# Pluriharmonic Mappings with the Convex Holomorphic Part

Authors: Ma Lihua; See Keong Lee
Journal: Journal of Mathematics
 (2022)

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## Abstract

In 2018, Partyka et al. established several equivalent conditions for a sense-preserving locally injective harmonic mappingf=h+g¯ in the unit disk D with convex holomorphic part h to be quasiconformal in terms of the relationships of two-point distortion of h, g, and f. In this study, we first generalize the above result to the case of pluriharmonic mappings fA=h+Ag¯, where h is a convex mapping in the unit ball Bn and A∈Lℂn,ℂn with A=1. Then, we establish a relationship of two-point distortion property between f and fA.


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## Body

## 1. Introduction and Main Results

Forn≥1, let ℂn denote the n-dimensional complex Euclidean space. Also, we identify each point z=z1,…,zn∈ℂn with a column vector. For two column vectors z,w∈ℂn, set z,w=∑j=1nzjw¯j and z=z,z1/2. We use Bnz0,r and Sn−1z0,r to denote the open ball z∈ℂn:z−z0<r and its boundary z∈ℂn:z−z0=r, respectively. In particular, let Bn0,1=Bn and Sn−10,1=Sn−1. Also, we identify B1 with the unit disk D.For ann×n complex matrix A, the operator norm of A is defined by(1)A=supAξ:ξ∈Sn−1.We useLℂn,ℂm to denote the space of continuous linear operators from ℂn into ℂm with the standard operator norm, and let In be the identity operator in Lℂn,ℂn.For two domainsΩ1,Ω2⊂ℂn, let f=f1,…,fn be a holomorphic mapping from Ω1 into Ω2. Then, the complex Jacobian matrix of f at z∈Ω1 is given by(2)Dfz=∂f1z∂z1⋯∂f1z∂zn⋮⋱⋮∂fnz∂z1⋯∂fnz∂zn=∇f1z,…,∇fnzT,where T means the matrix transpose, and ∇fj are understood as column vectors. Furthermore, let D¯fz be the conjugate of Jacobian matrix Dfzas follows:(3)D¯fz=∂f1z∂z¯1⋯∂f1z∂z¯n⋮⋱⋮∂fnz∂z¯1⋯∂fnz∂z¯n.IfdetDfz≠0 for every z∈Ω1, then we say that f is locally biholomorphic in Ω1 (cf. [1]). If f is one-to-one, onto and locally biholomorphic, then f is said to be biholomorphic (cf. ([2], Page 55)).A complex-valued functionf of class C2 in Bn is said to be pluriharmonic if its restriction to every complex line is harmonic, which is equivalent to the fact that for all z∈Bn and j, k∈1,2,…,n,(4)∂2∂zj∂z¯kfz≡0.Every pluriharmonic mappingf:Bn⟶ℂn can be written as f=h+g¯, where h, g are the holomorphic mappings, and this representation is unique if g0=0 (cf. [1–5]).Iff=h+g¯:Bn⟶ℂn is pluriharmonic and h is locally biholomorphic, we denote by(5)ωfz=DgzDhz−1,ωf∞=supz∈Bnωfz,and we use Jf to denote the real Jacobian matrix of f (cf. [4]). Then, for any z∈Bn,(6)Jfz=DhzDgz¯DgzDhz¯,and(7)detJfz=detDhz2detIn−ωfzωfz¯.Hence,f is sense-preserving, i.e., detJfz>0 in Bn, if and only if h is locally biholomorphic, and detIn−ωfzωfz¯>0.Iff=h+g¯ is a sense-preserving harmonic mapping from D into ℂ, it is known that detJfz=h′z2−g′z2 and the dilation ωfz=g′z/h′z is analytic with the property that ωfz<1 (see [6, 7]). Especially, if f is a diffeomorphism with ωf∞<1, then f is called a quasiconformal mapping.A domainΩ⊂ℂn is said to be M-linearly connected if there is a constant M>0 such that any two points w1, w2∈Ω can be connected by a smooth curve γ⊂Ω with length(8)lγ≤Mw1−w2.