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Some results of homogeneous expansions for a class of biholomorphic mappings defined on a Reinhardt domain in ℂn

  • Xiaoying Sima , Zhenhan Tu and Liangpeng Xiong EMAIL logo
Published/Copyright: July 11, 2023
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Demonstratio Mathematica
From the journal Demonstratio Mathematica Volume 56 Issue 1

Abstract

Let S γ , A , B ∗ ( D ) be the usual class of g -starlike functions of complex order γ in the unit disk D = { ζ ∈ C : ∣ ζ ∣ < 1 } , where g ( ζ ) = ( 1 + A ζ ) ∕ ( 1 + B ζ ) , with γ ∈ C \ { 0 } , − 1 ≤ A < B ≤ 1 , ζ ∈ D . First, we obtain the bounds of all the coefficients of homogeneous expansions for the functions f ∈ S γ , A , B ∗ ( D ) when ζ = 0 is a zero of order k + 1 of f ( ζ ) − ζ . Second, we generalize this result to several complex variables by considering the corresponding biholomorphic mappings defined in a bounded complete Reinhardt domain. These main theorems unify and extend many known results.

Keywords: biholomorphic mappings; coefficient problems; Reinhardt domain; g-starlike mappings of complex order gamma
MSC 2010: 32H02; 30C45; 26B10; 32A30

1 Introduction

Let C n be the n -dimensional complex variable space. Let D be the unit open disk in C 1 . Suppose that Ω 1 , Ω 2 ⊂ C n are two domains. Let H ( Ω 1 , Ω 2 ) be the family of all holomorphic mappings from Ω 1 into Ω 2 . Let ϕ 1 , ϕ 2 ∈ H ( D , C 1 ) . We say that ϕ 1 is subordinate to ϕ 2 , and write ϕ 1 ≺ ϕ 2 , if there exists a Schwarz function ω on D such that ϕ 1 = ϕ 2 ( ω ( z ) ) on D (see, Amini et al. [1]). Let Ω be a domain (connected open set) in C n , which contains 0. It is said that z = 0 is a zero of order k of f ( z ) if f ( 0 ) = 0 , … , D k − 1 f ( 0 ) = 0 , but D k f ( 0 ) ≠ 0 , where k ∈ N (see, Lin and Hong [2]). In one complex variable, the following Theorem A concerning starlike functions of order α is classical and well known.

Theorem A

(Boyd [3]) Let α ∈ ( 0 , 1 ) and k ∈ N + . If f ( z ) = z + ∑ m = k + 1 ∞ a m z m is a starlike function of order α on the unit disk D , then

∣ a m ∣ ≤ ∏ μ = 1 s ( ( μ − 1 ) k + 2 − 2 α ) s ! k s , s k + 1 ≤ m ≤ ( s + 1 ) k , s = 1 , 2 , … .

These estimates are sharp for m = s k + 1 , s = 1 , 2 , … . Especially, when k = 1 ,

∣ a m ∣ ≤ 1 ( m − 1 ) ! ∏ μ = 2 m ( μ − 2 α ) , m = 2 , 3 , … .

It is known that the coefficient inequalities are related to the Bieberbach conjectures [4], which was settled by de Branges [5]. However, Cartan [6] stated that the Bieberbach conjecture does not hold in several complex variables. In fact, only a few complete results are known for the inequalities of homogeneous expansions for subclasses of biholomorphic mappings in C n (see, e.g., Długosz and Liczberski [7], Hamada and Honda [8], Liu and Wu [9], Liu and Liu [10], Liu et al. [11], and Xu et al. [12]). Many works are concentrating on the bounds of second- and third-order terms of homogeneous expansions for starlike mappings and the sharp bounds of all homogeneous expansions for the special subclasses of starlike mappings with some restricted conditions (see, e.g., Hamada et al. [13], Liu and Liu [14], Tu and Xiong [15], Xiong [16], Xu et al. [17], and Xu et al. [18]). In Graham et al. [19], the estimate of the second-order coefficients of the first elements of g-Loewner chains in several complex variables was first obtained. In Bracci [20], a sharp estimate for the second-order coefficients for the first elements of g-Loewner chains on the Euclidean unit ball of C 2 , where g ( ζ ) = ( 1 + ζ ) ∕ ( 1 − ζ ) , which gives a support point for the family, was obtained. Generalizations of this result to the unit polydisk in C 2 and to bounded symmetric domains were considered in Graham et al. [21], Hamada and Kohr [22], respectively. In Xu and Liu [23], the Fekete-Szegö inequality for starlike mappings in several complex variables was first obtained. Very recent important results related to the Fekete-Szegö inequality in several complex variables were obtained in other articles (see, e.g., Długosz and Liczberski [24], Elin and Jacobzon [25], Hamada [26], and Lai and Xu [27]). In particular, the Fekete-Szegö inequality for univalent mappings in several complex variables was first obtained in Hamada et al. [28]. Also, the other related results may consult in Długosz and Liczberski [29], Graham and Kohr [30], and Nunokawa and Sokol [31]. Liu et al. [32] considered only the main coefficients that are analogous with the diagonal elements of a square matrix and generalized Theorem A to the case on a Reinhardt domain in C n from a new viewpoint. Let

D p 1 , p 2 , … , p n = z ∈ C n : ∑ l = 1 n ∣ z l ∣ p l < 1 , ( p l > 1 , l = 1 , 2 , … , n )

be a bounded complete Reinhardt domain in C n , its Minkowski function ρ ( z ) is C 1 except for some lower-dimensional manifolds in C n , and ρ : C n → [ 0 , + ∞ ) is defined by:

(1) ρ ( z ) = inf t > 0 : z t ∈ D p 1 , p 2 , … , p n , z ∈ C n .

