LOEWNER CHAINS APPLIED TO g -STARLIKE MAPPINGS OF COMPLEX ORDER OF COMPLEX BANACH SPACES

This paper is devoted to studying geometric and analytic properties of g - starlike mappings of complex order λ . By using Loewner chains, we obtain the growth theorems for g -starlike mappings of complex order λ on the unit ball in reflexive complex Banach spaces, which generalize some results of Graham, Hamada and Kohr. As applications, several different kinds of distortion theorems for g -starlike mappings of complex order λ are obtained. Finally, we prove that the Roper–Suffridge extension operators preserve the property of g -starlike mappings of complex order λ in complex Banach spaces, which generalizes many classical results.

However, in the case of several complex variables, H. Cartan pointed out that the growth theorem and distortion theorem do not hold for normalized biholomorphic mappings.In addition, he suggested that one should investigate the important geometrically defined subfamilies of convex and starlike mappings.As a matter of fact, there was little work in the geometric directions suggested by Cartan, until the 1970s, when a number of results dealing with the convex and starlike biholomorphic mappings appeared.As a direct generalization of the growth theorem for univalent function on the unit disk ‫,ބ‬ the growth theorem for normalized biholomorphic starlike mappings on the unit ball ‫ނ‬ n was obtained by Barnard, Fitzgerald and Gong [Barnard et al. 1991] using the analytical characterization of starlikeness, and by Kubicka and Poreda [1988] using the method of Loewner chains.
Theorem A [Barnard et al. 1991;Kubicka and Poreda 1988].Let f be a starlike mapping on the unit ball ‫ނ‬ n .Then, for any point z ∈ ‫ނ‬ n , we have Furthermore, the above estimates are sharp.
If the convexity restriction is attached to the family of normalized locally biholomorphic mapping f , the following growth theorem for convex mappings is due to Suffridge [1977], Thomas [1991], Liu [1989] or Liu and Ren [1998].
Theorem B. Let f be a convex mapping on the unit ball ‫ނ‬ n .Then, for any point z ∈ ‫ނ‬ n , we have Moreover, the above estimates are sharp.
In several complex variables, Barnard, Fitzgerald and Gong [Barnard et al. 1994] were the first to show that the version of the distortion theorem for the determinant of the Jacobian of normalized biholomorphic convex mappings holds on the unit ball ‫ނ‬ 2 in ‫ރ‬ 2 , but there does not exist a direct generalization of the distortion theorem in the case of the family of starlike mappings.The monograph of Graham and Kohr [2003, Chapter 7] and Gong [1998, Chapter 3, Chapter 4] contain a nice development of the growth theorem and distortion theorem for starlike mappings and convex mappings.And for a more classical results concerning starlike mappings and convex mappings in n-dimensional Euclidean space or complex Banach space; see [Gurganus 1975;Kikuchi 1973;Pfaltzgraff 1974;Poreda 1989;Roper and Suffridge 1995;Suffridge 1970;1973;1977].Hamada and Honda [2008] introduced a subfamily of starlike mappings on the unit ball in complex Banach spaces, which is called g-starlike mappings.They also obtained a sharp growth theorem for this mappings by using the method of parametric representation.Recently, the distortion theorem for g-starlike mappings on the unit ball ‫ނ‬ n was obtained by Graham, Hamada and Kohr [Graham et al. 2020a] using the Schwarz lemma at the boundary.As a generalization of spirallike mappings, Bȃlȃeţi and Nechita [2010] defined almost starlike mappings of complex order λ on the unit ball ‫ނ‬ n and gave an equivalent characterization in terms of Loewner chains.It is interesting to note that the family of g-starlike mappings gives a unified representation of some well-known subfamilies of starlike mappings, and the family of almost starlike mappings of complex order λ gives a unified expression of some well-known subfamilies of spirallike mappings of type β.There is a lot of results concerning g-starlike mappings and almost starlike mappings of complex order λ; see [Chirilȃ 2014;2015;Graham et al. 2002a;2020b;Hamada and Kohr 2004;Hamada et al. 2006;2021;Li and Zhang 2019;Zhang et al. 2018].
In view of the above results, the motivation for this paper can be summarized in terms of the following question: Question.Can we unify g-starlike mappings and complex order λ on the unit ball of complex Banach spaces and characterize their geometric and analytic properties?