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 f, if f maps Bn onto a convex domain, then we say that f is convex in Bn (cf. [13]).For sense-preserving harmonic mappingf=h+g¯ defined on D, Chuaqui and Hérnandez [10] showed that if fD is M-linearly connected and ωf<1/1+2M, then the deformation F=h+ag¯, a=1, is univalent. Kalaj [14] proved a more general result when fD is convex that for every a with a≤1, F is an a-quasiconformal close-to-convex harmonic mapping.We say that a mappingf: Ω⟶ℂn is in Lipα if there exist a constant c1 and an exponent α∈0,1 such that for all z, w∈Ω,(9)fz−fw≤c1z−wα.Such mappings are also calledα-Hölder continuous. In particular, if α=1, then we say that f is Lipschitz continuous. A mapping f: Ω⟶ℂn is said to be co-Lipschitz continuous if there exists a constant c2 such that for all z, w∈Ω,(10)fz−fw≥c2z−w.Iff is both Lipschitz continuous and co-Lipschitz continuous in Ω, then f is called bi-Lipschitz.In 2012, Partyka and Sakan established several equivalent conditions for a sense-preserving harmonic mappingf=h+g¯ from D onto a bounded convex domain to be quasiconformal in terms of the relationships of two-point distortion of h, g, and f (see [15], Theorem 3.8]). Later, Partyka et al. [12] generalized the result to the case where f is a sense-preserving and locally injective harmonic mapping and h is a convex holomorphic mapping.Theorem 1 (see [12], Theorem 3.3).
Letf=h+g¯ be a sense-preserving harmonic mapping in D such that h is convex. Then, f is injective, and the following five conditions are equivalent to each other:(1)
f is a quasiconformal mapping(2)
There exists a constantL1 such that L1∈1,2 and(11)fz2−fz1≤L1hz2−hz1,z1,z2∈D.(3)
There exists a constantl1 such that l1∈0,1 and(12)gz2−gz1≤l1hz2−hz1,z1,z2∈D.(4)
There exists a constantL2≥1 such that(13)hz2−hz1≤L2fz2−fz1,z1,z2∈D.(5)
h°f−1 and f°h−1 are bi-Lipschitz mappings.
Letf=h+g¯ be a pluriharmonic mapping in Bn. For simplicity, here and hereafter, we always use fA to denote the pluriharmonic mapping h+Ag¯, where A∈Lℂn,ℂn with A≤1. Obviously, ωfAz=A¯ωfz.
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.
Letf=h+g¯:Bn⟶ℂn be a pluriharmonic mapping, where h is convex in Bn and ωf∞≤1. Then, the following five statements are equivalent:(i)
There exists a constantN such that ωf∞≤N<1(ii)
There exists a constantL1∈1,2 such that for any z1,z2∈Bn and A∈Lℂn,ℂn with A=1,(14)fAz1−fAz2≤L1hz1−hz2.(iii)
There exists a constantl1∈0,1 such that for any z1,z2∈Bn,(15)gz1−gz2≤l1hz1−hz2.(iv)
There exists a constantL2≥1 such that for any z1,z2∈Bn and A∈Lℂn,ℂn with A=1,(16)hz1−hz2≤L2fAz1−fAz2.(v)
For anyA∈Lℂn,ℂn with A=1, h°fA−1 and fA°h−1 are bi-Lipschitz mappings.As the second aim of this study, we establish a relationship of two-point distortion property betweenf and fA. Our result is as follows.Theorem 3.
Letf=h+g¯:Bn⟶ℂn be a pluriharmonic mapping, where h is biholomorphic and hBn is M-linearly connected with constant M≥1. Suppose that there exists a constant N∈0,1/M such that ωf∞≤N. Then, for any z1,z2∈Bn and A∈Lℂn,ℂn with A≤1, there exist two positive constants c1 and c2 such that(17)c2fz1−fz2≤fAz1−fAz2≤c1fz1−fz2,where c1=M1+N2/1−N1−MN and c2=1−MN/1+MN. In particular,(18)1−MN1+MN2fz1−fz2≤hz1−hz2≤M1+N1−N1−MNfz1−fz2.The proofs of Theorems2 and 3 will be given in Section 2.