Let γ ∈ C ∗ = C \ { 0 } . Now, we introduce the following classes S g , γ ∗ ( D p 1 , p 2 , … , p n ) , which extend the usual class S ∗ ( γ ) of starlike functions of complex order γ on D in C to the classes of g -starlike mapping of complex order γ on the bounded Reinhardt domain D p 1 , p 2 , … , p n in C n . The function class S ∗ ( γ ) was considered earlier by Nasr and Aouf [33] (also, see Srivastava et al. [34]).

Definition 1.1

Suppose that γ ∈ C ∗ and g : D → C be a biholomorphic function such that g ( 0 ) = 1 , ℜ g ( ζ ) > 0 on D . A normalized locally biholomorphic mapping f : D p 1 , p 2 , … , p n → C n is called a g -starlike mappings of complex order γ on D p 1 , p 2 , … , p n if

(2) 1 + 1 γ ρ ( z ) 2 ∂ ρ ( z ) ∂ ( z ) [ D f ( z ) ] − 1 f ( z ) − 1 ∈ g ( D ) , ∀ z ∈ D p 1 , p 2 , … , p n \ { 0 } ,

where ρ is the Minkowski function of D p 1 , p 2 , … , p n . We denote by S g , γ ∗ ( D p 1 , p 2 , … , p n ) the set of all g -starlike mappings of complex order γ on D p 1 , p 2 , … , p n in C n .

Remark 1.1

(i) If g ( ζ ) = 1 + A ζ 1 + B ζ ( − 1 ≤ A < B ≤ 1 ,  ζ ∈ D ) in Definition 1.1, then we write S 1 + A ζ 1 + B ζ , γ ∗ ( D p 1 , p 2 , … , p n ) by S γ , A , B ∗ ( D p 1 , p 2 , … , p n ) .

(ii) If n = 1 and g ( ζ ) = 1 + A ζ 1 + B ζ in Definition 1.1, then it is obvious that Condition (2) is equivalent to

1 + 1 γ ζ f ′ ( ζ ) f ( ζ ) − 1 ≺ 1 + A ζ 1 + B ζ , − 1 ≤ A < B ≤ 1 , γ ∈ C ∗ , ζ ∈ D .

We denote by S γ , A , B ∗ ( D ) the set of all g -starlike mappings of complex order γ on D in C , where g ( ζ ) = ( 1 + A ζ ) ∕ ( 1 + B ζ ) , ζ ∈ D . In particular, S 1 , A , B ∗ ( D ) is identical with the well-known class of Janowski starlike functions (see, Janowski [35]) and S γ , − 1 , 1 ∗ ( D ) is the set of all starlike functions of complex order γ in D .

(iii) If γ = 1 and A = 2 α − 1 ( α ∈ [ 0 , 1 ) ) , B = 1 in Definition 1.1 (the case g is defined as (ii)), then we obtain the starlike mappings of order α on D p 1 , p 2 , … , p n (see, e.g., Liu et al. [32]).

(iv) By choosing the suitable functions g and parameters γ in Definition 1.1, we can obtain kinds of subclasses of starlike mappings defined on the Reinhardt domain D p 1 , p 2 , … , p n in C n .

In this article, we first extend the definition of g -starlike mappings of complex order γ from the case of one-dimensional space to the case of higher-dimensional space (see Definition 1.1). Next, we obtain the bound of all coefficients of homogeneous expansions for the functions f ∈ S γ , A , B ∗ ( D ) when ζ = 0 is a zero of order k + 1 of f ( ζ ) − ζ (see Lemma 2.2). Finally, by applying the results in Section 2, we consider the bound of main coefficients of the homogeneous expansions for the functions f ∈ S γ , A , B ∗ ( D p 1 , p 2 , … , p n ) in several complex variables (Theorems 3.1 and 3.2). Also, our results extend some theorems given in the previous literature (see Remarks 1.1–3.1).

2 Preliminaries

The following lemmas are needed in order to prove our estimates. Actually, we may use the similar way to those in the proof of Liu and Liu [36] (Lemmas 2.1 and 2.3). Here, we give the proof for the sake of completeness.

Lemma 2.1

Let k ∈ N + , C ≥ 0 , γ ∈ C ∗ . Then, for q = 2 , 3 , … , we have

(3) C 2 ∣ γ ∣ 2 + C ∣ γ ∣ ∑ m = 1 q − 1 ( 2 m k + C ∣ γ ∣ ) 1 m ! ∏ μ = 0 m − 1 μ + C ∣ γ ∣ k 2 = k ( q − 1 ) ! ∏ μ = 0 q − 1 μ + C ∣ γ ∣ k 2 .

Proof

We try to prove this lemma by mathematical induction. First, if q = 2 , it is easy to see that (3) is true. Next, for all q = 2 , 3 , … l , assume that (3) holds true. Then, we need to show that (3) holds true when q = l + 1 . By a simple computation, we have

C 2 ∣ γ ∣ 2 + C ∣ γ ∣ ∑ m = 1 l ( 2 m k + C ∣ γ ∣ ) 1 m ! ∏ μ = 0 m − 1 μ + C ∣ γ ∣ k 2 = k ( l − 1 ) ! ∏ μ = 0 l − 1 μ + C ∣ γ ∣ k 2 + C ∣ γ ∣ ( 2 l k + C ∣ γ ∣ ) 1 l ! ∏ μ = 0 l − 1 μ + C ∣ γ ∣ k 2 = k l ! ∏ μ = 0 l μ + C ∣ γ ∣ k 2 .

