KERNEL CONVERGENCE AND BIHOLOMORPHIC MAPPINGS IN SEVERAL COMPLEX VARIABLES

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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 =n

j=1zjwjand the Euclidean normz = z,z^1/2,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 ), B_r is denoted byU_r, and the unit discU1byU. IfGis an open set inCⁿ, let H(G)be the set of holomorphic mappings fromGintoCⁿ. If{g_k}k∈N is a se- quence of holomorphic mappings from a domainΩ⊆CⁿintoC^m, we will write g_k→gto mean that{g_k}k∈N converges (simply or locally uniformly onΩ) to gask→ ∞.

ByL(Cⁿ,C^m)we denote the space of continuous linear operators fromCⁿ intoC^m 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.

In this paper, we discuss the connection between an unusual notion of convergence of domains inCⁿ and biholomorphic mappings which satisfy a growth result. In the case of one variable, the notion of kernel convergence

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 {G_k}k∈N be a sequence of domains inCⁿ such that 0∈Gkfork∈N. If 0 is an interior point of

k∈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⊂G_kfork≥k0(in other words,Kis contained in all but finitely many of the setsG_k). If 0 is not an interior point of

k∈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 of

k∈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 of

k∈NGk.

Definition1.2. We say that the{G_k}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{G_k}k∈Nis an increasing sequence of domains inCⁿ, that is,G_k⊆G_k+1, k∈N, such that 0∈G_k, k∈N, thenG=

k∈NG_k is the kernel of{Gk}k∈N and{Gk}k∈N converges toGin the sense of kernel convergence.

LetS^c(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 ∈S^c(B). On the other hand, if z0∈ B\ {0} is fixed, then the functional f (z0) is continuous on S^c(B)with respect to the topology of locally uni- form convergence, and hence attains its infimum for somef0∈S^c(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 ∈S^c(B). (It is also easy to see thatm(r )is a strictly in- creasing function by the maximum principle for holomorphic mappings and lim_r→0⁺m(r )=0.) Consequently, we have proved that

m(r )≤f (z)≤M(r ), z =r , ∀f∈S^c(B). (1.1)

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,S^c(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 f_k/α_k∈S^c(B), k∈N. Also letG_k=f_k(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∈S^c(B), whereα=lim_k→∞α_k, andf (B)=G.

In the latter case,f_k⁻¹→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 eitherJ_f ≡0, or elsefis biholomorphic onB.

Case1. First, assume thatJf ≡0. Since fk→f locally uniformly onB, it follows that lim_k→∞J_fk(0)=J_f(0)=0, that is,

k→∞limα_k=0. (2.1)

Sincefk/αk∈S^c(B), we deduce in view of relations (1.1) and (2.1) thatfk→0 locally uniformly onBask→ ∞.

Next, we show thatG= {0}andG_k→ {0}in the sense of kernel convergence.

Letgk=f_k⁻¹fork∈N. Suppose thatG≠{0}. Then there isε >0 such that

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 thatg_k(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{G_k}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=lim_k→∞α_k>0 and, taking into account the fact thatf_k/α_k∈S^c(B) andS^c(B)is compact, we easily deduce thatf /α∈S^c(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=∂B_r 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 f_k→f uniformly onγask→ ∞, there is somek0=k0(γ)∈Nsuch that

f_k(z)−f (z)< η, z∈γ, k≥k0. (2.3) Hence, ifk≥k0andz∈γ, we obtain

f_k(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 onB_r sincef is biholomorphic on B, and thus for eachk≥k0, there is a unique point z_k∈ B_r 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_k⁻_p¹→f⁻¹locally uniformly onΩ. Indeed, the inverse functions gk=f_k⁻¹are well defined on any fixed compact subset ofΩforksufficiently large, since Ω⊆G, and moreovergk(w)<1 forklarge. By Montel’s theorem, there is

a subsequence{gkp}p∈Nsuch thatgkp→glocally uniformly onΩ. Thengis a holomorphic mapping onΩ,g(0)=0, and

Dg(0)=lim

p→∞Dg_kp(0)=lim

p→∞

Df_kp(0)₋₁

=lim

p→∞

1

α_kpI. (2.6) Sincefis biholomorphic onB, we must have lim_p→∞α_kp>0. Hence,J_g(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 thatf_k⁻¹→f⁻¹locally uniformly onΩask→ ∞ andΩ=G.

The argument in the second step implies that each subsequence of{g_k}k∈N

contains a further subsequence which converges locally uniformly onΩtof⁻¹. Since the sequence{g_k}k∈Nis locally uniformly bounded, a further application of Montel’s theorem yields that the whole sequence{g_k}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{f_kp}p∈N of{f_k}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 thatG_k→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,J_fk(0)→0 ask→ ∞. Otherwise, if{α_k}k∈Ndoes not converge to zero, then there exist someε >0 and a subsequence{α_kp}p∈Nof{α_k}k∈Nsuch thatα_kp≥ εforp∈N.