## 2. Proofs of Main Results

The aim of this section is to prove Theorems2 and 3. Before proving Theorem 2, we need some preparation, which consists of three lemmas.Lemma 1.
Letf=h+g¯:Bn⟶ℂn be a pluriharmonic mapping, where h is biholomorphic and hBn is M-linearly connected with constant M≥1. Suppose that there exists a constant N∈0,1 such that ωf∞≤N. Then, for any z1, z2∈Bn,(19)gz1−gz2≤MNhz1−hz2,and for any A∈Lℂn,ℂn with A≤1,(20)fAz1−fAz2≤1+MNhz1−hz2.Proof.
The proof of (19) is based upon the ideas from Theorem 2.1 [4]. The details are as follows.
For any distinct pointsz1, z2∈Bn, let w1=hz1 and w2=hz2. Then, the M-linear connectivity of hBn implies that there exists a smooth curve γ:0,1⟶hBn between w1 and w2 such that γ0=w1, γ1=w2, and lγ≤Mw1−w2. Since h is a biholomorphic mapping, we see that σ=h−1°γ is a curve in Bn joining z1 and z2. Then, the assumption ωf∞≤N implies(21)gz1−gz2=∫01Dg°σsds=∫01DgσsDhσs−1γ′sds≤N∫01dγs=Nlγ≤MNhz1−hz2,which yields (19). Moreover, for any A∈Lℂn,ℂn with A≤1, we have that(22)fAz1−fAz2≤hz1−hz2+A·gz1−gz2≤1+MNhz1−hz2,and so the proof of this lemma is complete.
The following result is a converse of Lemma1.Lemma 2.
Letf=h+g¯:Bn⟶ℂn be a pluriharmonic mapping, where h is biholomorphic. Suppose that ωf∞≤1 and there exists a constant N≥1 such that for any z1, z2∈Bn and A∈Lℂn,ℂn with A=1,(23)hz1−hz2≤NfAz1−fAz2.
Then,ωf∞≤1−1/N.Proof.
For any distinct pointsw1,w2∈hBn, let z1=h−1w1 and z2=h−1w2, where z1,z2∈Bn and z1≠z2. Then, for any w1∈Bnw2,dhBnw2, we have(24)gh−1w1−gh−1w2=Dg°h−1w2·w1−w2+ow1−w2=Dgz2Dhz2−1·w1−w2+ow1−w2,where dhBnw2 denotes the distance of w2 to the boundary of hBn and ow1−w2 denotes a vector in ℂn with limw1⟶w2ow1−w2/w1−w2=0. It follows from (23) and (24) that(25)w1−w2≤Nw1−w2+Agh−1w−gh−1w2¯≤Nw1¯−w2¯+A¯Dgz2Dhz2−1·w1−w2+ow1−w2.
For fixedz2∈Bn, we choose some w1∈Bnw2,dhBnw2\w2 such that(26)Dgz2Dhz2−1·w1−w2w1−w2=Dgz2Dhz2−1.
Then, there existsη∈Sn−1 such that(27)Dgz2Dhz2−1·w1−w2w1−w2=η·Dgz2Dhz2−1.
ChooseA∈Lℂn,ℂn satisfying A=1 and(28)Aη¯=−w1−w2w1−w2.
Then, it follows from (25), (27) and (28) that(29)1N≤limw1⟶w2w1¯−w2¯w1−w2−w1¯−w2¯w1−w2·Dgz2Dhz2−1+ow1−w2w1−w2=1−Dgz2Dhz2−1,which, together with the arbitrary of z2, shows that ωf∞≤1−1/N, as needed.Lemma 3.
Letf=h+g¯:Bn⟶ℂn be a pluriharmonic mapping, where h is biholomorphic and hBn is M-linearly connected with constant M≥1. Suppose that there exists a constant N∈0,1 such that for any z1, z2∈Bn and A∈Lℂn,ℂn with A=1,(30)fAz1−fAz2≤1+Nhz1−hz2.
Then,(31)gz1−gz2≤MNhz1−hz2.Proof.