We manage to answer the above questions affirmatively in the case of the unit ball of some complex Banach space.In Section 2, the definition of g-starlike mappings of complex order λ is given by combining the definition of g-starlike mappings with the definition of almost starlike mappings of complex order λ on the unit ball in complex Banach spaces.As mentioned in Remark 2.4, it gives a unified expression of a variety of biholomorphic mappings, which includes gstarlike mappings, almost starlike mappings of complex order λ as the special case.In Section 3, by using Loewner chains idea, we establish a growth theorem of g-starlike mappings of complex order λ in reflexive complex Banach spaces, which is a generalization of [Hamada and Honda 2008, Theorem 3.1].Because the family of g-starlike mappings of complex order λ contains most of the biholomorphic mappings that have geometry meaning in higher dimensions, this result essentially corresponds to giving a unified form of the growth theorems for some subfamilies of starlike mappings and spirallike mappings.As applications, in Section 4, we obtain distortion theorems for g-starlike mappings of complex order λ on the unit polydisk ‫ބ‬ n and the unit ball ‫ނ‬ n respectively, which is a generalization of [Graham et al. 2020a, Theorem 5.6, Theorem 5.11;Liu et al. 2015, Theorem 4.2;2011, Theorem 3.1, Theorem 3.2].In Section 5, we will prove that the Roper-Suffridge type extension operator and the Muir type extension operator preserve g-starlike mappings of complex order λ on domain r respectively, where g is a univalent convex function on ‫.ބ‬In particular, if λ = 0, then the results obtained in this paper are generalizations of results in [Graham et al. 2020b;Muir 2005].

Preliminaries
2A. Notations and definitions.Let ‫ބ‬ r = {ζ ∈ ‫ރ‬ : |ζ | < r } be the disk of radius r in the complex plane ‫,ރ‬ and let ‫ބ‬ 1 = ‫.ބ‬Let ‫ރ‬ n denote the space of n complex variables u = (u 1 , . . ., u n ) ′ equipped with inner product ⟨u, v⟩ = n k=1 u k v k , and the Euclidean norm ∥u∥ = n k=1 |u k | 2 , the symbol ′ means the transpose of vectors and matrices.The open ball centered at zero and radius r in ‫ރ‬ n is denoted by ‫ނ‬ n (0, r ) = {u ∈ ‫ރ‬ n : ∥u∥ < r }, the closed ball is denoted by ‫ނ‬ n (0, r ), the unit ball is denoted by . ., n} be the polydisk of radius r .The unit polydisk is denoted by ‫ބ‬ n .The boundary of ‫ނ‬ n is denoted by Let X be a complex Banach space with respect to the norm ∥•∥ X .Let B r = {x ∈ X : ∥x∥ X < r } be the open ball centered at zero and of radius r , and let B be the open unit ball in X .Let B r be the closed ball centered at zero and of radius r .Let ⊆ X be a domain which contains the origin, we denote by H ( ) the set of holomorphic mappings from to X .If f ∈ H ( ), and f (0) = 0, D f (0) = I , then we say that f is normalized, where D f (0) is the Fréchet derivative of f at 0, I is the identity operator on X .A holomorphic mapping f ∈ H ( ) is said to be biholomorphic if the inverse f −1 exists and it is holomorphic on the open set f ( ).A mapping f ∈ H ( ) is said to be locally biholomorphic if each x ∈ has a neighborhood Let T : X → ‫ރ‬ be a continuous linear functional.Then ∥T ∥ = sup{|T x| : x ∈ ∂B}.
For each x ∈ X \{0}, we define In particular, T r x = T x when r > 0.
The following elementary definitions are used: • If for any x ∈ , t ∈ [0, 1], (1 − t)x ∈ holds, then is said to be starlike (with respect to the origin).
• A domain ⊆ X is said to be convex if given • A domain ⊆ X is said to be ε-starlike if there exists a positive number ε ∈ [0, 1], such that for any z, w ∈ , one has In particular, if ε = 0 or ε = 1, then the ε-starlike domain reduces to starlike domain with respect to the origin or convex domain, respectively.