Since{f_kp/α_kp}p∈N⊂S^c(B), it follows in view of (1.1) that αkpm z

≤fkp(z), z∈B, p∈N, (2.7) and thusf_kp(B)⊇B_εµforp∈N, where 0< µ=lim_r→1⁻m(r ). (Clearly,µ <∞ sincefkp∈S^c(B).) However, this is a contradiction to the fact thatGkp→ {0}. Hence, we must haveα_k→0 ask→ ∞. Using an argument similar to that in Case 1of the proof of necessity, we deduce thatf_k→0 locally uniformly onB ask→ ∞.

Case2. We now assume thatG≠{0}andG≠ Cⁿ. We first prove that the sequence{α_k}k∈Nis bounded. Otherwise, there is a subsequence{k_p}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 {G_kp}p∈Nhas the kernelCⁿ. This contradiction shows that there isL >0 such that αk≤Lfork∈N. Taking into account the relation (1.1), we can easily

obtain

f_k(z)≤α_kM 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. IfJ_f ≡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{G_kp}p∈Nby the necessary part of the proof applied to the sequences{f_kp}p∈Nand{G_kp}p∈N. But the kernel of{Gkp}p∈Nis the same as the kernel of{Gk}k∈N, that isG, since G_k→G. Therefore,f (B)=G. Further, since{f_kp/α_kp}p∈N⊂S^c(B)andS^c(B) is compact, it follows thatf /α∈S^c(B)too andf_k⁻¹_p→f⁻¹locally uniformly on Gby the necessary part of the proof.

We next prove thatf_k→f locally uniformly onB ask→ ∞. To this end, it suffices to prove thatf_k(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{f_k(z0)}k∈N is a bounded sequence, there exist two subsequences {f_kp(z0)}p∈N and {f_kp(z0)}p∈N of{fk(z0)}k∈N, which converge to some dis- tinct limits denoted by w0and w₀. Since{f_kp}p∈N and {f_kp}p∈N are locally uniformly bounded families, we may extract two subsequences of these se- quences, again denoted by{f_kp}p∈Nand{f_kp}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=lim_p→∞α_kp and 0 < γ=lim_p→∞α_kp. It is also obvious that w0=h1(z0)andw₀=h2(z0). Moreover, sinceG_kp→GandG_kp→Gand by the necessary part of the proof,h1(B)=h2(B)=G. Next, letq=h⁻₂¹◦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 {f_k}k∈N be a sequence of mappings in S^∗(B) and let Gk=fk(B). Also letGbe the kernel of{Gk}k∈N. Then{fk}k∈Nconverges locally

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_k⁻¹→f⁻¹locally uniformly onGask→ ∞.

Theorem3.2. Let{f_k}k∈Nbe a sequence of mappings inK(B)and letG_k= 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_k⁻¹→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)=e^tI, 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)=e^s−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

f z,t1

−f z,t2≤M(r ,T )t1−t2, z ≤r , t1,t2∈[0,T ]. (3.3)

LetS^c(B)be the subclass ofS(B)consisting of all mappings inS(B)which satisfy the 1/4-growth result. That is,f∈S^c(B)if and only iff∈S(B)and

z

1+z2≤f (z)≤ z

1−z2, z∈B. (3.4) Also letg_t(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)∈ S^c(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 t_k

→G t0

ift_k →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⁻₀¹g(z,β⁻¹(t))is a Loewner chain andf (B,t)=α⁻¹₀ 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

α(s)I=Dg(0,s)=Dg(0,t)Dv(0,s,t)=α(t)Dv(0,s,t), (3.9)

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 astk→ ∞, we must haveα(t)→ ∞ast→ ∞. On the other hand, from Theorem 2.1, we know thatg_tk→g_t locally uniformly onB ast_k→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 e^t=α(τ)/α0. Consequently, we deduce that

Df (0,t)=α⁻₀¹Dg(0,τ)=α₀⁻¹α(τ)I=e^tI, t≥0. (3.11) We conclude thatf (z,t)is a Loewner chain. Clearly,f (B,t)=α⁻¹₀ G(β⁻¹(t)), t≥0. Further,{e^−tf (z,t)}t≥0 is a normal family since g_t/α(t)∈S^c(B)for t∈[0,∞).

We now prove part (ii). To this end, letf_t(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=f_t⁻¹◦fsis a biholomorphic mapping ofBontoBsuch thatqs,t(0)= 0. SinceBis a circular domain, it follows thatq_s,tis the restriction of a unitary linear operator. On the other hand, sinceDq_s,t(0)=e^s−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)⊇B_et/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.

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Gabriela Kohr: Faculty of Mathematics and Computer Science, Babe¸s-Bolyai Univer- sity, 1 M. Kog˘alniceanu Street, 3400 Cluj-Napoca, Romania

E-mail address:[email protected].