By Lemma1, we know that to prove (31), it suffices to prove(32)ωf∞≤N.
For any distinct pointsw1,w2∈hBn, let z1=h−1w1 and z2=h−1w2, where z1,z2∈Bn and z1≠z2. Then, for any w1∈Bnw2,dhBnw2\w2, it follows from (24) and (30) that(33)w1−w2¯w1−w2+A¯Dgz2Dhz2−1·w1−w2w1−w2+ow1−w2w1−w2≤1+N.
For fixedz2∈Bn, similar to (27), we know that there exist some η∈Sn−1 and w1∈Bnw2,dhBnw2\w2 such that(34)Dgz2Dhz2−1·w1−w2w1−w2=η·Dgz2Dhz2−1.
ChooseA∈Lℂn,ℂn satisfying A=1 and(35)Aη¯=w1−w2w1−w2.
Combining (33)–(35) and letting w1⟶w2, we get ωf∞≤N, as required. In addition, the inequality (31) can be derived from the proof of ([4], Theorem 1.2).
Now, we are ready to prove Theorem2.Proof of Theorem 2.
Since every convex domain is a 1-linearly connected domain, the implication from Statement (i) to (ii) follows from Lemma1, 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 gz1−gz2≤l1hz1−hz2 for any z1,z2∈Bn implies(36)fAz1−fAz2≥hz1−hz2−A·gz1−gz2≥1−l1hz1−hz2,where l1∈0,1 and A∈Lℂn,ℂn with A=1. 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 constantL2≥1 such that for any z1,z2∈Bn and A∈Lℂn,ℂn with A=1,(37)hz2−hz1≤L2fAz2−fAz1.
Then, we infer from Lemma2 that ωf∞≤1−1/L2. This, together with Lemma 1, implies that(38)fAz1−fAz2≤2−1L2hz1−hz2.
Note that the assumption “h is convex in Bn” implies that h is a biholomorphic mapping. This, together with the above two inequalities, yields that both h and fA are univalent in Bn and(39)1L2hz1−hz2≤fAz1−fAz2≤2−1L2hz1−hz2.
Letw1=fAz1 and w2=fAz2. Obviously, we have that(40)1L2h°fA−1w1−h°fA−1w2≤w1−w2≤2−1L2h°fA−1w1−h°fA−1w2,which means that h°fA−1 is bi-Lipschitz continuous in fABn. Similarly, we see that fA°h−1 is bi-Lipschitz continuous in hBn. These show that Statement (v) is true. The proof of the theorem is complete.
In order to prove Theorem3, 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).
LetA be an n×n complex matrix with A<1. Then, In±A are nonsingular matrices and(41)In±A−1≤11−A.Proof.
For a complex matrixA, any kind of operator norm has the property. Let x be any nonzero vector, then(42)In−Ax=x−Ax≥x−Ax≥x−Ax=1−Ax>0.
Ifx≠0, then In−Ax≠0. So, for In−Ax≠0, no more than zero solution, then the matrix In−A is nonsingular.
WhenIn−A is nonsingular, we have(43)In−AIn−A−1=In,In−A−1=In−A+AIn−A−1=In−AIn−A−1+AIn−A−1=In+AIn−A−1.
Now,(44)In−A−1≤In+AIn−A−1=1+AIn−A−1,In−A−1≤11−A.Lemma 4.
