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KERNEL CONVERGENCE AND BIHOLOMORPHIC
MAPPINGS IN SEVERAL COMPLEX VARIABLES
GABRIELA KOHR
Received 27 March 2003
We deal with kernel convergence of domains inCnwhich are biholomorphically equivalent to the unit ballB. We also prove that there is an equivalence between the convergence on compact sets of biholomorphic mappings onB, which satisfy a growth theorem, and the kernel convergence. Moreover, we obtain certain con-sequences of this equivalence in the study of Loewner chains and of starlike and convex mappings onB.
2000 Mathematics Subject Classification: 32H02, 30C45.
1. Introduction and preliminaries. LetCnbe the space ofncomplex vari-ablesz=(z1,...,zn)with the usual inner productz,w =nj=1zjwjand the
Euclidean normz = z,z1/2,z∈Cn. LetB(a,r )be the open ball of radius
r centered ata∈Cn. The ballB(0,r )will be denoted byB
r and the unit ball
B1will be denoted byB. Also the closed ball of centeraand radiusr will be denoted byB(a,r ). In the case of one variable,B(a,r )is denoted byU(a,r ),
Br is denoted byUr, and the unit discU1byU. IfGis an open set inCn, let
H(G)be the set of holomorphic mappings fromGintoCn. If{gk}k∈
N is a
se-quence of holomorphic mappings from a domainΩ⊆CnintoCm, we will write
gk→gto mean that{gk}k∈N converges (simply or locally uniformly onΩ) to gask→ ∞.
ByL(Cn,Cm)we denote the space of continuous linear operators fromCn intoCm with the standard operator norm. LetI be the identity inL(Cn,Cn). Iff∈H(B), we say thatf is normalized iff (0)=0 andDf (0)=I. We also say thatf∈H(B)is locally biholomorphic onBiff has a local holomorphic inverse at eachz∈B. This is equivalent toJf(z)≠0 forz∈B, whereJf(z)= detDf (z)is the complex Jacobian determinant offatz∈B. A biholomorphic mapping ofBwill also be called a univalent mapping. LetS(B)be the subset ofH(B)consisting of normalized univalent mappings onB. In the case of one variable,S(B) is denoted byS. LetS∗(B)and K(B)be the subsets ofS(B), consisting, respectively, of starlike and convex mappings onB.
was introduced and studied by Carathéodory [2] (see also [5,8,15,24,27]). He proved a fundamental result of independent interest, which was later used to prove certain important results in the theory of univalent functions, especially in the study of Loewner chains and the Loewner differential equation. His re-sult is a complete geometric characterization of the convergence of univalent functions in terms of the convergence of their image domains. Gehring [7] de-fined the notions of kernel and kernel convergence in the case of domains in
R3, and obtained an analogue of the Carathéodory kernel convergence result
in the case ofK-quasiconformal mappings inR3. Other results in this direction
were obtained by Reshetnyak [25] in the case of quasiconformal mappings in
Rn (see also [30, pages 72–75]). We mention that a metric space analogue of the Carathéodory kernel convergence result was obtained in [19].
We begin with the following definitions.
Definition 1.1. Let {Gk}k∈N be a sequence of domains inCn such that 0∈Gkfork∈N. If 0 is an interior point ofk∈NGk, we define thekernelGof
{Gk}k∈Nto be the largest domain which contains 0 such that ifKis a compact
subset ofG, then there is a positive integerk0such thatK⊂Gkfork≥k0(in other words,Kis contained in all but finitely many of the setsGk). If 0 is not an interior point ofk∈NGk, we define the kernel to be{0}.
LetᏳbe the set of all domainsΩinCnsuch that 0∈Ωand each compactK ofΩis contained in all but finitely many of the setsGk. We assume that 0 is an interior point ofk∈NGk. An application of the Heine-Borel theorem shows that ifD=Ω∈ᏳΩ, thenD∈Ᏻ, and it is clear that no larger domain can belong toᏳ. This yields the existence of the kernel of any domainsG1,...,Gk,...,such that 0 is an interior point ofk∈NGk.
Definition1.2. We say that the{Gk}k∈N kernel converges toGand write
Gk→G, if each subsequence of{Gk}k∈N has the same kernelG.