• Let f ∈ H ( ) be biholomorphic mapping with 0 ∈ f ( ).If f ( ) is starlike (with respect to the origin), then f is said to be starlike.If f ( ) is convex, then f is said to be convex.If f ( ) is ε-starlike, then f is said to be ε-starlike, where ε ∈ [0, 1].
• Let g : ‫ބ‬ → ‫ރ‬ be a holomorphic univalent function, g(0) = 1 and ℜg(ζ ) > 0. Furthermore, let g be symmetric along the real axis, i. 2B.Loewner chains.We next recall the notions of subordination and Loewner chains on the unit ball B in X .Some results may be found in [Graham et al. 2013;2020b].
If f, g ∈ H (B), and there exists a Schwarz mapping v such that f = g • v, then we say that f is subordinate to g, denoted by f ≺ g.
If g is biholomorphic on B, then f ≺ g is equivalent to requiring that f (0) = g(0) and f (B) ⊆ g(B).
Definition 2.1.Let B be the unit ball of a complex Banach space X .A mapping The subordination condition of Loewner chain is equivalent to the existence of a unique biholomorphic Schwarz mapping v = v( • , s, t), called the transition mapping associated with f (x, t), such that f (x, s) = f (v(x, s, t), t) for x ∈ B and 0 ≤ s ≤ t.
Let g ∈ G(‫)ބ‬ be defined as above.The family M g (B) of holomorphic mappings h : B → X that is analogous to the analytic functions on the unit disk in the complex plane, with positive real part, is defined as follows.
reduces to the Carathéodory family M(B) on the unit ball B in a complex Banach space.
We know that both M(B) and M g (B) consist of so-called holomorphically accretive mappings, which were intensively studied in Euclidean space ‫ރ‬ n or complex Banach spaces during the last decades.Some related results may be found in [Duren et al. 2010;Elin et al. 2019;Graham et al. 2002a;2013;Hamada and Kohr 2004;Pfaltzgraff 1974;Reich and Shoikhet 1996;2005;Suffridge 1973].
Definition 2.2 [Bracci et al. 2009;Duren et al. 2010;Graham et al. 2002a].A Herglotz vector field associated with the family M(B) on B is a mapping h = h(x, t) : B × [0, ∞) → X satisfying the following conditions: Hamada and Kohr [2004] proved that if X is a reflexive complex Banach space, and h(x, t) : B × [0, ∞) → X is a Herglotz vector field, then for each s ≥ 0 and x ∈ B, the initial value problem exists uniformly on each closed ball B r for r ∈ (0, 1), s ∈ [0, ∞).And f (x, t) is a univalent subordination chain.
then f is called a g-starlike mapping of complex order λ.
We denote by S * g,λ (B) the family of g-starlike mapping of complex order λ on B. Obviously, for the case of X = ‫,ރ‬ B = ‫,ބ‬ the above definition shows that B) is a g-starlike mapping on the unit ball B, some results of g-starlike mappings may be found in [Chirilȃ 2014;2015;Graham et al. 2002a;Hamada et al. 2021].
, the definition on the unit ball ‫ނ‬ n in Euclidean space can be found in [Tu and Xiong 2019].
maps the unit ball B\{0} into the right half plane, i.e., This is the definition of almost starlike mappings of complex order λ; see [Bȃlȃeţi and Nechita 2010;Zhang et al. 2018]. (iv is the Janowski-starlike mappings of complex order λ on the unit ball B; see [Li and Zhang 2019]. Let α ∈ (0, 1), This is the definition of spirallike mappings of type β and order α; see [Feng et al. 2007].
where the branch of the logarithm function is chosen such that log 1 = 0, which reduces to the definition of parabolic spirallike mappings of type β and order ρ; see [Zhang and Yan 2016].
Next, we give two examples in higher dimensions.
By some elementary calculations, we get Growth theorems for g-starlike mappings of complex order λ In the next subsection, we utilize the method of Loewner chains to deal with the growth theorem of g-starlike mappings of complex order λ on the unit ball in a reflexive complex Banach space X .The family of g-starlike mappings of complex order λ unifies the family of almost starlike mappings of complex order λ and the family of g-starlike mappings, and the result in the forthcoming subsection will lead to a number of well known statements.
3A.Several lemmas.We begin this subsection with the following equivalent characterization for almost starlike mappings of complex order λ in terms of Loewner chains on the unit ball B.