Letf=h+g¯:Bn⟶ℂn be a univalent pluriharmonic mapping, where h is locally biholomorphic and fBn is M-linearly connected with constant M≥1. Suppose that there exists a constant N∈0,1 such that ωf∞≤N. Then, for any z1, z2∈Bn,(45)gz1−gz2≤MN1−Nfz1−fz2,(46)hz1−hz2≤M1−Nfz1−fz2.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 w1,w2∈fBn, it follows from the M-linear connectivity of fBn that there exists a smooth curve γ:0,1⟶fBn connecting w1 and w2 with γ0=w1, γ1=w2, and lγ≤Mw1−w2. Since f is univalent, we assume that z1=f−1w1 and z2=f−1w2, and hence σ=f−1°γ is a curve in Bn joining z1 and z2.
By the assumptionωf∞≤N<1, the inverse mapping theorem and Theorem 4, we know that f−1 is differentiable (cf. [3]). Moreover, by [3], (28), we have(47)Df−1=Dh−1In−Dg¯Dh¯−1DgDh−1−1,(48)D¯f−1=−Dh−1In−Dg¯Dh¯−1DgDh−1−1Dg¯Dh¯−1.
Fors∈0,1, let z=σs=f−1°γs∈Bn. Therefore, it follows from (47) and (48) that(49)gz1−gz2=∫σDgzdz=∫01Dgσs·σ′sds=∫01Dgσs·Df−1γs·γ′s+D¯f−1γs·γ′s¯ds≤∫01ωfσsIn−ωfσs¯·ωfσs−1·γ′sds+∫01ωfσsIn−ωfσs¯·ωfσs−1·ωfσs¯·γ′s¯ds.
Furthermore, by Theorem4, the assumption ωf∞≤N and the M-linear connectivity of fBn, we know that(50)gz1−gz2≤∫01ωfσs1−ωfσs2+ωfσs21−ωfσs2·γ′sds=∫01ωfσs1−ωfσsdγs≤N1−Nlγ≤MN1−Nfz1−fz2,which yields the inequality (45).
Similarly, by applying (47), (48), Theorem 4, the assumption ωf∞≤N and the M-linear connectivity of fBn, we get(51)hz1−hz2=∫σDhzdz=∫01Dhσs·σ′sds=∫01Dhσs·Df−1γs·γ′s+D¯f−1γs·γ′s¯ds≤∫01In−ωfσs¯·ωfσs−1·γ′sds+∫01In−ωfσs¯·ωfσs−1·ωfσs¯·γ′s¯ds≤∫0111−ωfσs2+ωfσs1−ωfσs2·γ′sds=∫0111−ωfσsdγs≤11−Nlγ≤M1−Nfz1−fz2,which leads to (46), and thus, the proof of this lemma is complete.
Based on Lemma4, we have the following result.Lemma 5.
Letf=h+g¯:Bn⟶ℂn be a pluriharmonic mapping, where h is biholomorphic and hBn is M-linearly connected with constant M≥1. Suppose that there exists a constant N∈0,1/M such that ωf∞≤N. Then, f is a univalent and sense-preserving mapping, and for any z1, z2∈Bn,(52)gz1−gz2≤N1+NM1−N1−MNfz1−fz2,hz1−hz2≤1+NM1−N1−MNfz1−fz2.Proof.
By the proof of Theorem 2.1 [4], we know that f is a univalent and sense-preserving mapping and fBn is a 1+NM/1−MN-linearly connected domain. Then, the result of this lemma follows from Lemma 4.
Based on Lemmas1 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 pointsz1, z2∈Bn and A∈Lℂn,ℂn with A≤1, we see from Lemma 5 that(53)fAz1−fAz2≤hz1−hz2+gz1−gz2≤1+N2M1−N1−MNfz1−fz2.
On the other hand, by Lemma1 and the assumption “MN<1,” we get(54)fAz1−fAz2≥hz1−hz2−A·gz1−gz2≥hz1−hz2−gz1−gz2≥hz1−hz2−MNhz1−hz2≥1−MN1+MNfz1−fz2.
Hence, (17) follows, and the proof of this theorem is complete [16].



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