This completes the proof.□

Lemma 2.2

Let f ( z ) = z + ∑ m = k + 1 ∞ a m z m ∈ S γ , A , B ∗ ( D ) with k ∈ N + , γ ∈ C ∗ , − 1 ≤ A < B ≤ 1 . Then, for s = 1 , 2 , … , we have

∣ a m ∣ ≤ ∏ μ = 1 s ( ( μ − 1 ) k + ∣ A − B ∣ ∣ γ ∣ ) s ! k s , s k + 1 ≤ m ≤ ( s + 1 ) k .

In particular, if k = 1 , then ∣ a m ∣ ≤ 1 ( m − 1 ) ! ∏ μ = 0 m − 2 ( μ + ∣ A − B ∣ ∣ γ ∣ ) , m = 2 , 3 , … .

Proof

Since f ∈ S γ , A , B ∗ ( D ) , so we can write that

1 + 1 γ z f ′ ( z ) f ( z ) − 1 ≺ 1 + A z 1 + B z , z ∈ D .

Thus, there is a function φ ∈ ℋ ( D , D ) with ∣ φ ( z ) ∣ < 1 , such that

(4) 1 + A ( φ ( z ) ) 1 + B ( φ ( z ) ) = 1 + 1 γ z f ′ ( z ) f ( z ) − 1 , z ∈ D .

Using (4), a simple computation shows that

(5) φ ( z ) = z f ′ ( z ) − f ( z ) ( ( A − B ) γ + B ) f ( z ) − B z f ′ ( z ) = b k z k + b k + 1 z k + 1 + ⋯ , z ∈ D .

(5) is equal to

(6) z f ′ ( z ) − f ( z ) = φ ( z ) { ( ( A − B ) γ + B ) f ( z ) − B z f ′ ( z ) }

and

(7) ( m − 1 ) a m = ( A − B ) γ b m − 1 , m = k + 1 , k + 2 , … , 2 k .

In view of the relations ∣ φ ( z ) ∣ < 1 and ∑ m = k ∞ ∣ b m ∣ 2 ≤ 1 , then we have

(8) ∑ m = k 2 k − 1 ∣ b m ∣ 2 ≤ 1 .

Using (7) and (8), it can be easily shown that

(9) ∑ m = k + 1 2 k ( m − 1 ) 2 ∣ a m ∣ 2 ≤ ∣ A − B ∣ 2 ∣ γ ∣ 2 .

By applying (6), then it can also be written as:

(10) ∑ m = k + 1 ∞ ( m − 1 ) a m z m = φ ( z ) ( A − B ) γ z + ∑ m = k + 1 ∞ ( ( A − B ) γ + B − m B ) a m z m = φ ( z ) ( A − B ) γ z + ∑ m = k + 1 p − k ( ( A − B ) γ + B − m B ) a m z m + ∑ m = p + 1 ∞ c m z m .

Using the equality in (10), we have

(11) ∑ m = k + 1 p ( m − 1 ) a m z m + ∑ m = p + 1 ∞ d m z m = φ ( z ) ( A − B ) γ z + ∑ m = k + 1 p − k ( ( A − B ) γ + B − m B ) a m z m .

Taking r → 1 in (11) and applying the similar argument being used in Theorem 1 by Boyd [3], we obtain

(12) ∑ m = k + 1 p ( m − 1 ) 2 ∣ a m ∣ 2 ≤ ∣ A − B ∣ 2 ∣ γ ∣ 2 + ∑ m = k + 1 p − k ∣ ( A − B ) γ + B − m B ∣ 2 ∣ a m ∣ 2 .

In fact, with (12), a simple computation shows that

(13) ∑ m = p − k + 1 p ( m − 1 ) 2 ∣ a m ∣ 2 ≤ ∣ A − B ∣ 2 ∣ γ ∣ 2 + ∑ m = k + 1 p − k ( ∣ ( A − B ) γ + B − m B ∣ 2 − ( m − 1 ) 2 ) ∣ a m ∣ 2 .

Furthermore, in view of (13), we have

(14) ∑ m = p − k + 1 p ( m − 1 ) 2 ∣ a m ∣ 2 ≤ 2 ∣ A − B ∣ ∣ γ ∣ ∣ A − B ∣ ∣ γ ∣ 2 + ∑ m = k + 1 p − k m − 1 + ∣ A − B ∣ ∣ γ ∣ 2 ∣ a m ∣ 2 .

Here, we will prove that the following inequalities

(15) ∑ m = s k + 1 ( s + 1 ) k ( m − 1 ) 2 ∣ a m ∣ 2 ≤ k ( s − 1 ) ! ∏ μ = 0 s − 1 μ + ∣ A − B ∣ ∣ γ ∣ k 2

and

(16) ∑ m = s k + 1 ( s + 1 ) k m − 1 + ∣ A − B ∣ ∣ γ ∣ 2 ∣ a m ∣ 2 ≤ s k + ∣ A − B ∣ ∣ γ ∣ 2 1 s ! ∏ μ = 0 s − 1 μ + ∣ A − B ∣ ∣ γ ∣ k 2

hold true for s = 1 , 2 , … .

If s = 1 , then (15) holds from (9). Moreover, by using Lemma 2.2 in Liu and Liu [36] and (9), we obtain

(17) ∑ m = k + 1 2 k m − 1 + ∣ A − B ∣ ∣ γ ∣ 2 ∣ a m ∣ 2 = k + ∣ A − B ∣ ∣ γ ∣ 2 k 2 ∑ m = k + 1 2 k k 2 k + ∣ A − B ∣ ∣ γ ∣ 2 m − 1 + ∣ A − B ∣ ∣ γ ∣ 2 ∣ a m ∣ 2 ≤ k + ∣ A − B ∣ ∣ γ ∣ 2 k 2 ∑ m = k + 1 2 k ( m − 1 ) 2 ∣ a m ∣ 2 ≤ k + ∣ A − B ∣ ∣ γ ∣ 2 k 2 ∣ A − B ∣ 2 ∣ γ ∣ 2 = k + ∣ A − B ∣ ∣ γ ∣ 2 ∣ A − B ∣ ∣ γ ∣ k 2 .

