A Note on Strongly Starlike Mappings in Several Complex Variables

and Applied Analysis 3 Since |z 1 z2 2 | ≤ 2/(3√3), for z ∈ ∂B, we obtain that |bz 1 z2 2 | ≤ sin(απ/2)‖z‖3 for z ∈ B. This implies that ⟨[Df(z)]−1f(z), z⟩ lies in the disc of center ‖z‖2 and radius sin(απ/2)‖z‖2 for each z ∈ B2 \ {0} and thus 󵄨󵄨󵄨󵄨arg ⟨[Df (z)] −1 f (z) , z⟩ 󵄨󵄨󵄨󵄨 < α π 2 , z ∈ B2 \ {0} . (16) Therefore, f = f α is strongly starlike of order α in the sense of Definition 1. On the other hand, ([Df (z)] −1 ) ∗ z = (z 1 , z 2 − 2bz 2 z 1 ) . (17) So, for z0 = (1/√3,√2/√3), we have ⟨[Df (z0)] −1 f (z0) , z0⟩ = 1 − m, 󵄩󵄩󵄩󵄩󵄩 ([Df (z0)] −1 ) ∗ z0 󵄩󵄩󵄩󵄩󵄩 2 = 1 3 + 2 3 (1 − 3m) 2, 󵄩󵄩󵄩󵄩f (z 0) 󵄩󵄩󵄩󵄩 2 = 1 3 (1 + 3m) 2 + 2 3 , sin((1 − α) π 2 ) = √1 − m, (18) where m = sin(α 2 ) . (19) Then, we obtain 󵄩󵄩󵄩󵄩󵄩 ([Df (z0)] −1 ) ∗ z0 󵄩󵄩󵄩󵄩󵄩 2󵄩󵄩󵄩󵄩f (z 0) 󵄩󵄩󵄩󵄩 2 sin2 ((1 − α) π 2 ) − (R⟨[Df (z0)] −1 f (z0) , z0⟩) 2 = (1 − m) {[ 1 3 + 2 3 (1 − 3m) 2] [ 1 3 (1 + 3m) 2 + 2 3 ] × (1 + m) − (1 − m) } . (20)

If Ω is a domain in C  , let (Ω) be the set of holomorphic mappings from Ω to C  .If Ω is a domain in C  which contains the origin and  ∈ (Ω), we say that  is normalized if (0) = 0 and (0) =   , where   is the identity matrix.
A normalized mapping  ∈ (B  ) is said to be starlike if  is biholomorphic on B  and (B  ) ⊂ (B  ) for  ∈ [0, 1], where the last condition says that the image (B  ) is a starlike domain with respect to the origin.For a normalized locally biholomorphic mapping  on B  ,  is starlike if and only if (see [1][2][3][4] and the references therein, cf.[5]).
Let  ∈ (0, 1].A function  ∈ (), normalized by (0) = 0 and   (0) = 1, 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  ∈ (0, 1), then a domain Ω ̸ = C which contains the origin is accessible if and only if Ω = (), where  is the unit disc in C and  is a strongly starlike function of order 1 −  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 B  .Definition 1.Let 0 <  ≤ 1.A normalized locally biholomorphic mapping  ∈ (B  ) is said to be strongly starlike of order  if Obviously, if  is strongly starlike of order , then  is also starlike, and if  = 1 in (3), one obtains the usual notion of starlikeness on the unit ball B  .
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 B  in C  , where  ∈ (0, 1], and proved that a normalized biholomorphic mapping  on B  is strongly starlike of order 1 −  if and only if (B  ) is an -accessible domain in C  for  ∈ (0, 1).Their definition is as follows.
Definition 2. Let 0 <  ≤ 1.A normalized locally biholomorphic mapping  ∈ (B  ) is said to be strongly starlike of order  (in the sense of Liczberski and Starkov) if In the case  = 1, it is obvious that both notions of strong starlikeness of order  are equivalent.Thus, the following natural question arises in dimension  ≥ 2.
Question 1.Let  ∈ (0, 1).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 B  in C  and let  ∈ (0, 1).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][12][13][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.
For fixed  ∈ B  \ {0}, let  = /‖‖ and Then  is a holomorphic function on  with | arg ()| ≤ /2 for  ∈ .Since arg  is a harmonic function on  and arg (0) = 0, by applying the maximum and minimum principles for harmonic functions, we obtain | arg ()| < /2 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  ≥ 2.