Pluriharmonic Mappings with the Convex Holomorphic Part

Abstract

In 2018, Partyka et al. established several equivalent conditions for a sense-preserving locally injective harmonic mapping in the unit disk with convex holomorphic part to be quasiconformal in terms of the relationships of two-point distortion of , , and . In this study, we first generalize the above result to the case of pluriharmonic mappings , where is a convex mapping in the unit ball and with . Then, we establish a relationship of two-point distortion property between and .

1. Introduction and Main Results

For , let denote the -dimensional complex Euclidean space. Also, we identify each point with a column vector. For two column vectors , set and . We use and to denote the open ball and its boundary , respectively. In particular, let and . Also, we identify with the unit disk .

For an complex matrix , the operator norm of is defined by

We use to denote the space of continuous linear operators from into with the standard operator norm, and let be the identity operator in .

For two domains , let be a holomorphic mapping from into . Then, the complex Jacobian matrix of at is given bywhere means the matrix transpose, and are understood as column vectors. Furthermore, let be the conjugate of Jacobian matrix as follows:

If for every , then we say that is locally biholomorphic in (cf. [1]). If is one-to-one, onto and locally biholomorphic, then is said to be biholomorphic (cf. ([2], Page 55)).

A complex-valued function of class in is said to be pluriharmonic if its restriction to every complex line is harmonic, which is equivalent to the fact that for all and , ,

Every pluriharmonic mapping can be written as , where , are the holomorphic mappings, and this representation is unique if (cf. [1–5]).

If is pluriharmonic and is locally biholomorphic, we denote byand we use to denote the real Jacobian matrix of (cf. [4]). Then, for any ,and

Hence, is sense-preserving, i.e., in , if and only if is locally biholomorphic, and .

If is a sense-preserving harmonic mapping from into , it is known that and the dilation is analytic with the property that (see [6, 7]). Especially, if is a diffeomorphism with , then is called a quasiconformal mapping.

A domain is said to be -linearly connected if there is a constant such that any two points , can be connected by a smooth curve with length

It is clear that any convex domain is 1-linearly connected. For extensive discussion on linearly connected domains, see [4, 8–12]. For a biholomorphic mapping , if maps onto a convex domain, then we say that is convex in (cf. [13]).

For sense-preserving harmonic mapping defined on , Chuaqui and Hérnandez [10] showed that if is -linearly connected and , then the deformation , , is univalent. Kalaj [14] proved a more general result when is convex that for every with , is an -quasiconformal close-to-convex harmonic mapping.

We say that a mapping : is in if there exist a constant and an exponent such that for all , ,

Such mappings are also called -Hölder continuous. In particular, if , then we say that is Lipschitz continuous. A mapping : is said to be co-Lipschitz continuous if there exists a constant such that for all , ,

If is both Lipschitz continuous and co-Lipschitz continuous in , then is called bi-Lipschitz.

In 2012, Partyka and Sakan established several equivalent conditions for a sense-preserving harmonic mapping from onto a bounded convex domain to be quasiconformal in terms of the relationships of two-point distortion of , , and (see [15], Theorem 3.8]). Later, Partyka et al. [12] generalized the result to the case where is a sense-preserving and locally injective harmonic mapping and is a convex holomorphic mapping.

Theorem 1 (see [12], Theorem 3.3). Let be a sense-preserving harmonic mapping in such that is convex. Then, is injective, and the following five conditions are equivalent to each other:(1) is a quasiconformal mapping(2)There exists a constant such that and(3)There exists a constant such that and(4)There exists a constant such that(5) and are bi-Lipschitz mappings.Let be a pluriharmonic mapping in . For simplicity, here and hereafter, we always use to denote the pluriharmonic mapping , where with . Obviously, .
As the first aim of this study, we establish the following counterpart of ([12], Theorem 3.3) in the setting of pluriharmonic mappings.

Theorem 2. Let be a pluriharmonic mapping, where is convex in and . Then, the following five statements are equivalent:(i)There exists a constant such that (ii)There exists a constant such that for any and with ,(iii)There exists a constant such that for any ,(iv)There exists a constant such that for any and with ,(v)For any with , and are bi-Lipschitz mappings.

As the second aim of this study, we establish a relationship of two-point distortion property between and . Our result is as follows.

Theorem 3. Let be a pluriharmonic mapping, where is biholomorphic and is -linearly connected with constant . Suppose that there exists a constant such that . Then, for any , and with , there exist two positive constants and such thatwhere and . In particular,

The proofs of Theorems 2 and 3 will be given in Section 2.

2. Proofs of Main Results

The aim of this section is to prove Theorems 2 and 3. Before proving Theorem 2, we need some preparation, which consists of three lemmas.