Thus, (17) implies that (16) is true for s = 1 .

At last, assume that (15) and (16) are valid for s = 1 , 2 , … , q − 1 . Taking p = ( q + 1 ) k in (14) and using Lemma 2.1, it gives that

∑ m = q k + 1 ( q + 1 ) k ( m − 1 ) 2 ∣ a m ∣ 2 ≤ 2 ∣ A − B ∣ ∣ γ ∣ ∣ A − B ∣ ∣ γ ∣ 2 + ∑ m = k + 1 q k m − 1 + ∣ A − B ∣ ∣ γ ∣ 2 ∣ a m ∣ 2 = 2 ∣ A − B ∣ ∣ γ ∣ ∣ A − B ∣ ∣ γ ∣ 2 + ∑ s = 1 q − 1 ∑ m = s k + 1 ( s + 1 ) k m − 1 + ∣ A − B ∣ ∣ γ ∣ 2 ∣ a m ∣ 2 ≤ 2 ∣ A − B ∣ ∣ γ ∣ ∣ A − B ∣ ∣ γ ∣ 2 + ∑ s = 1 q − 1 s k + ∣ A − B ∣ ∣ γ ∣ 2 1 s ! ∏ μ = 0 s − 1 μ + ∣ A − B ∣ ∣ γ ∣ k 2 = k ( q − 1 ) ! ∏ μ = 0 q − 1 μ + ∣ A − B ∣ ∣ γ ∣ k 2 .

The aforementioned inequality shows that (15) holds for s = q . On the other hand, when s = q , we find that

∑ m = q k + 1 ( q + 1 ) k m − 1 + ∣ A − B ∣ ∣ γ ∣ 2 ∣ a m ∣ 2 = q k + ∣ A − B ∣ ∣ γ ∣ 2 ( q k ) 2 ∑ m = q k + 1 ( q + 1 ) k ( q k ) 2 q k + ∣ A − B ∣ ∣ γ ∣ 2 m − 1 + ∣ A − B ∣ ∣ γ ∣ 2 ∣ a m ∣ 2 ≤ q k + ∣ A − B ∣ ∣ γ ∣ 2 ( q k ) 2 ∑ n = q k + 1 ( q + 1 ) k ( m − 1 ) 2 ∣ a m ∣ 2 ≤ q k + ∣ A − B ∣ ∣ γ ∣ 2 ( q k ) 2 k ( q − 1 ) ! ∏ μ = 0 q − 1 μ + ∣ A − B ∣ ∣ γ ∣ k 2 = q k + ∣ A − B ∣ ∣ γ ∣ 2 1 q ! ∏ μ = 0 q − 1 μ + ∣ A − B ∣ ∣ γ ∣ k 2 .

The aforementioned inequality shows that (16) holds for s = q .

In view of (15), for s k + 1 ≤ m ≤ ( s + 1 ) k , we have

∣ a m ∣ ≤ k ( m − 1 ) ( s − 1 ) ! ∏ μ = 0 s − 1 μ + ∣ A − B ∣ ∣ γ ∣ k ≤ 1 s ! ∏ μ = 0 s − 1 μ + ∣ A − B ∣ ∣ γ ∣ k = 1 s ! ∏ μ = 1 s μ − 1 + ∣ A − B ∣ ∣ γ ∣ k = ∏ μ = 1 s ( ( μ − 1 ) k + ∣ A − B ∣ ∣ γ ∣ ) s ! k s .

This completes the proof.□

Remark 2.1

If A = 2 α − 1 ( α ∈ [ 0 , 1 ) ) , B = 1 , and γ = 1 in Lemma 2.2, then it reduces to Theorem A.

3 Main results

The following theorems give the estimates of main coefficients of homogeneous expansions for the class of g -starlike mappings of complex order γ defined on the bounded complete Reinhardt domain D p 1 , p 2 , … , p n in C n , where g ( ζ ) = ( 1 + A ζ ) ∕ ( 1 + B ζ ) , − 1 ≤ A < B ≤ 1 , ζ ∈ D . Theorems 3.1 and 3.2 will give generalizations of the results in Boyd [3] and Liu et al. [32].

Theorem 3.1

Suppose that f ( z ) = ( f 1 ( z ) , f 2 ( z ) , … , f n ( z ) ) ′ ∈ S γ , A , B ∗ ( D p 1 , p 2 , … , p n ) and z = 0 is a zero of order k + 1 of f ( z ) − z . Define

D m f q ( 0 ) ( z m ) m ! = ∑ l 1 , l 2 , … , l m = 1 n a q l 1 , l 2 , … , l m z l 1 z l 2 … z l m ,

where q = 1 , 2 , … , n , s k + 1 ≤ m ≤ ( s + 1 ) k , s = 1 , 2 , … , and k ∈ N + . Let a q t m be the a q t , t ⋯ , t ︸ m ( t = { 1 , 2 , … , n } ) . If a q j m = 0  ( q ≠ j ), then we have

∣ a j j m ∣ ≤ ∏ μ = 1 s ( ( μ − 1 ) k + ∣ A − B ∣ ∣ γ ∣ ) s ! k s , s k + 1 ≤ m ≤ ( s + 1 ) k , s = 1 , 2 , … , j ∈ { 1 , 2 , … , n } .

