m 2 + 1 ) ( 3 m 2 + 1 ) ( 1 - m 2 ) - 1 = m 2 ( - 18 m 4 + 9 m 2 + 8 ) . Since - 18 m 4 + 9 m 2 + 8 is positive for m [ 0,1 / 3 ] , we have (25) [ D f ( z ~ 0 ) ] - 1 f ( z ~ 0 ) , z ~ 0 < ( [ D f ( z ~ 0 ) ] - 1 ) * z ~ 0 × f ( z ~ 0 ) sin ( ( 1 - α ) π 2 ) for m ( 0,1 / 3 ] .

Thus, f = f α is not strongly starlike of order α in the sense of Definition 2 for α ( 0,1 ) .

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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