It is not difficult to see that if{Gk}k∈Nis an increasing sequence of domains
inCn, that is,G
k⊆Gk+1, k∈N, such that 0∈Gk, k∈N, thenG=k∈NGk is the kernel of{Gk}k∈N and{Gk}k∈N converges toGin the sense of kernel
convergence.
LetSc(B)be a compact subset ofS(B). Then it is clear that for eachr∈[0,1), there exists some M=M(r )≥0 such thatf (z) ≤M(r ) for z =r for
f ∈Sc(B). On the other hand, if z0∈ B\ {0} is fixed, then the functional f (z0) is continuous on Sc(B)with respect to the topology of locally uni-form convergence, and hence attains its infimum for somef0∈Sc(B). Since
f0is biholomorphic onB, this infimum cannot be zero. Therefore, there exists a functionm(r )which is positive forr∈(0,1)such thatm(r )≤ f (z)for z =r <1 andf ∈Sc(B). (It is also easy to see thatm(r )is a strictly in-creasing function by the maximum principle for holomorphic mappings and limr→0+m(r )=0.) Consequently, we have proved that
In [10] it is shown that the setS0(B), consisting of all mappings inS(B)
which have parametric representation, is also a compact subset ofS(B)since any mapping in the classS0(B)satisfies the 1/4-growth result. Moreover,S0(B)
contains the setS∗(B)as a proper subset (see also [1,9]). On the other hand, the setK(B)is also a compact subset ofS(B)since any mapping inK(B)satisfies the 1/2-growth result (see [6,26,29]).
It is known that in the case of one variable, the classSis compact; however, in several variables, the classS(B)is not compact, and there exist mappingsf
inS(B)which do not satisfy the above growth result, that is,Sc(B)S(B)in dimensionn≥2 (see [9,10]).
In the next section, we will prove that there is an equivalence between the kernel convergence and the convergence on compact sets of biholomorphic mappings on the unit ballBwhich satisfy the growth result (1.1). In the last section, we will obtain some consequences of this result in the case of the ker-nel convergence and the convergence on compact sets of normalized starlike and normalized convex mappings onB. Also, we will prove that there is an equivalence between the notions of a Loewner chain, which satisfies a certain normality condition, and kernel convergence.
2. Kernel convergence and biholomorphic mappings. In this section, we prove the main result of this paper, which is an analogue of the Carathéodory kernel convergence theorem [2], on the convergence of conformal functions of one variable for biholomorphic mappings which satisfy the growth result (1.1).
Theorem 2.1. Let{fk}k∈N be a sequence of biholomorphic mappings on
B such thatfk(0)=0andDfk(0)=αkI, whereαk>0, k∈N. Assume that
fk/αk∈Sc(B), k∈N. Also letGk=fk(B), k∈N, and letGbe the kernel of {Gk}k∈N. Then{fk}k∈Nconverges locally uniformly onBto a mappingfif and only ifGk→G≠ Cn. In the case of convergence, eitherf≡0andG= {0}, or elsefis biholomorphic onB,f /α∈Sc(B), whereα=lim
k→∞αk, andf (B)=G. In the latter case,f−1
k →f−1locally uniformly onGask→ ∞.
Proof
Necessity. First, assume thatfk→f locally uniformly on B ask→ ∞. In view of a version of Hurwitz’s theorem in higher dimensions, we deduce that eitherJf ≡0, or elsefis biholomorphic onB.
Case1. First, assume thatJf ≡0. Since fk→f locally uniformly onB, it follows that limk→∞Jfk(0)=Jf(0)=0, that is,
lim
k→∞αk=0. (2.1)
Sincefk/αk∈Sc(B), we deduce in view of relations (1.1) and (2.1) thatfk→0 locally uniformly onBask→ ∞.
Bε⊆Gkfork∈N. Thengk is a biholomorphic mapping onGk, and thus in
Bε such thatgk(0)=0 andgk(w)<1,w∈Bε. By the Schwarz lemma for holomorphic mappings, we deduce thatgk(w) ≤(1/ε)wforw< εand Dgk(0) ≤1/ε fork∈N. Consequently, we deduce thatαk≥εfork∈N. However, this is a contradiction to (2.1), and thus we must haveG= {0}. Fur-ther, since each subsequence of{Gk}k∈Nhas the same kernel{0}by a similar
argument as above, we deduce thatGk→ {0}.
Case2. We next assume thatJf ≡0, and thusf is biholomorphic on B. Thenα=limk→∞αk>0 and, taking into account the fact thatfk/αk∈Sc(B) andSc(B)is compact, we easily deduce thatf /α∈Sc(B)too.