Let f be a normalized locally biholomorphic mapping on B, and let λ ∈ ‫ރ‬ with ℜλ ≤ 0. Then f is an almost starlike mapping of complex order λ on B if and only if The following lemma is due to Kato.
Lemma 3.2 [Kato 1967].Let x : [0, +∞) → X be differentiable at the point s ∈ (0, +∞), and let ∥x(t)∥ be also differentiable at the point s with respect to t.Then In fact, the following lemma shows that Loewner chain is generated by its transition mapping.It is due to Graham et al. [2013].
Lemma 3.3.Suppose that X is a reflexive complex Banach space.Let f (x, t) : B × [0, ∞) → X be a Loewner chain.And let v(x, s, t) be the transition mapping associated with f (x, t).If for each r ∈ (0, 1), there exists M = M(r ) > 0 such that uniformly on B r for r ∈ (0, 1).
In fact, the following lemma plays an important role in the proof of growth theorem.
Lemma 3.4.Let λ ∈ ‫ރ‬ with ℜλ ≤ 0, g ∈ G(‫,)ބ‬ and let f : B → X be a g-starlike mapping of complex order λ on B. Then Then the holomorphic function is well defined on the unit disk ‫.ބ‬ Since from Definition 2.3, it yields that q(0) = g(0) = 1, q(‫)ބ‬ ⊆ g(‫,)ބ‬ i.e., q ≺ g.By the subordination principle, it follows that q(r ‫)ބ‬ ⊆ g(r ‫,)ބ‬ r ∈ (0, 1).Hence , then the inequality (3.1) is equivalent to the following form: 3B.Growth theorems of the classes S * g,λ (B).The method to approach the following theorem is analogous to that of [Zhang et al. 2018], although we are now considering normalized biholomorphic mappings on the unit ball B in an infinite dimensional complex Banach space.
Theorem 3.6.Let λ ∈ ‫ރ‬ with ℜλ ≤ 0, g ∈ G(‫,)ބ‬ and let f : B → X be a g-starlike mapping of complex order λ on B in reflexive complex Banach space.Then is also an almost starlike mapping of complex order λ.By Lemma 3.1 we know that is a Loewner chain, hence we have i.e., there is a Schwarz mapping v(x, s, t) such that F(x, s) = F(v(x, s, t), t).By some calculation, we obtain ∂ F ∂t (x, t) = (1 − λ)e t e −λt f (e λt x) + λe t D f (e λt x)x, D F(x, t) = e t D f (e λt x).
Since for all x ∈ B, we have here we use the fact that f is also an almost starlike mapping of complex order λ and the upper bound of ∥ f (x)∥ X ; see Furthermore, by Lemmas 3.2 and 3.4 we see that By Lemma 3.4 and equality (3.2), we have Since we have Integrating on both sides of the inequality (3.4) with respect to τ ∈ [s, t], we infer that By using Lemma 3.3 we have ∥e t v(t)∥ → ∥e (1−λ)s f (e λs x)∥ as t → +∞.Because lim t→+∞ ∥e λt v(t)∥ = 0, then taking t → +∞ on the both sides of the inequality (3.5), and taking s = 0, we see that By using the same method for obtaining inequality (3.3), we get Remark 3.7.In particular, if g ∈ G(‫)ބ‬ and λ ∈ ‫ރ‬ with ℜλ ≤ 0 are some special functions and special complex number, such as in Remark 2.4, we can get the growth theorems of starlike mappings, spirallike mappings of type β, etc.This is one of the reasons for the interest in this normalized biholomorphic mappings.
4. Distortion theorems for g-starlike mappings of complex order λ 4A.Distortion theorems along a unit direction.In this subsection, we obtain the distortion theorems for g-starlike mappings of complex order λ along a unit direction on the unit polydisk ‫ބ‬ n and the unit ball B, respectively.Proof.The proof is divided into the following two steps: Step By using Lemma 3.4, we have Since ∥T z ∥ ≤ 1, by Lemma 3.4 we get Hence, by Theorem 3.6, (4.1) and (4.2), we have r ∈ (0, 1), and let f : B → X be a g-starlike mapping of complex order λ in reflexive complex Banach spaces.Then, for all x ∈ B\{0}, there exists a unit vector ζ (x) such that By using Lemma 3.4, we get Hence, by Theorem 3.6, we have Remark 4.3.If λ = 0 and g is some biholomorphic function in Definition 2.3, we can get the results in [Liu et al. 2011;2012] from Theorems 4.1 and 4.2.