The aforementioned estimates are sharp for 0 < γ ≤ 1 , A = − 1 , B = 1 , m = s k + 1 , s = 1 , 2 , … , and 0 < γ ≤ 1 , A = 2 α − 1 ( α ∈ [ 0 , 1 ) ) , B = 1 , m = s k + 1 , s = 1 , 2 , … . In particular, if k = 1 , then

∣ a j j m ∣ ≤ ∏ μ = 0 m − 2 ( μ + ∣ A − B ∣ ∣ γ ∣ ) ( m − 1 ) ! , j ∈ { 1 , 2 , … , n } ; m = 2 , 3 , … .

Proof

Since f ( z ) = ( f 1 ( z ) , f 2 ( z ) , … , f n ( z ) ) ′ , then

D f ( z ) = ∂ f 1 ( z ) ∂ z 1 ⋯ ∂ f 1 ( z ) ∂ z j ⋯ ∂ f 1 ( z ) ∂ z n ⋮ ⋱ ⋮ ⋱ ⋮ ∂ f j ( z ) ∂ z 1 ⋯ ∂ f j ( z ) ∂ z j ⋯ ∂ f j ( z ) ∂ z n ⋮ ⋱ ⋮ ⋱ ⋮ ∂ f n ( z ) ∂ z 1 ⋯ ∂ f n ( z ) ∂ z j ⋯ ∂ f n ( z ) ∂ z n .

For ( 0 , … , z j , … , 0 ) ′ ∈ D p 1 , p 2 , … , p n , it is easy to see that D f ( ( 0 , … , z j , … , 0 ) ′ )

(18) = ∂ f 1 ( ( 0 , … , z j , … , 0 ) ′ ) ∂ z 1 ⋯ ∂ f 1 ( ( 0 , … , z j , … , 0 ) ′ ) ∂ z j ⋯ ∂ f 1 ( ( 0 , … , z j , … , 0 ) ′ ) ∂ z n ⋮ ⋱ ⋮ ⋱ ⋮ ∂ f j ( ( 0 , … , z j , … , 0 ) ′ ) ∂ z 1 ⋯ ∂ f j ( ( 0 , … , z j , … , 0 ) ′ ) ∂ z j ⋯ ∂ f j ( ( 0 , … , z j , … , 0 ) ′ ) ∂ z n ⋮ ⋱ ⋮ ⋱ ⋮ ∂ f n ( ( 0 , … , z j , … , 0 ) ′ ) ∂ z 1 ⋯ ∂ f n ( ( 0 , … , z j , … , 0 ) ′ ) ∂ z j ⋯ ∂ f n ( ( 0 , … , z j , … , 0 ) ′ ) ∂ z n .

Since a q j m = 0 ( q ≠ j ) , it follows that

(19) ∂ f q ( ( 0 , … , z j , … , 0 ) ′ ) ∂ z j = 0 , q = 1 , 2 , … , n , q ≠ j .

From (18) and (19), then D f ( ( 0 , … , z j , … , 0 ) ′ )

(20) = ∂ f 1 ( ( 0 , … , z j , … , 0 ) ′ ) ∂ z 1 ⋯ 0 ⋯ ∂ f 1 ( ( 0 , … , z j , … , 0 ) ′ ) ∂ z n ⋮ ⋱ ⋮ ⋱ ⋮ ∂ f j ( ( 0 , … , z j , … , 0 ) ′ ) ∂ z 1 ⋯ ∂ f j ( ( 0 , … , z j , … , 0 ) ′ ) ∂ z j ⋯ ∂ f j ( ( 0 , … , z j , … , 0 ) ′ ) ∂ z n ⋮ ⋱ ⋮ ⋱ ⋮ ∂ f n ( ( 0 , … , z j , … , 0 ) ′ ) ∂ z 1 ⋯ 0 ⋯ ∂ f n ( ( 0 , … , z j , … , 0 ) ′ ) ∂ z n .

In view that D f ( ( 0 , … , z j , … , 0 ) ′ ) is invertible, thus, it implies that

(21) ∂ f j ( ( 0 , … , z j , … , 0 ) ′ ) ∂ z j ≠ 0 , j ∈ { 1 , 2 , … , n } .

Using (20) and (21), then we obtain

(22) ( D f ( ( 0 , … , z j , … , 0 ) ′ ) ) − 1 = ∗ ⋯ 0 ⋯ ∗ ⋮ ⋱ ⋮ ⋱ ⋮ ∗ ⋯ 1 ∂ f j ( ( 0 , … , z j , … , 0 ) ′ ) ∂ z j ⋯ ∗ ⋮ ⋱ ⋮ ⋱ ⋮ ∗ ⋯ 0 ⋯ ∗ ,

where the symbol ∗ means the unknown term. Furthermore, a simple computation in (22) shows that

( D f ( ( 0 , … , z j , … , 0 ) ′ ) ) − 1 f ( ( 0 , … , z j , … , 0 ) ′ ) = ∗ ⋯ 0 ⋯ ∗ ⋮ ⋱ ⋮ ⋱ ⋮ ∗ ⋯ 1 ∂ f j ( ( 0 , … , z j , … , 0 ) ′ ) ∂ z j ⋯ ∗ ⋮ ⋱ ⋮ ⋱ ⋮ ∗ ⋯ 0 ⋯ ∗ 0 ⋮ f j ( ( 0 , … , z j , … , 0 ) ′ ) ⋮ 0 = 0 ⋮ f j ( ( 0 , … , z j , … , 0 ) ′ ) ∂ f j ( ( 0 , … , z j , … , 0 ) ′ ) ∂ z j ⋮ 0 .