LetΩ=f (B). We prove thatG=ΩandGk→Gin the sense of kernel con-vergence.
First step.We prove thatΩ⊆G. To this end, it suffices to prove that ifKis a compact subset ofΩ, thenK⊂Gkfor sufficiently largek. Indeed, ifKis such a compact subset ofΩ,f−1(K)is a compact subset ofB, and thus there is some r∈(0,1)such thatf−1(K)⊂B
r. Letγ=∂Br andΓ =f (γ). It is obvious that
K∩Γ = ∅sincef is biholomorphic. Further, letηbe the Euclidean distance betweenΓ andK. Thenη >0 and clearly
η=minf (z)−w:w∈K, z =r. (2.2)
If v0∈K, thenf (z)−v0 ≥ηforz∈γ. On the other hand, since fk→f uniformly onγask→ ∞, there is somek0=k0(γ)∈Nsuch that
fk(z)−f (z)< η, z∈γ, k≥k0. (2.3)
Hence, ifk≥k0andz∈γ, we obtain
fk(z)−f (z)<f (z)−v0, (2.4)
and in view of Rouché’s theorem (see [18, Theorem 3] and also [3, 17]), we deduce that both equations
fk(z)−v0=0, f (z)−v0=0 (2.5)
have the same number of solutions insideγ, that is, onBr, fork≥k0. But the equationf (z)−v0=0 has only one solution onBr sincef is biholomorphic on B, and thus for eachk≥k0, there is a unique point zk∈ Br such that
v0=fk(zk). Hence,v0∈fk(B)fork≥k0. Also sincek0does not depend onv0 (k0depends only onK) andGk=fk(B), we deduce thatK⊆Gkfor sufficiently largek. We have therefore proved thatΩ⊆G.
Second step. We prove that there is a subsequence {kp}p∈N such that
f−1
kp→f−1locally uniformly onΩ. Indeed, the inverse functions gk=fk−1are well defined on any fixed compact subset ofΩforksufficiently large, since
a subsequence{gkp}p∈Nsuch thatgkp→glocally uniformly onΩ. Thengis a
holomorphic mapping onΩ,g(0)=0, and
Dg(0)=lim
p→∞Dgkp(0)=p→∞lim
Dfkp(0)−1=p→∞lim α1
kpI. (2.6)
Sincefis biholomorphic onB, we must have limp→∞αkp>0. Hence,Jg(0)≠ 0, and thusgis biholomorphic onΩ.
Next, we can prove thatg=f−1by an argument based again on the Rouché
theorem.
Third step. We next prove thatfk−1→f−1locally uniformly onΩask→ ∞ andΩ=G.
The argument in the second step implies that each subsequence of{gk}k∈N
contains a further subsequence which converges locally uniformly onΩtof−1.
Since the sequence{gk}k∈Nis locally uniformly bounded, a further application
of Montel’s theorem yields that the whole sequence{gk}k∈Nconverges locally
uniformly onΩtof−1. In fact, the same argument combined with Vitali’s
the-orem (see, e.g., [20]) yields that{gk}k∈Nconverges locally uniformly onGto a
biholomorphic mappingφofGontoB. Sinceφ|Ω=gandgis a biholomorphic mapping ofΩontoG, we must haveΩ=G.
We have therefore proved that the kernel of{Gk}k∈Nisf (B), and since each
subsequence{fkp}p∈N of{fk}k∈N converges locally uniformly onB tof, the
corresponding subsequence{Gkp}p∈N of{Gk}k∈N has the same kernelf (B).
HenceGk→GandG=f (B).
Sufficiency. We now assume thatGk→G≠ Cnin the sense of kernel con-vergence and prove that{fk}k∈Nconverges locally uniformly onB.
Case1. First, assume thatG= {0}. We show thatαk→0 ask→ ∞, that is,Jfk(0)→0 ask→ ∞. Otherwise, if{αk}k∈Ndoes not converge to zero, then
there exist someε >0 and a subsequence{αkp}p∈Nof{αk}k∈Nsuch thatαkp≥
εforp∈N.