4B. Distortion theorems on the unit ball ‫ނ‬ n .In this subsection, the distortion theorems for g-starlike mappings of complex order λ at extreme points are established on the unit ball ‫ނ‬ n in ‫ރ‬ n .Denote by T (1,0) has the following properties: where c = f (0).
By Lemmas 4.4 and 3.4, there exists a µ ∈ ‫ޒ‬ such that By Lemma 4.4 we have where we have used Theorem 3.6, i.e., Theorem 4.7.Let λ ∈ ‫ރ‬ with ℜλ ≤ 0, g ∈ G(‫,)ބ‬ and let f : ‫ނ‬ n → ‫ރ‬ n be a g-starlike mapping of complex order λ: (1) If z ∈ ‫ނ‬ n satisfies max ( Proof. (1) From the proof of Theorem 4.5, we know that if z ∈ ‫ނ‬ n is the maximum module point of f in the ball ‫ނ‬ n (0, ∥z∥), there exists a real number µ > 0 such that and (2) From the proof of Theorem 4.5, we know that if z ∈ ‫ނ‬ n is the minimum module point of f in the ball ‫ނ‬ n (0, ∥z∥), there is a real number µ > 0 such that By Lemma 4.4 we have we know that Replacing D f (z)β with γ in the inequality (4.3),where for all β ∈ T (1,0) z ‫ނ∂(‬ n (0, ∥z∥)).By Theorem 3.6 we see that  1995].This operator is used to construct starlike mappings and convex mappings in higher dimensions via starlike functions and convex functions in the unit disk, respectively.In the same paper, Roper and Suffridge proved that if f is a normalized locally biholomorphic convex function on the unit disk ‫,ބ‬ then is a normalized locally biholomorphic convex mapping on the Euclidean unit ball ‫ނ‬ n , where Graham and Kohr [2000] used the analytic definition of starlike mappings to prove that if f is a starlike function on ‫,ބ‬ then n ( f ) is a starlike mapping on the unit ball ‫ނ‬ n .Furthermore, Graham and Hamada et al. [2002b] proved that if f is a normalized locally biholomorphic starlike function on ‫,ބ‬ then n,α,β ( f ) is a normalized locally biholomorphic starlike mapping on ‫ނ‬ n for α ∈ [0, 1], β ∈ 0, 1 2 , α + β ≤ 1; if f is a normalized locally biholomorphic convex function on ‫,ބ‬ then n,α,β ( f ) is a normalized locally biholomorphic convex mapping on ‫ނ‬ n if and only if (α, β) = 0, 1 2 , where , and the branches of the power functions are chosen such that ( f (u 1 )/u 1 ) α | u 1 =0 = 1, ( f ′ (u 1 )) β | u 1 =0 = 1.In the above, the Roper-Suffridge operator is only defined on the unit ball ‫ނ‬ n .Graham and Kohr [2000], raised the following question: extend convex functions on ‫ބ‬ to convex mappings on the egg domain 2, p ? Gong and Liu [2002] gave an affirmative answer to the above question.They used the contractive property of Carathéodory metric under holomorphic mappings to show that if f is a normalized locally biholomorphic ε starlike function on ‫,ބ‬ then is a normalized locally biholomorphic ε starlike mapping on p , where p = (u 1 , . . ., Muir [2005] introduced an extension operator from a new perspective as follows: where f is a normalized locally biholomorphic function on ‫ބ‬ and P : ‫ރ‬ n−1 → ‫ރ‬ is a homogeneous polynomial mapping of degree 2, and √ f ′ (0) = 1.Furthermore, he showed that if f is a normalized locally biholomorphic starlike function on ‫,ބ‬ then n,P ( f ) is a normalized locally biholomorphic starlike mapping on ‫ނ‬ n if and only if ∥P∥ ≤ 1 4 ; if f is a normalized locally biholomorphic convex function on ‫,ބ‬ then n,P ( f ) is a normalized locally biholomorphic convex mapping on ‫ނ‬ n if and only if ∥P∥ ≤ 1 2 .Recently, Graham and Hamada et al. [Graham et al. 2020b] consider the extension operator α,β and P r on some unit ball in the complex Banach space Z = ‫ރ‬ × Y , where f is a normalized locally biholomorphic function on ‫,ބ‬ the