Lemma 1. Let be a pluriharmonic mapping, where is biholomorphic and is -linearly connected with constant . Suppose that there exists a constant such that . Then, for any , ,and for any with ,

Proof. The proof of (19) is based upon the ideas from Theorem 2.1 [4]. The details are as follows.
For any distinct points , , let and . Then, the -linear connectivity of implies that there exists a smooth curve between and such that , , and . Since is a biholomorphic mapping, we see that is a curve in joining and . Then, the assumption implieswhich yields (19). Moreover, for any with , we have thatand so the proof of this lemma is complete.
The following result is a converse of Lemma 1.

Lemma 2. Let be a pluriharmonic mapping, where is biholomorphic. Suppose that and there exists a constant such that for any , and with ,Then, .

Proof. For any distinct points , let and , where and . Then, for any , we havewhere denotes the distance of to the boundary of and denotes a vector in with . It follows from (23) and (24) thatFor fixed , we choose some such thatThen, there exists such thatChoose satisfying andThen, it follows from (25), (27) and (28) thatwhich, together with the arbitrary of , shows that , as needed.

Lemma 3. Let be a pluriharmonic mapping, where is biholomorphic and is -linearly connected with constant . Suppose that there exists a constant such that for any , and with ,Then,

Proof. By Lemma 1, we know that to prove (31), it suffices to proveFor any distinct points , let and , where and . Then, for any , it follows from (24) and (30) thatFor fixed , similar to (27), we know that there exist some and such thatChoose satisfying andCombining (33)–(35) and letting , we get , as required. In addition, the inequality (31) can be derived from the proof of ([4], Theorem 1.2).
Now, we are ready to prove Theorem 2.

Proof of Theorem 2. Since every convex domain is a 1-linearly connected domain, the implication from Statement (i) to (ii) follows from Lemma 1, and the implication from Statement (ii) to (iii) follows from Lemma 3. Furthermore, the implication from Statement (iii) to (iv) follows from the triangle inequality since the assumption that for any implieswhere and with . The implication from Statement (v) to (i) follows from Lemma 2. Hence, to prove the theorem, it remains to prove the implication from Statement (iv) to (v).
Assume that Statement (iv) holds true, which means that there exists a constant such that for any and with ,Then, we infer from Lemma 2 that . This, together with Lemma 1, implies thatNote that the assumption “ is convex in ” implies that is a biholomorphic mapping. This, together with the above two inequalities, yields that both and are univalent in andLet and . Obviously, we have thatwhich means that is bi-Lipschitz continuous in . Similarly, we see that is bi-Lipschitz continuous in . These show that Statement (v) is true. The proof of the theorem is complete.
In order to prove Theorem 3, we also need some preparation. First, let us recall a known result, which is useful for the Proof of Theorem 3.

Theorem 4 (see [3], Lemma A). Let be an complex matrix with . Then, are nonsingular matrices and

Proof. For a complex matrix A, any kind of operator norm has the property. Let be any nonzero vector, thenIf , then . So, for , no more than zero solution, then the matrix is nonsingular.
When is nonsingular, we haveNow,

Lemma 4. Let be a univalent pluriharmonic mapping, where is locally biholomorphic and is -linearly connected with constant . Suppose that there exists a constant such that . Then, for any , ,

Proof. The proof of the lemma is based upon the ideas from ([4], Theorem 2.3). The details are as follows.
First, we prove inequality (45). For any distinct points , it follows from the -linear connectivity of that there exists a smooth curve connecting and with , , and . Since is univalent, we assume that and , and hence is a curve in joining and .
By the assumption , the inverse mapping theorem and Theorem 4, we know that is differentiable (cf. [3]). Moreover, by [3], (28), we haveFor , let . Therefore, it follows from (47) and (48) thatFurthermore, by Theorem 4, the assumption and the -linear connectivity of , we know thatwhich yields the inequality (45).
Similarly, by applying (47), (48), Theorem 4, the assumption and the -linear connectivity of , we getwhich leads to (46), and thus, the proof of this lemma is complete.
Based on Lemma 4, we have the following result.

Lemma 5. Let be a pluriharmonic mapping, where is biholomorphic and is -linearly connected with constant . Suppose that there exists a constant such that . Then, is a univalent and sense-preserving mapping, and for any , ,

Proof. By the proof of Theorem 2.1 [4], we know that is a univalent and sense-preserving mapping and is a -linearly connected domain. Then, the result of this lemma follows from Lemma 4.
Based on Lemmas 1 and 5, we can give the Proof of Theorem 3.

Proof of Theorem 2. The inequalities in (18) follows from (17), (20), and (52). Hence, it remains to prove the inequalities in (17).
For any distinct points , and with , we see from Lemma 5 thatOn the other hand, by Lemma 1 and the assumption “,” we getHence, (17) follows, and the proof of this theorem is complete [16].

Data Availability

No data are available for this article.

Conflicts of Interest

The authors declare that they have no conflicts of interest.

Acknowledgments

The first author is partly supported by the NSFS of China (Grant nos. 11822105 and 11801166).