Since{fkp/αkp}p∈N⊂Sc(B), it follows in view of (1.1) that
αkpm z≤fkp(z), z∈B, p∈N, (2.7)
and thusfkp(B)⊇Bεµforp∈N, where 0< µ=limr→1−m(r ). (Clearly,µ <∞
sincefkp∈Sc(B).) However, this is a contradiction to the fact thatGkp→ {0}. Hence, we must haveαk→0 ask→ ∞. Using an argument similar to that in Case 1of the proof of necessity, we deduce thatfk→0 locally uniformly onB ask→ ∞.
Case2. We now assume thatG≠{0}andG≠ Cn. We first prove that the sequence{αk}k∈Nis bounded. Otherwise, there is a subsequence{kp}p∈Nsuch
thatαkp≥pforp∈N. Using again an argument similar to that in the previ-ous case, we deduce thatGkp=fkp(B)⊇Bpµ, p∈N, and thus the sequence {Gkp}p∈Nhas the kernelCn. This contradiction shows that there isL >0 such
obtain
fk(z)≤αkM z≤LM z, z∈B, k∈N, (2.8)
and thus{fk}k∈N is a locally uniformly bounded sequence on B. In view of
Montel’s theorem, there is a subsequence{fkp}p∈Nof{fk}k∈Nwhich converges
locally uniformly to a holomorphic mappingf. IfJf ≡0, then using a similar argument as inCase 1of the proof of necessity, applied to the subsequence {fkp}p∈N, we deduce thatf≡0 and henceG= {0}. However, this is impossible,
and thusfis a biholomorphic mapping ofBonto the kernel of{Gkp}p∈Nby the
necessary part of the proof applied to the sequences{fkp}p∈Nand{Gkp}p∈N.
But the kernel of{Gkp}p∈Nis the same as the kernel of{Gk}k∈N, that isG, since Gk→G. Therefore,f (B)=G. Further, since{fkp/αkp}p∈N⊂Sc(B)andSc(B)
is compact, it follows thatf /α∈Sc(B)too andf−1
kp→f−1locally uniformly on
Gby the necessary part of the proof.
We next prove thatfk→f locally uniformly onB ask→ ∞. To this end, it suffices to prove thatfk(z)→f (z)ask→ ∞, for allz∈B, in view of Vitali’s theorem and the fact that{fk}k∈Nis a locally uniformly bounded family onB.
Suppose that there is somez0 ∈B such that {fk(z0)}k∈N is not
conver-gent. Since{fk(z0)}k∈N is a bounded sequence, there exist two subsequences
{fkp(z0)}p∈N and {fkp(z0)}p∈N of{fk(z0)}k∈N, which converge to some
dis-tinct limits denoted by w0and w
0. Since{fkp}p∈N and {fkp}p∈N are locally uniformly bounded families, we may extract two subsequences of these se-quences, again denoted by{fkp}p∈Nand{fkp}p∈N, which converge locally uni-formly onBtoh1andh2, respectively. It is easy to see thath1andh2are biholo-morphic mappings onB, h1(0)=h2(0)=0,Dh1(0)=βI, andDh2(0)=γI, where 0< β=limp→∞αkp and 0 < γ=limp→∞αkp. It is also obvious that
w0=h1(z0)andw
0=h2(z0). Moreover, sinceGkp→GandGkp→Gand by the necessary part of the proof,h1(B)=h2(B)=G. Next, letq=h−1
2 ◦h1:B→B.
Thenqis a biholomorphic mapping ofBontoB,q(0)=0, and sinceBis a cir-cular domain, we deduce thatqis the restriction of a unitary linear operator (see [28, Theorem 2.1.3]). This yieldsβ=γ. Consequently,q(0)=0,Dq(0)=I, and in view of a uniqueness result due to Cartan (see, e.g., [28, Theorem 2.1.1]), we conclude thatq(z)=zforz∈B, that is,h1≡h2. However, this is a contra-diction toh1(z0)≠h2(z0). Thus, we must havefk(z)→f (z)ask→ ∞, for all
z∈B. This completes the proof.
3. Applications. We first apply the result ofTheorem 2.1to obtain the fol-lowing connections between the kernel convergence and locally uniform con-vergence of normalized starlike and convex mappings.
Theorem 3.1. Let {fk}k∈N be a sequence of mappings in S∗(B) and let
uniformly onBtof if and only ifGk→G≠ Cn. Moreover,f∈S∗(B),G=f (B) (thusGis a starlike domain with respect to the origin), and f−1
k →f−1locally uniformly onGask→ ∞.