Let ( D f ( z ) − 1 f ( z ) ) = W ( z ) = ( W 1 ( z ) , … , W j ( z ) , … , W n ( z ) ) ′ and h j ( z j ) = f j ( ( 0 , … , z j , … , 0 ) ′ ) ∈ H ( D ) , z j ∈ D . Since f ( z ) ∈ S γ , A , B ∗ ( D p 1 , p 2 , … , p n ) and z = 0 is a zero of order k + 1 of f ( z ) − z , it follows that

(23) 1 + 1 γ z j h j ′ ( z j ) h j ( z j ) − 1 = 1 + 1 γ z j ∂ f j ( ( 0 , … , z j , … , 0 ) ′ ) ∂ z j f j ( ( 0 , … , z j , … , 0 ) ′ ) − 1 = 1 + 1 γ ρ ( ( 0 , … , z j , … , 0 ) ′ ) 2 ∂ ρ ( ( 0 , … , z j , … , 0 ) ′ ) ∂ z j W ( ( 0 , … , z j , … , 0 ) ′ ) − 1 ∈ g ( D ) , z j ≠ 0 .

From (23), we find that h j ∈ S γ , A , B ∗ ( D ) , and z j = 0 is a zero of order k ˜ + 1 of h j ( z j ) − z j , where k ˜ ≥ k . We note that

(24) a j j m = h j ( 0 ) ( m ) m ! , m = 2 , 3 , … .

Thus, in view of Lemma 2.2 and (24), we obtain the desired results.

Furthermore, let 0 < γ ≤ 1 and

(25) f ( z ) = z 1 ( 1 − z 1 k ) 2 ( 1 − α ) γ k , z 2 ( 1 − z 2 k ) 2 ( 1 − α ) γ k , … , z n ( 1 − z n k ) 2 ( 1 − α ) γ k ′ , z = ( z 1 , z 2 , … z n ) ∈ D p 1 , p 2 , … , p n \ { 0 } .

It is easy to check that f ∈ S γ , 2 α − 1 , 1 ∗ ( D p 1 , p 2 , … , p n ) . Thus, we have

f j ( ( 0 , 0 , … , z j , … , 0 ) ′ ) = z j + ∑ s = 1 ∞ a j j s k + 1 z j s k + 1 , j ∈ { 1 , 2 , … , n }

and

∣ a j j s k + 1 ∣ = ∏ μ = 1 s ( ( μ − 1 ) k + 2 γ ( 1 − α ) ) s ! k s , s = 1 , 2 , … , j ∈ { 1 , 2 , … , n } .

This completes the proof.□

Theorem 3.2

Suppose that f ( z ) = ( f 1 ( z ) , f 2 ( z ) , … , f n ( z ) ) ′ ∈ S γ , A , B ∗ ( D p 1 , p 2 , … , p n ) and z = 0 is a zero of order k + 1 of f ( z ) − z . Define

D m f q ( 0 ) ( z m ) m ! = ∑ l 1 , l 2 , … , l m = 1 n a q l 1 , l 2 , … , l m z l 1 z l 2 ⋯ z l m ,

where q = 1 , 2 , … , n , s k + 1 ≤ m ≤ ( s + 1 ) k , s = 1 , 2 , … , and k ∈ N + . Let a t q m be the a t q t , t ⋯ , t ︸ m − 1 ( t = { 1 , 2 , … , n } ) . If a j q m = 0  ( q ≠ j ), then we have

∣ a j j m ∣ ≤ ∏ μ = 1 s ( ( μ − 1 ) k + ∣ A − B ∣ ∣ γ ∣ ) s ! k s , s k + 1 ≤ m ≤ ( s + 1 ) k , s = 1 , 2 , … , j ∈ { 1 , 2 , … , n } .

The aforementioned estimates are sharp for 0 < γ ≤ 1 , A = − 1 , B = 1 , m = s k + 1 , s = 1 , 2 , … and 0 < γ ≤ 1 , A = 2 α − 1 ( α ∈ [ 0 , 1 ) ) , B = 1 , m = s k + 1 , s = 1 , 2 , … . In particular, if k = 1 , then

∣ a j j m ∣ ≤ ∏ μ = 0 m − 2 ( μ + ∣ A − B ∣ ∣ γ ∣ ) ( m − 1 ) ! , j ∈ { 1 , 2 , … , n } ; m = 2 , 3 , … .

Proof

Since a j q m = 0  ( q ≠ j ), it follows that

(26) ∂ f j ( ( 0 , … , z j , … , 0 ) ′ ) ∂ z q = 0 .

Using (18) and (26), we have D f ( ( 0 , … , z j , … , 0 ) ′ )

(27) = ∂ f 1 ( ( 0 , … , z j , … , 0 ) ′ ) ∂ z 1 ⋯ ∂ f 1 ( ( 0 , … , z j , … , 0 ) ′ ) ∂ z j ⋯ ∂ f 1 ( ( 0 , … , z j , … , 0 ) ′ ) ∂ z n ⋮ ⋱ ⋮ ⋱ ⋮ 0 ⋯ ∂ f j ( ( 0 , … , z j , … , 0 ) ′ ) ∂ z j ⋯ 0 ⋮ ⋱ ⋮ ⋱ ⋮ ∂ f n ( ( 0 , … , z j , … , 0 ) ′ ) ∂ z 1 ⋯ ∂ f n ( ( 0 , … , z j , … , 0 ) ′ ) ∂ z j ⋯ ∂ f n ( ( 0 , … , z j , … , 0 ) ′ ) ∂ z n .

