extremal properties associated with univalent subordination chains in $$\mathbb {c}^n$$ c n

$: Extremal properties associated with univalent subordination chains in $$\mathbb {C}^n$$ C n$
Math. Ann.DOI 10.1007/s00208-013-0998-y Mathematische Annalen

Extremal properties associated with univalentsubordination chains in C

n

Ian Graham · Hidetaka Hamada ·Gabriela Kohr · Mirela Kohr

Received: 18 December 2012 / Revised: 31 October 2013© Springer-Verlag Berlin Heidelberg 2013

Abstract For a linear operator A ∈ L(Cn), let k+(A) be the upper exponentialindex of A and let m(A) = min{�〈A(z), z〉 : ‖z‖ = 1}. Under the assumptionk+(A) < 2m(A), we consider the family S0

A(Bn) of mappings which have A-parametric representation on the Euclidean unit ball Bn in C

n , i.e. f ∈ S0A(Bn) if

and only if there exists an A-normalized univalent subordination chain f (z, t) suchthat f = f (·, 0) and {e−t A f (·, t)}t≥0 is a normal family on Bn . We prove thatif f = f (·, 0) is an extreme point (respectively a support point) of S0

A(Bn), thene−t A f (·, t) is an extreme point of S0

A(Bn) for t ≥ 0 (respectively a support point ofS0

A(Bn) for t ≥ 0). These results generalize to higher dimensions related results due to

I. Graham was partially supported by the Natural Sciences and Engineering Research Council of Canadaunder Grant A9221. H. Hamada was partially supported by JSPS KAKENHI Grant Number 25400151. G.Kohr was supported by a grant of the Romanian National Authority for Scientific Research,CNCS-UEFISCDI, project number PN-II-ID-PCE-2011-3-0899. M. Kohr was supported by a grant of theRomanian National Authority for Scientific Research, CNCS-UEFISCDI, project numberPN-II-ID-PCE-2011-3-0994.

I. Graham (B)Department of Mathematics, University of Toronto, Toronto, ON M5S 2E4, Canadae-mail: [email protected]

H. HamadaFaculty of Engineering, Kyushu Sangyo University, 3-1 Matsukadai 2-Chome, Higashi-ku,Fukuoka 813-8503, Japane-mail: [email protected]

G. Kohr · M. KohrFaculty of Mathematics and Computer Science, Babes-Bolyai University, 1 M. Kogalniceanu Str.,400084 Cluj-Napoca, Romaniae-mail: [email protected]

M. Kohre-mail: [email protected]

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Pell and Kirwan. We also deduce an n-dimensional version of an extremal principle dueto Kirwan and Schober. In the second part of the paper, we consider extremal problemsrelated to bounded mappings in S0

A(Bn). To this end, we use ideas from control the-ory to investigate the (normalized) time-log M-reachable family Rlog M (idBn ,NA) of(4.1) generated by the Carathéodory mappings, where M ≥ 1 and k+(A) < 2m(A).We prove that each mapping f in the above reachable family can be imbedded asthe first element of an A-normalized univalent subordination chain f (z, t) such that{e−t A f (·, t)}t≥0 is a normal family and f (·, log M) = eA log M idBn . We also provethat the family Rlog M (idBn ,NA) is compact and we deduce a density result related tothe same family, which involves the subset ex NA of NA consisting of extreme points.These results are generalizations to C

n of related results due to Roth. Finally, we areconcerned with extreme points and support points associated with compact familiesgenerated by extension operators.

Mathematics Subject Classification (2000) 32H02 · 30C45

1 Introduction

The study of the family S(Bn) of normalized biholomorphic mappings on the unitball Bn in C

n , which is the analog of the family S (the family of normalized univalentfunctions on the unit disc U ) in several complex variables, was first suggested byCartan (see [11]). He studied biholomorphic mappings on the unit polydisc in C

2 andgave a counterexample which shows that the growth theorem for the family S failsin dimension n ≥ 2. Cartan also suggested that particular subfamilies of normalizedbiholomorphic mappings, such as the starlike and convex mappings, should be singledout for further development. Indeed, many of the results in univalent function theoryhave extensions to higher dimensions for these families of mappings. Suffridge (see[59]) obtained analytical characterizations of starlikeness and convexity on the unit ballin C

n and in complex Banach spaces. Gong and others (see [19,28], and the referencestherein) obtained growth results for normalized starlike and convex mappings in severalcomplex variables.