Theorem3.2. Let{fk}k∈Nbe a sequence of mappings inK(B)and letGk=
fk(B). Also letGbe the kernel of{Gk}k∈N. Then{fk}k∈N converges locally uni-formly onBtofif and only ifGk→G≠ Cn. Moreover,f∈K(B),G=f (B)(thus
Gis a convex domain), andf−1
k →f−1locally uniformly onGask→ ∞.
Next we useTheorem 2.1to prove that there is an equivalence between the notions of a Loewner chainf (z,t), such that{e−tf (z,t)}t≥0is a normal family, and the kernel convergence of the family{f (B,t)}t≥0. To this end, we recall
some notions and results which are useful in the proof ofTheorem 3.5. Iff ,g∈H(B), we say thatfis subordinate togif there is a Schwarz mapping
v(i.e.,v∈H(B),v(0)=0, andv(z)<1,z∈B) such thatf (z)=g(v(z)),
z∈B. We will writef≺gto mean thatfis subordinate tog.
A mapping f :B×[0,∞)→Cn is called a Loewner chain if the following conditions hold:
(i) f (·,t)is univalent onB,f (0,t)=0, andDf (0,t)=etI, for eacht≥0; (ii) f (·,s)≺f (·,t)whenever 0≤s≤t <∞.
Condition (ii) is equivalent to the fact that there is a unique univalent Schwarz mappingv=v(z,s,t)called the transition mapping associated tof (z,t)such that
f (z,s)=f v(z,s,t),t, z∈B,0≤s≤t <∞. (3.1)
Note thatDv(0,s,t)=es−tI, 0≤s≤t <∞, in view of the normalization of
f (z,t).
Recently, in [10,14], the authors have proved the following growth result for Loewner chains f (z,t)such that {e−tf (z,t)}≥0 is a normal family. Still this result does not hold for an arbitrary Loewner chain (see [10]).
Lemma 3.3. Letf (z,t) be a Loewner chain such that{e−tf (z,t)}t≥0 is a
normal family onB. Then
z
1+z2≤e−tf (z,t)≤
z
1−z2, z∈B, t≥0. (3.2)
On the other hand, in [12] (see also [13,14]), Graham and Kohr proved the following absolute continuity result for Loewner chains.
Lemma3.4. Letf (z,t)be a Loewner chain. Then, for eachr ∈(0,1)and
T >0, there isM=M(r ,T ) >0such that
LetSc(B)be the subclass ofS(B)consisting of all mappings inS(B)which satisfy the 1/4-growth result. That is,f∈Sc(B)if and only iff∈S(B)and
z
1+z2≤f (z)≤
z
1−z2, z∈B. (3.4)
Also letgt(z)=g(z,t)be a biholomorphic mapping ofBonto a domainG(t) such thatgt(0)=0,Dgt(0)=α(t)I, whereα(t) >0 fort≥0, andgt/α(t)∈
Sc(B), t≥0. Also let α0=α(0). Further, assume that the family {G(t)}t≥
0
satisfies the following conditions:
G(s)G(t), 0≤s < t <∞, (3.5)
G tk →G t0 iftk →t0<∞,
G tk →Cn iftk → ∞. (3.6)
The convergence in question is the kernel convergence. Then we obtain the fol-lowing result (cf. [24, Chapter 6] and [4]).Theorem 3.5(i) provides an example of a Loewner chain associated to a given family of domains which are biholo-morphically equivalent to the unit ball and converge in the sense of kernel con-vergence. On the other hand,Theorem 3.5(ii) shows that given a Loewner chain
f (z,t)such that{e−tf (z,t)}t≥
0is a normal family, the associated family of
do-mains satisfies conditions (3.5) and (3.6). For further applications of Loewner chains in several complex variables, see [10,11,12,13,14,16,21,22,23].
Theorem3.5. (i)LetgtandG(t)satisfy the conditions in the previous para-graph.
(a) Thenαis a strictly increasing continuous function, andα(t)→ ∞as t→ ∞.
(b) Ifβ(t)=log[α(t)/α0], then f (z,t)=α−1
0 g(z,β−1(t))is a Loewner chain andf (B,t)=α−1
0 G(β−1(t)). Further,{e−tf (z,t)}t≥0is a nor-mal family onB.
(ii)Conversely, letf (z,t)be a Loewner chain such that{e−tf (z,t)}t≥
0is a normal family onB. Also letG(t)=f (B,t),t≥0. Then the family of domains {G(t)}t≥0satisfies conditions (3.5) and (3.6).
