Hongyan LIU(劉紅炎)Zhenhan TU(涂振漢)School of Mathematics and Statistics，Wuhan University，Wuhan 430072，China E-mail: hongyanliu@whu.edu.cn; zhhtu.math@whu.edu.cn

Liangpeng XIONG (熊良鵬)i School of Mathematics and Computer Science,Jiangri Science and Technology Normal Universitgy，Nanchang 330038，China E-mail: lpxionq2016@whu.edu.cn

In the geometric theory of one complex variable, the following distortion theorem for biholomorphic functions is well known:

Theorem A (Duren [5]) Let f be a biholomorphic function on the unit disk D = {z ∈C:|z|<1}, and let f(0)=f′(0)-1=0.Then

However, for the case of several complex variables, Cartan [2] has pointed out that the above theorem does not hold.Thus,it is necessary to study the properties for some special subclasses of biholomorphic mappings defined on different domains, for example, starlike mappings, convex mappings, close-to-convex mappings, Bloch mappings, and so on (see, e.g., Chu-Hamada-Honda-Kohr [4], Graham-Hamada-Kohr [9], Hamada-Honda-Kohr [10], Liu [15], Wang [20]).Barnard-FitzGerald-Gong [1] first studied the distortion theorems of Jacobi-determinant type for convex mappings defined on the Euclidean unit ball, and subsequently, there appeared a lot of important results regarding distortion theorems for convex mappings(see,e.g.,Chu-Hamada-Honda-Kohr [3], Gong-Wang-Yu [6], Gong-Liu [7], Xu-Liu [22], Zhu-Liu [24]).Compared with the case of convex mappings, we know that the geometric properties of biholomorphic starlike mappings are weaker, and thus it is much more difficult to obtain the corresponding distortion theorems for certain kinds of subclasses of starlike mappings.For the space of n-dimensional(n ≥2) complex variables, Poreda[17] discussed the biholomorphic mappings of the unit polydisk in Cnwhich have a parametric representation, After that, some further results regarding subclasses of biholomorphic mappings which have a parametric representation were obtained with different targets (see, e.g., Kohr-Liczberski[11], Kohr [12], Tu-Xiong [19]).Recently, Liu-Liu[14]discussed the distortion theorems with respect to a subclass of biholomorphic mappings which have a parametric representation in several complex variables (also see, e.g., Graham-Hamada-Kohr [8], Xiong [23] ).This paper will make further progress along these lines.

First, we introduce some notations and definitions.

Let A denote the class of all analytic functions on the unit disk D = {z : |z| < 1} in C.Denote by S the subclass of A consisting of functions that are univalent.If f and g are analytic in D, we say that f is subordinate to g,writing this as f ?g, provided that there exist analytic functions ω(z) defined on D with ω(0)=0 and |ω(z)|<1 satisfying that f(z)=g(ω(z)).

A function f ∈A is said to belong to the class S?(γ) of starlike functions of complex order γ if it satisfies the following inequality:

The function class S?(γ) was considered by Nasr-Aouf [16] (also see Srivastava et al., [18]).In particular, this is the usual class of starlike functions in D when γ = 1.The following natural questions arise regarding the dimensions when n ≥2:

Question 1.1 Can we extend the definition of the class S?(γ) from the case in onedimensional space to the case in n-dimensional space (n ≥2)?

Question 1.2 Can we establish the distortion theorems of the class of starlike mappings of complex order γ in the n-dimensional complex variables space (n ≥2)?

We shall try to give affirmative answers to the above questions.

Let g ∈H(D) be a univalent function such that g(0)=1,g(ζ)=g(ζ) for ζ ∈D (i.e., g has real coefficients) and ?g(ζ) > 0 on D, and assume that g satisfies the following conditions for r ∈(0,1) (see Xu-Liu [22]):

We define the following class of g-parametric starlike mappings of real order γ on BX(γ ∈(0,1]),this is closely related to the g-parametric starlike mappings on BX:

Definition 1.3 Let f ∈H(BX),f(0) = 0,Df(0) = I,0 < γ ≤1 and let the function g satisfy the condition (1.2).Then

where x ∈BX{0}, Tx∈T(x).

Remark 1.4 (i) If we choose the parametric γ ∈C?in Definition 1.3, then the existing critical Lemma 2.1 looks like failure.Thus, we cannot make sure that the main theorems hold true in this situation.

(ii) If we take γ =1 and replace the function g by G in Definition 1.3, where G=1g, then it reduces to the class defined by Liu-Liu [14].

(iii) Let Cnbe the space of n-dimensional complex variables z =(z1,z2,··· ,zn) with the maximum norm‖z‖=‖z‖∞=max{|z1|,|z2|,··· ,|zn|}.Denote by Dnthe unit polydisc in Cn.Therefore, if we take X=Cnand BX=Dnin Definition 1.3, then

where 0 < |zj| = ‖z‖∞< 1, h(z) = [Df(z)]-1f(z) = (h1(z),h2(z),··· ,hn(z)) and g satisfies the condition(1.2).As usual,we write a point z ∈Cnas a column vector in the following n×1 matrix form:

The derivative of f ∈H(Dn) at a point a ∈Dnis the complex Jacobian matrix of f given by

We denote by det Jf(z) the Jacobi determinant of the holomorphic mapping f.

2 Preliminaries

In this section,we give some lemmas which play a key role in the proof of our main theorems.

In (2.2), setting ζ =‖x‖ yields the desired inequalities.□

3 Main Results

We derive the desired results from (3.21) and (3.24).□

4 Some Corollories

In this section, we give some corollaries by using Theorems 3.1 to 3.4, which are the corresponding distortion results for subclasses of g-starlike mappings defined on B (resp.Dn).If we replace the function g by G in Corollaries 4.1 to 4.4, where G =1g, then we see that these results were proven by Liu-Liu [14].

Corollary 4.1 Suppose that the function g satisfies the condition (1.2) and that fl:Dml→C are some holomorphic functions, l=1,2,··· ,n.Let

where α ∈[0,1), β ∈(0,1], c ∈(0,1), ξ ∈D.It is easy to verify that every function g conforms to the condition (1.2) whenever g ∈M.Therefore, if we take a certain function g ∈M in Theorems 3.1 to 3.4, then the distortion theorems for certain kinds of subclasses of biholomorphic g-starlike mappings of real order γ defined on Dn(resp.B) can be obtained immediately.

Conflict of InterestThe authors declare no conflict of interest.