In view that D f ( ( 0 , … , z j , … , 0 ) ′ ) is invertible, thus, it implies that

(28) ∂ f j ( ( 0 , … , z j , … , 0 ) ′ ) ∂ z j ≠ 0 .

Thus, (27) and (28) show that

(29) ( D f ( ( 0 , … , z j , … , 0 ) ′ ) ) − 1 = ∗ ⋯ ∗ ⋯ ∗ ⋮ ⋱ ⋮ ⋱ ⋮ 0 ⋯ 1 ∂ f j ( ( 0 , … , z j , … , 0 ) ′ ) ∂ z j ⋯ 0 ⋮ ⋱ ⋮ ⋱ ⋮ ∗ ⋯ ∗ ⋯ ∗ ,

where the symbol ∗ means the unknown term. Furthermore, a simple computation in (29) shows that

( D f ( ( 0 , … , z j , … , 0 ) ′ ) ) − 1 f ( ( 0 , … , z j , … , 0 ) ′ ) = ∗ ⋯ ∗ ⋯ ∗ ⋮ ⋱ ⋮ ⋱ ⋮ 0 ⋯ 1 ∂ f j ( ( 0 , … , z j , … , 0 ) ′ ) ∂ z j ⋯ 0 ⋮ ⋱ ⋮ ⋱ ⋮ ∗ ⋯ ∗ ⋯ ∗ f 1 ( ( 0 , … , z j , … , 0 ) ′ ) ⋮ f j ( ( 0 , … , z j , … , 0 ) ′ ) ⋮ f n ( ( 0 , … , z j , … , 0 ) ′ ) = ∗ ⋮ f j ( ( 0 , … , z j , … , 0 ) ′ ) ∂ f j ( ( 0 , … , z j , … , 0 ) ′ ) ∂ z j ⋮ ∗ .

Define ( D f ( z ) − 1 f ( z ) ) = W ( z ) = ( W 1 ( z ) , … , W j ( z ) , … , W n ( z ) ) ′ and h j ( z j ) = f j ( ( 0 , … , z j , … , 0 ) ′ ) ∈ H ( D ) . Since f ( z ) ∈ S γ , A , B ∗ ( D p 1 , p 2 , … , p n ) and z = 0 is a zero of order k + 1 of f ( z ) − z , it follows that

(30) 1 + 1 γ z j h j ′ ( z j ) h j ( z j ) − 1 = 1 + 1 γ z j ∂ f j ( ( 0 , … , z j , … , 0 ) ′ ) ∂ z j f j ( ( 0 , … , z j , … , 0 ) ′ ) − 1 = 1 + 1 γ ρ ( ( 0 , … , z j , … , 0 ) ′ ) 2 ∂ ρ ( ( 0 , … , z j , … , 0 ) ′ ) ∂ z j W ( ( 0 , … , z j , … , 0 ) ′ ) − 1 ∈ g ( D ) , z j ≠ 0 .

From (30), we find that h j ∈ S γ , A , B ∗ ( D ) , and z j = 0 is a zero of order k ˜ + 1 of h j ( z j ) − z j , where k ˜ ≥ k . We note that

(31) a j j m = h j ( m ) ( 0 ) m ! , m = 2 , 3 , … .

Thus, in view of Lemma 2.2 and (31), we obtain the desired results. We note that the sharpness of the estimates of Theorem 3.2 is similar to that of Theorem 3.1. This completes the proof.□

Remark 3.1

  1. If n = 1 , then Theorem 3.1 (resp. Theorem 3.2) reduces to Lemma 2.2.

  2. If γ = 1 , A = 2 α − 1 ( α ∈ [ 0 , 1 ) ) , and B = 1 in Theorems 3.1 and 3.2, then we obtain Theorems 3.1 and 3.2 in Liu et al. [32], respectively.

  3. If we take different functions g in Theorem 3.1 (resp. Theorem 3.2), then the bounds of homogeneous expansions for kinds of subclasses of g -starlike mappings of complex order γ defined on D p 1 , p 2 , … , p n can be obtained immediately.

Acknowledgement

The authors would like to thank the referees for their helpful comments.

  1. Funding information: The project is supported by the National Natural Science Foundation of China (12071354, 12061035), Jiangxi Provincial Natural Science Foundation (20212BAB201012), Research Foundation of Jiangxi Provincial Department of Education (GJJ201104), and Research Foundation of Jiangxi Science and Technology Normal University (2021QNBJRC003).

  2. Author contributions: All authors have accepted responsibility for the entire content of this manuscript and approved its submission.

  3. Conflict of interest: The authors state no conflict of interest.

  4. Data availability statement: Data sharing is not applicable to this article as no datasets were generated or analyzed during this current study.

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Received: 2023-02-02
Revised: 2023-04-19
Accepted: 2023-04-25
Published Online: 2023-07-11

© 2023 the author(s), published by De Gruyter

This work is licensed under the Creative Commons Attribution 4.0 International License.