Proof. First we prove part (i). Using the relation (3.5), we have
g(z,s)≺g(z,t), 0≤s≤t <∞, (3.7)
and therefore there is a Schwarz mappingv=v(z,s,t)such that
g(z,s)=g v(z,s,t),t, z∈B,0≤s≤t <∞. (3.8)
Differentiating both sides of the above relation with respect toz, we obtain
and thusα(s)/α(t)= Dv(0,s,t) ≤1, that is,α(s)≤α(t). Since g(B,s) g(B,t),s < t, by (3.5), we deduce thatα(s)≠α(t)fors < t. Otherwise, ifα(s)=
α(t)for somes < t, thenDv(0,s,t)=I. Sincev(B,s,t)⊆B,v(0,s,t)=0, and
Dv(0,s,t)=I, we deduce in view of a uniqueness result due to Cartan (see [28, Theorem 2.1.1]) thatv(z,s,t)≡z. Hence,g(z,s)=g(z,t),z∈B. However, this is a contradiction to (3.5). Thus,α(s)≠α(t)fors≠t, and, consequently,αis a strictly increasing function from[0,∞)into(0,∞). Moreover, sinceG(tk)→
Cn ast
k→ ∞, we must haveα(t)→ ∞ast→ ∞. On the other hand, from Theorem 2.1, we know thatgtk→gt locally uniformly onB astk→t <∞, so that the functionαis continuous. These arguments prove (a).
We next prove assertion (b). To this end, it suffices to observe that α :
[0,∞)→[α0,∞)is strictly increasing and continuous, hence one-to-one. Con-sequently,βis also a strictly increasing function from[0,∞)onto[0,∞). Using relation (3.7) and the above argument, we obtain
f (z,s)≺f (z,t), z∈B,0≤s≤t <∞, (3.10)
and sinceg(·,t)is univalent, we deduce thatf (·,t)is also univalent fort≥0. Moreover, ifτ=β−1(t), thent=β(τ)and et=α(τ)/α0. Consequently, we deduce that
Df (0,t)=α−1
0 Dg(0,τ)=α0−1α(τ)I=etI, t≥0. (3.11)
We conclude thatf (z,t)is a Loewner chain. Clearly,f (B,t)=α−1
0 G(β−1(t)), t≥0. Further,{e−tf (z,t)}t≥0 is a normal family since gt/α(t)∈Sc(B)for
t∈[0,∞).
We now prove part (ii). To this end, letft(z)=f (z,t)forz∈Bandt≥0. Obviously,G(s)⊆G(t)for 0≤s≤t <∞. SupposeG(s)=G(t)for somes < t. Thenqs,t=ft−1◦fsis a biholomorphic mapping ofBontoBsuch thatqs,t(0)= 0. SinceBis a circular domain, it follows thatqs,tis the restriction of a unitary linear operator. On the other hand, sinceDqs,t(0)=es−tI, we must haves=t. However, this is a contradiction. The claimed conclusion now follows. This implies (3.5). Further, since{e−tf (z,t)}t≥0is a normal family, we deduce in
view ofLemma 3.3thatf (B,t)⊇Bet/4fort≥0. Hence,G(t)=f (B,t)→Cnas
t→ ∞. This proves the second condition in (3.6). The first part in (3.6) follows fromTheorem 2.1andLemma 3.4. This completes the proof.
Acknowledgment. The author thanks the referee for helpful comments and suggestions which improved the aspects of the paper.
References
[1] R. W. Barnard, C. H. FitzGerald, and S. Gong,The growth and1/4-theorems for starlike mappings inCn, Pacific J. Math.150(1991), no. 1, 13–22. [2] C. Carathéodory,Untersuchungen über die konformen Abbildungen von festen
[3] B. Chabat,Introduction à l’Analyse Complexe. Tome 2[Introduction to Complex Analysis. Vol. 2], Translated from Russian: Mathématiques, Mir, Moscow, 1990 (French).
[4] J. B. Conway,Functions of One Complex Variable. II, Graduate Texts in Mathemat-ics, vol. 159, Springer-Verlag, New York, 1995.
[5] P. L. Duren, Univalent Functions, Grundlehren der mathematischen Wis-senschaften, vol. 259, Springer-Verlag, New York, 1983.