branches of the power functions are chosen such that ( f (z 1 )/z 1 ) α | z 1 =0 = 1, P r : Y → ‫ރ‬ is a homogeneous polynomial mapping of degree r , 2 ≤ r and where Y is a complex Banach space.The Minkowski function of r is a complete norm ∥•∥ Z on Z , r is the unit ball of Z with respect to this norm.They proved that these two extension operators had the following properties: Let g ∈ G(‫)ބ‬ be a convex function, and the normalized locally biholomorphic function f can be embedded as the first element of a g-Loewner chain on ‫.ބ‬ Then α,β ( f ) can be embedded as the first element of a g-Loewner chain on r .If ∥P r ∥ ≤ 1 4 dist(1, ∂g(‫,))ބ‬ then P r ( f ) can be embedded as the first element of a g-Loewner chain on r .The extension operators for normalized locally biholomorphic functions on the unit disk ‫ބ‬ to higher dimensional spaces have been extensively studied in the literature, see, e.g., [Elin 2011;Elin and Levenshtein 2014;Feng and Liu 2008;Gong and Liu 2003;Graham et al. 2012;Liu et al. 2019;Liu and Xu 2006;Wang 2013;Wang and Liu 2010;2018].
In the next subsection, we study the extension operators α,β and P r associated with the g-starlike mappings of complex order λ on r by using two different methods.
5B.Some lemmas.In order to prove the main theorems in this subsection, we need the following lemmas.
where z = (z 1 , w) ∈ r and the branches of the power functions are chosen such that is a g-Loewner chain, since f is a g-starlike function of complex order λ on the unit disk ‫.ބ‬According to a result of Graham et al. [2020b, Theorem 3.1], we know that Theorem 5.7.Let Y be a complex Banach space and let k ≥ 2 be an integer.Denote r ≥ 1, and let P k : Y → ‫ރ‬ be a homogeneous polynomial mapping of degree k, r ≤ k.Assume that f is a g-starlike function of complex order λ on ‫,ބ‬ where g ∈ G(‫)ބ‬ is a convex function.If where (z 1 , w) ∈ r and the branch of the power function is chosen such that Hence, (5.1) Next, we will show that Indeed, if z = (z 1 , 0) ∈ ∂ r , then + λ ∈ g(‫.)ބ‬ If z = (z 1 , w) ∈ ∂ r with w ̸ = 0, then by using Lemma 5.1 and (5.1), we have Since f is a g-starlike function of complex order λ on ‫,ބ‬ we have Therefore, we have Since g(‫)ބ‬ contains a disk with g(a) as center and 1 2 |g ′ (a)|(1 − |a| 2 ) as radius, where a = v(ρz 1 ), and  In particular, when Y = ‫ރ‬ n−1 , we do have the following corollary, which is a generalization of [Muir 2005, Theorem 4.1].
theorems for g-starlike mappings of complex order λ 409 1. Introduction Let f = z + ∞ n=2 a n z n be a normalized univalent function on the unit disk ‫ބ‬ in ‫.ރ‬The growth theorem shows that the modulus of a normalized univalent function | f | has a finite upper and positive lower bound depending only on the modulus of the variable |z|, and the image of f contains a disk centered at origin with radius 1 4 .The distortion theorem gives explicit upper and lower bounds on | f ′ (z)| in terms of |z|.The term distortion arises from the geometric interpretation of | f ′ (z)| as the infinitesimal magnification factor of arc length and the interpretation of the square of | f ′ (z)| as the infinitesimal magnification factor of area.
t) exists for a.e.t ≥ 0 and for all x ∈ B, and there exists a Herglotz vector field h = h(x, t) : B × [0, ∞) → X with h( • , t) ∈ M g (B) for a.e.t ≥ 0 Let g ∈ G(‫.)ބ‬It is not difficult to deduce that f : B → X is a g-starlike mapping of complex order λ if and only if F(x, t) = e (1−λ)t f (e λt x), Let g ∈ G(‫)ބ‬ be a convex function , and let Y be a complex Banach space.Denote r