The growth theorems for subclasses of biholomorphic mappings in several complex variables

It is natural to extend the above beautiful results to higher dimensions. However, Cartan (1933) pointed out the above theorem for normalized biholomorphic mappings would not hold in several complex variables. And he also suggested to study the star-like mappings and convex mappings as appropriate topics for generalization. Untill 1991, Barnard, Fitzgerald, and Gong (1991) firstly established the growth and 1 4 -theorems for normalized biholomorphic star-like mappings on the unit ball


Introduction
In geometric theory of one complex variable, the following growth theorem for biholomorphic functions was well known. Theorem 1.1 (Duren, 1983) Let f be a biholomorphic function on the unit disk U = { ∈ ℂ:| | < 1}, and f (0) = f � (0) − 1 = 0. Then It is natural to extend the above beautiful results to higher dimensions. However, Cartan (1933) pointed out the above theorem for normalized biholomorphic mappings would not hold in several complex variables. And he also suggested to study the star-like mappings and convex mappings as appropriate topics for generalization. Untill 1991, Barnard, Fitzgerald, and Gong (1991) firstly established the growth and 1 4 -theorems for normalized biholomorphic star-like mappings on the unit ball B n = {z ∈ ℂ n :‖z‖ = � ∑ n j=1 �z j � 2 � 1 2 < 1}. And after that, a lot of researchers came to study the | | (1 + | |) 2 ≤ |f ( )| ≤

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It is an important work to establish the growth theorems for subclasses of normalized biholomorphic mappings in geometric function theory of several complex variables. As well-known results, many researchers have studied the growth theorems for star-like mappings, covex mappings, and their subclasses. The aim of this article is to discuss growth theorems on the Euclidean unit ball for the generalized star-like mappings and the generalized spiral-like mappings that are defined by Loewner chains.
The subject of Loewner chains in higher dimensions was initiated by Pfaltzgraff (1974). He generalized to higher dimensions the Loewner differential equation and developed existence and uniqueness theorems for its solutions on the Euclidean unit ball B n in ℂ n . Poreda (1987aPoreda ( , 1987b obtained some applications of parametric representation to growth theorems and coefficient estimates on the unit polydisk in ℂ n . Poreda (1989) also deduced certain generalizations on the unit ball of finite dimensional complex Banach space. The existence and regularity of the theory of Loewner chains in higher dimensions were considered by Duren, Graham, Hamada, and Kohr (2010), Graham, Hamada, and Kohr (2002), Hamada and Kohr (2000), etc. Many details and applications of the theory of Loewner chains in several complex variables may be found in the monograph of Graham and Kohr (2003). Chirilă (2014a) used the method of Loewner chains to generate certain subfamilies of normalized biholomorphic mappings on the Euclidean unit ball B n in ℂ n , which have interesting geometric characterizations. In this paper, we will continue to study these biholomorphic mappings introduced by Chirilă. Furthermore, we will obtain the growth theorems for g-almost star-like mapping of order (0 ≤ < 1) and g-spiral-like mapping of type − 2 < < 2 on the unit ball B n using the method of parametric representation. As the application, some well-known results can be got when special functions g are taken on the unit disk in the complex plane.
In the following, we will give some notations and definitions. Let ℂ be the complex plane and U r = { ∈ ℂ:| | < r}. The unit disk in ℂ is denoted by U. Let ℂ n be the space of n complex variables z = (z 1 , … , z n ) � with the Euclidean inner product ⟨z, w⟩ = ∑ n i=1 z i w i and the Euclidean norm ‖z‖ = ⟨z, z⟩ 1 2 , where z, w ∈ ℂ n and the symbol "ʹ" means transpose. The unit ball B n = {z ∈ ℂ n :‖z‖ < 1}. Let X denote the complex Banach space with norm ‖ ⋅ ‖,  = {x ∈ X:‖x‖ < 1} be the unit ball in X. Let Ω be a domain in X, f :Ω → X. If for any x ∈ Ω, there is a linear mapping Df(x) from X to X such that then f is said to be holomorphic on Ω. The linear map Df(x) is called the Fréchet derivative of f at x. In ℂ n , Df(x) is the Jacobian, always written by J f (z). We denote by H(Ω) the set of holomorphic mappings from Ω into ℂ n . Let f :Ω → X be a holomorphic mapping; if its Fréchet derivative Df(x) is nonsingular at each x ∈ Ω, then f is said to be locally biholomorphic on Ω. If f −1 means the inverse of f exists and it is holomorphic on the open set f (Ω), then f is said to be biholomorphic. If f (0) = 0 and Df (0) = I, then f is called normalized, where I is the identity operator.
The following families play a key role in our discussion:

Definition and lemmas
Definition 2.1 (Hamada et al., 2006) Let g ∈ H(U) be a biholomorphic function such that g(0) = 1, g( ) = g( ) for ∈ U (i.e. g has real coefficients in its power series expansion), Reg( ) ≥ 0 on U and assume that g satisfies the following conditions for r ∈ (0, 1) where g satisfies the requirements of Definition 2.1.
Hence, the g-almost star-like mapping of order is a subclass of almost star-like mapping of order on B n . And hence biholomorphic on B n .
this class reduces to the class of almost star-like mappings of order on B n . If g( ) = 1− 1+(1−2 ) , ∈ U, ∈ (0, 1), the almost star-like mappings of order and type on B n can be got (see Chirilă, 2014b).
This shows that if f is a parabolic star-like of order , then � is a mapping from B n onto the parabolic region in the right half-plane Ω , where Definition 2.4 (Chirilă, 2014a) Let ∈ − 2 , 2 . A normalized biholomorphic mapping f :B n → ℂ n is said to be g-spiral-like mapping of type if where g satisfies the requirements of Definition 2.1.

Remark 2
(1) Obviously, if f is g-spiral-like of type , then Hence, f is also a normalized biholomorphic spiral-like mapping of type on B n .
Lemma 2.7 (Liu, Zhang, & Lu, 2006) Let ∈ − 2 , 2 , and let f :B n → ℂ n be a normalized biholomorphic spiral-like mapping of type . Then, where v(z, t) is the solution of the initial value problem (2.2), and the corresponding h(z) = e −i J −1 f (z)f (z).
Lemma 2.8 Let ∈ [0, 1) and let f :B n → ℂ n be a g-almost star-like of order . Then, Proof For z ∈ B n �{0}, let z 0 = z ‖z‖ . Then, is well defined on the unit disk U, and p is biholomorphic on U. By Definition 2.2, we know that Since p(0) = g(0) = 1, we have p ≺ g. By the maximum and minimum principles for harmonic functions, we have Let = ‖z‖. Then,

Main results
Theorem 3.1 Let ∈ [0, 1) and let f :B n → ℂ n be a g-almost star-like of order . Then, And let v(z, t) be the solution of the initial value problem (2.2) corresponding to the above function h. For any 0 ≤ t < t ′ , let v(t) = v(z, t). Then, Since ‖v(t)‖ is continuous, the above inequality implies that ‖v(t)‖ is absolutely continuous for t ∈ [0, +∞), and thus ‖v(t)‖ is differentiable almost everywhere on [0, +∞). Since , v(t)⟩, and by Lemma 2.8, Thus ‖v(t)‖ is a decreasing function on [0, +∞). From the right-hand side of the above inequality, It yields that i.e.
Let t → +∞, by Lemma 2.6, Using the same arguments, ✷ In particular, we can get the growth theorem for almost star-like mappings of order on the unit ball B n when g( ) = 1− 1+ , ∈ U.
Corollary 3.2 Let ∈ [0, 1) and let f :B n → ℂ n be a almost star-like of order . Then, We can also obtain the growth theorem for almost star-like mappings of order and type on B n if let g( ) = 1− 1+(1−2 ) , ∈ U, ∈ (0, 1) in Theorem 3.1.
Corollary 3.3 Let ∈ [0, 1), ∈ (0, 1) and let f :B n → ℂ n be a almost star-like of order and type . Then, Corollary 3.4 Let ∈ (0, 1) and let f :B n → ℂ n be a star-like mapping of order . Then, Proof Let g( ) = 1− 1+(1−2 ) , ∈ U, ∈ (0, 1). Then, g satisfies the conditions of Definition 2.1. Let = 0 in Definition 2.2. We have thus, it is that So f is a star-like mapping of order . And letting = 0 in Theorem 3.1, the result can be obtained. ✷ The following corollary is due to Zhang and Feng (2013).
Corollary 3.5 Let ∈ [0, 1) and let f :B n → ℂ n be a parabolic star-like mapping of order . Then, where we choose the branch of the square root such that √ 1 = 1, and the branch of the logarithm function such that log 1 = 0.
Then, g is a biholomorphic function from U onto the parabolic region Ω in the right half-plane (see Ali, 2005), where And g satisfies the conditions of Definition 2.1.
It yields that i.e.
Let t → +∞, by Lemma 2.7, Using the same arguments,

✷
In particular, letting g( ) = 1− 1+ , ∈ U in Theorem 3.6, we can obtain the growth theorem for spiral-like mappings of type on B n . This result is due to Hamada and Kohr, and we can refer to the Theorem 7.3.6 in Kohr and Liczberski (1998).