[6] C. H. FitzGerald and C. R. Thomas,Some bounds on convex mappings in several complex variables, Pacific J. Math.165(1994), no. 2, 295–320.
[7] F. W. Gehring,The Carathéordory convergence theorem for quasiconformal map-pings in space, Ann. Acad. Sci. Fenn. Ser. A I No.336/11(1963), 3–21. [8] G. M. Goluzin,Geometric Theory of Functions of a Complex Variable,
Transla-tions of Mathematical Monographs, vol. 26, American Mathematical Soci-ety, Rhode Island, 1969.
[9] S. Gong,Convex and Starlike Mappings in Several Complex Variables, Mathemat-ics and Its Applications, vol. 435, Kluwer Academic Publishers, Dordrecht, 1998.
[10] I. Graham, H. Hamada, and G. Kohr,Parametric representation of univalent map-pings in several complex variables, Canad. J. Math.54(2002), no. 2, 324– 351.
[11] I. Graham, H. Hamada, G. Kohr, and T. J. Suffridge,Extension operators for locally univalent mappings, Michigan Math. J.50(2002), no. 1, 37–55.
[12] I. Graham and G. Kohr,Geometric Function Theory in One and Higher Dimensions, Marcel Dekker, New York, 2003.
[13] I. Graham, G. Kohr, and M. Kohr,Basic properties of Loewner chains in several complex variables, submitted.
[14] ,Loewner chains and parametric representation in several complex vari-ables, J. Math. Anal. Appl.281(2003), no. 2, 425–438.
[15] P. Henrici,Applied and Computational Complex Analysis, III, Wiley Classical Li-brary, John Wiley & Sons, New York, 1993.
[16] G. Kohr,Using the method of Löwner chains to introduce some subclasses of bi-holomorphic mappings inCn, Rev. Roumaine Math. Pures Appl.46(2001), no. 6, 743–760.
[17] N. G. Lloyd,Degree Theory, Cambridge Tracts in Mathematics, vol. 73, Cambridge University Press, Cambridge, 1978.
[18] ,Remarks on generalising Rouché’s theorem, J. London Math. Soc. (2)20
(1979), no. 2, 259–272.
[19] V. P. Luferenko and G. D. Suvorov,Families of homeomorphisms, relative met-rics and the Carathéodory theorem, Sibirsk. Mat. Zh.13(1972), 368–383 (Russian).
[20] R. Narasimhan,Several Complex Variables, Chicago Lectures in Mathematics, The University of Chicago Press, Chicago, 1971.
[21] J. A. Pfaltzgraff,Subordination chains and univalence of holomorphic mappings inCn, Math. Ann.210(1974), 55–68.
[22] ,Subordination chains and quasiconformal extension of holomorphic maps inCn, Ann. Acad. Sci. Fenn. Ser. A I Math.1(1975), no. 1, 13–25.
[23] J. A. Pfaltzgraff and T. J. Suffridge,Close-to-starlike holomorphic functions of sev-eral variables, Pacific J. Math.57(1975), no. 1, 271–279.
[25] Yu. G. Reshetnyak, Space Mappings with Bounded Distortion, Translations of Mathematical Monographs, vol. 73, American Mathematical Society, Rhode Island, 1989.
[26] K. A. Roper and T. J. Suffridge,Convexity properties of holomorphic mappings in
Cn, Trans. Amer. Math. Soc.351(1999), no. 5, 1803–1833.
[27] M. Rosenblum and J. Rovnyak,Topics in Hardy Classes and Univalent Functions, Birkhäuser Advanced Texts: Basler Lehrbücher, Birkhäuser Verlag, Basel, 1994.
[28] W. Rudin,Function Theory in the Unit Ball ofCn, Grundlehren der mathematis-chen Wissenschaften, vol. 241, Springer-Verlag, New York, 1980. [29] T. J. Suffridge,Biholomorphic mappings of the ball onto convex domains, Abstract
of papers presented to AMS,11(66)(1990), 46.
[30] J. Väisälä,Lectures onn-Dimensional Quasiconformal Mappings, Lecture Notes in Mathematics, vol. 229, Springer-Verlag, New York, 1971.
Gabriela Kohr: Faculty of Mathematics and Computer Science, Babe¸s-Bolyai Univer-sity, 1 M. Kog˘alniceanu Street, 3400 Cluj-Napoca, Romania