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https://doi.org/10.1515/dema-2022-0242
Keywords for this article
biholomorphic mappings; coefficient problems; Reinhardt domain; g-starlike mappings of complex order gamma
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  29. Regularity criteria via horizontal component of velocity for the Boussinesq equations in anisotropic Lorentz spaces
  30. Generalized Stević-Sharma operators from the minimal Möbius invariant space into Bloch-type spaces
  31. On initial value problem for elliptic equation on the plane under Caputo derivative
  32. A dimension expanded preconditioning technique for block two-by-two linear equations
  33. Asymptotic behavior of Fréchet functional equation and some characterizations of inner product spaces
  34. Small perturbations of critical nonlocal equations with variable exponents
  35. Dynamical property of hyperspace on uniform space
  36. Some notes on graded weakly 1-absorbing primary ideals
  37. On the problem of detecting source points acting on a fluid
  38. Integral transforms involving a generalized k-Bessel function
  39. Ruled real hypersurfaces in the complex hyperbolic quadric
  40. On the monotonic properties and oscillatory behavior of solutions of neutral differential equations
  41. Approximate multi-variable bi-Jensen-type mappings
  42. Mixed-type SP-iteration for asymptotically nonexpansive mappings in hyperbolic spaces
  43. On the equation fn + (f″)m ≡ 1
  44. Results on the modified degenerate Laplace-type integral associated with applications involving fractional kinetic equations
  45. Characterizations of entire solutions for the system of Fermat-type binomial and trinomial shift equations in ℂn#
  46. Commentary
  47. On I. Meghea and C. S. Stamin review article “Remarks on some variants of minimal point theorem and Ekeland variational principle with applications,” Demonstratio Mathematica 2022; 55: 354–379
  48. Special Issue on Fixed Point Theory and Applications to Various Differential/Integral Equations - Part II
  49. On Cauchy problem for pseudo-parabolic equation with Caputo-Fabrizio operator
  50. Fixed-point results for convex orbital operators
  51. Asymptotic stability of equilibria for difference equations via fixed points of enriched Prešić operators
  52. Asymptotic behavior of resolvents of equilibrium problems on complete geodesic spaces
  53. A system of additive functional equations in complex Banach algebras
  54. New inertial forward–backward algorithm for convex minimization with applications
  55. Uniqueness of solutions for a ψ-Hilfer fractional integral boundary value problem with the p-Laplacian operator
  56. Analysis of Cauchy problem with fractal-fractional differential operators
  57. Common best proximity points for a pair of mappings with certain dominating property
  58. Investigation of hybrid fractional q-integro-difference equations supplemented with nonlocal q-integral boundary conditions
  59. The structure of fuzzy fractals generated by an orbital fuzzy iterated function system
  60. On the structure of self-affine Jordan arcs in ℝ2
  61. Solvability for a system of Hadamard-type hybrid fractional differential inclusions
  62. Three solutions for discrete anisotropic Kirchhoff-type problems
  63. On split generalized equilibrium problem with multiple output sets and common fixed points problem
  64. Special Issue on Computational and Numerical Methods for Special Functions - Part II
  65. Sandwich-type results regarding Riemann-Liouville fractional integral of q-hypergeometric function
  66. Certain aspects of Nörlund ℐ-statistical convergence of sequences in neutrosophic normed spaces
  67. On completeness of weak eigenfunctions for multi-interval Sturm-Liouville equations with boundary-interface conditions
  68. Some identities on generalized harmonic numbers and generalized harmonic functions
  69. Study of degenerate derangement polynomials by λ-umbral calculus
  70. Normal ordering associated with λ-Stirling numbers in λ-shift algebra
  71. Analytical and numerical analysis of damped harmonic oscillator model with nonlocal operators
  72. Compositions of positive integers with 2s and 3s
  73. Kinematic-geometry of a line trajectory and the invariants of the axodes
  74. Hahn Laplace transform and its applications
  75. Discrete complementary exponential and sine integral functions
  76. Special Issue on Recent Methods in Approximation Theory - Part II
  77. On the order of approximation by modified summation-integral-type operators based on two parameters
  78. Bernstein-type operators on elliptic domain and their interpolation properties
  79. A class of strongly convergent subgradient extragradient methods for solving quasimonotone variational inequalities
  80. Special Issue on Recent Advances in Fractional Calculus and Nonlinear Fractional Evaluation Equations - Part II
  81. Application of fractional quantum calculus on coupled hybrid differential systems within the sequential Caputo fractional q-derivatives
  82. On some conformable boundary value problems in the setting of a new generalized conformable fractional derivative
  83. A certain class of fractional difference equations with damping: Oscillatory properties
  84. Weighted Hermite-Hadamard inequalities for r-times differentiable preinvex functions for k-fractional integrals
  85. Special Issue on Recent Advances for Computational and Mathematical Methods in Scientific Problems - Part II
  86. The behavior of hidden bifurcation in 2D scroll via saturated function series controlled by a coefficient harmonic linearization method
  87. Phase portraits of two classes of quadratic differential systems exhibiting as solutions two cubic algebraic curves
  88. Petri net analysis of a queueing inventory system with orbital search by the server
  89. Asymptotic stability of an epidemiological fractional reaction-diffusion model
  90. On the stability of a strongly stabilizing control for degenerate systems in Hilbert spaces
  91. Special Issue on Application of Fractional Calculus: Mathematical Modeling and Control - Part I
  92. New conticrete inequalities of the Hermite-Hadamard-Jensen-Mercer type in terms of generalized conformable fractional operators via majorization
  93. Pell-Lucas polynomials for numerical treatment of the nonlinear fractional-order Duffing equation
  94. Impacts of Brownian motion and fractional derivative on the solutions of the stochastic fractional Davey-Stewartson equations
  95. Some results on fractional Hahn difference boundary value problems
  96. Properties of a subclass of analytic functions defined by Riemann-Liouville fractional integral applied to convolution product of multiplier transformation and Ruscheweyh derivative
  97. Special Issue on Development of Fuzzy Sets and Their Extensions - Part I
  98. The cross-border e-commerce platform selection based on the probabilistic dual hesitant fuzzy generalized dice similarity measures
  99. Comparison of fuzzy and crisp decision matrices: An evaluation on PROBID and sPROBID multi-criteria decision-making methods
  100. Rejection and symmetric difference of bipolar picture fuzzy graph
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