Subordination chains in several complex variables were first studied by Pfaltzgraffin the 1970s [44,45]. He generalized to higher dimensions the Loewner differentialequation and developed existence and uniqueness theorems for its solutions on the unitball Bn in C

n . The existence and regularity theory (including changes in normalization)has been considered by several authors, and various applications have been given (see[14,20–25,28,30,50,51,60]). A new approach to Loewner theory in the unit disc andcomplete hyperbolic complex manifolds may be found in [1–4,7–9,34]. On the otherhand, recent contributions in the theory of Loewner chains and the Loewner differentialequations in the case of reflexive Banach spaces were given in [26,27].

Significant differences between the one complex variable and the several complexvariables Loewner theory have arisen (see [20,28]). For example, it is well known thatevery function f ∈ S can be embedded in a Loewner chain. Moreover, if f ∈ S, then fhas parametric representation, i.e. f (z) = limt→∞ etv(z, t), z ∈ U , where v = v(z, t)is the unique Lipschitz continuous solution on [0,∞) of the initial value problem

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Extremal properties and univalent subordination chains

∂v

∂t= −vp(v, t) a.e. t ≥ 0, v(z, 0) = z,

for some choice of p = p(z, t) such that p(·, t) ∈ P (the Carathéodory family ofholomorphic functions q on U with positive real part such that q(0) = 1) for almostall t ∈ [0,∞) and p(z, ·) is measurable on [0,∞) for z ∈ U (see [48]). On the otherhand, Becker [6] studied the general form of solutions to the Loewner differentialequation in one complex variable, i.e.

∂ f

∂t(z, t) = z f ′(z, t)p(z, t) a.e. t ≥ 0, ∀z ∈ U,

where p(·, t) ∈ P for any fixed t ∈ [0,∞), and p(z, ·) is measurable on [0,∞) for z ∈U . In one complex variable there exists a unique univalent solution f (z, t) = et z+· · ·of the above Loewner differential equation, called the canonical solution, and given by

f (z, s) = limt→∞ etv(z, s, t)

locally uniformly on U , for each s ≥ 0, where the transition function v = v(z, s, t)is the unique Lipschitz continuous solution on [s,∞) of the initial value problem

∂v

∂t= −vp(v, t) a.e. t ≥ s, v(z, s, s) = z.

For what follows, all necessary definitions are given in Sect. 2 and the beginningof Sect. 4.

In higher dimensions, the above uniqueness result related to the normalized solu-tions of the Loewner differential equation does not hold (see [14,20]). Indeed, iff (z, t) = et z + · · · is a Loewner chain on Bn × [0,∞) that satisfies the Loewnerdifferential equation

∂ f

∂t(z, t) = D f (z, t)h(z, t) a.e. t ≥ 0, ∀z ∈ Bn,

where h(·, t) ∈ M for t ∈ [0,∞), and h(z, ·) is measurable on [0,∞) for z ∈ Bn ,and if � is a normalized biholomorphic mapping on C

n , then g(z, t) = �( f (z, t))is another Loewner chain, which satisfies the same Loewner differential equation asf (z, t).

In dimension n ≥ 2, the family S0(Bn) of normalized biholomorphic mappingson Bn which have parametric representation is a proper subfamily of S(Bn) (see[20]). We remark that S0(Bn) may be characterized as the subfamily of S(Bn) whichconsists of those mappings f that can be embedded in Loewner chains f (z, t) suchthat {e−t f (·, t)}t≥0 is a normal family on Bn (see [29]). The family S0(Bn) is compact,while S(Bn) is not for n ≥ 2.

Now, suppose that we allow the subordination chains to be non-normalized. Ifa : [0,∞) → C is a function of class C1 such that a(t) �= 0 for t ≥ 0 and |a(·)|is strictly increasing on [0,∞), and if f (z, t) = a(t)z + · · · is a non-normalized

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univalent subordination chain on the unit disc U , then f∗(z, t∗) = f (e−iθ(t)z, t)/a(0)

is a Loewner chain, where t∗ = log(|a(t)/a(0)|) and θ(t) = arg(a(t)/a(0)). Thatis, there is a Loewner chain with essentially the same geometric properties as theoriginal chain. In dimension n ≥ 2, there exist non-normalized subordination chainsf (z, t) = et Az + · · · which cannot be normalized by an analogous change of vari-able. On the other hand, there exist mappings f ∈ S(Bn)\S0(Bn) which have usefulembeddings in non-normalized subordination chains. For example, if A ∈ L(Cn)

is such that m(A) > 0 and if f is a spirallike mapping with respect to A, thenf (z, t) = et A f (z) is an A-normalized univalent subordination chain. However,there exist spirallike mappings f which do not belong to S0(Bn) (see e.g. [28]).In connection with this observation, we introduced the family S0

A(Bn) of mappingswhich have A-parametric representation (see [22]). We proved that if A ∈ A , whereA = {A ∈ L(Cn) : k+(A) < 2m(A)}, then S0

A(Bn) consists of all mappings f thatcan be imbedded in A-normalized univalent subordination chains f (z, t) = et Az+· · ·such that {e−t A f (·, t)}t≥0 is a normal family on Bn (see [22, Corollary 2.9]). Also, ifA ∈ A , then S0

A(Bn) is a compact subset of S(Bn) (see [22, Theorem 2.15]). Sincethe family SA(Bn) of spirallike mappings with respect to A is a closed subfamily ofS0

A(Bn), it follows that SA(Bn) is also compact for A ∈ A (see [22,60]). However,some of the above results fail in the case k+(A) = 2m(A) (see [22,60]). For exam-ple, the family S0

A(Bn) is not compact if k+(A) = 2m(A) and the family SA(Bn)

is compact if and only if A is nonresonant (see [60, Example 3.7, Theorem 3.1]).Also, we introduced the notion of A-asymptotic spirallikeness, a geometric character-ization of the image domain in terms of differential equations (see [22]). We provedthat if A ∈ A , then f ∈ S0

A(Bn) if and only if f ∈ S(Bn) and f (Bn) is an A-asymptotically spirallike domain (see [22, Theorem 3.6]). In other words, if Sa

A(Bn)

is the family of A-asymptotically spirallike mappings on Bn , then SaA(Bn) = S0

A(Bn)

whenever A ∈ A (see [22]). If k+(A) = 2m(A), then SaA(Bn) is a normal but not

necessarily compact family (see [60, Proposition 3.12, Remark 3.13]). On the otherhand, unlike the case k+(A) < 2m(A), S0

A(Bn) �= SaA(Bn) (see [60, Proposition 3.8,

Remark 3.9]).Further, Duren et al. [14] (see also [2,33,60]) studied the general solutions of the

Loewner differential equation

∂ f

∂t(z, t) = D f (z, t)h(z, t) a.e. t ≥ 0, ∀z ∈ Bn, (1.1)

where h(·, t) ∈ NA for t ∈ [0,∞), and h(z, ·) is measurable on [0,∞) for z ∈ Bn .In [14] (see also [33]) it was proved that if A ∈ A , then the Loewner differentialequation (1.1) has a unique normalized solution f (z, t) = et Az + · · · (the canonicalsolution) such that

f (z, s) = limt→∞ et Av(z, s, t) (1.2)

locally uniformly on Bn , for each s ≥ 0, where the transition mapping v = v(z, s, t)is the unique Lipschitz continuous solution on [s,∞) of the initial value problem

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∂v

∂t= −h(v, t) a.e. t ≥ s, v(z, s, s) = z.

Moreover, any other standard solution g(z, t) of (1.1) is given by g(z, t) = �( f (z, t)),for some mapping � ∈ H(Cn) with �(0) = 0 (see [14,33,60]). However, the aboveuniqueness result and the relation (1.2) do not hold if k+(A) = 2m(A) (see [14,33,60]).

Based on the above arguments, it is therefore of interest to consider A-normalizedunivalent subordination chains and mappings which have A-parametric representationon Bn , in the case that A ∈ A and n ≥ 2.

The main results of this paper can be summarized as follows. The notation inTheorem 1.1 is explained in the next section. The additional notation in Theorem 1.2is explained in Sect. 4.

Theorem 1.1 Let A ∈ A . Also, let f ∈ ex S0A(Bn) (respectively f ∈ supp S0

A(Bn))

and let f (z, t) be an A-normalized univalent subordination chain such that f =f (·, 0) and {e−t A f (·, t)}t≥0 is a normal family on Bn. Then e−t A f (·, t) ∈ ex S0

A(Bn)

(respectively e−t A f (·, t) ∈ supp S0A(Bn)) for t ≥ 0.

Theorem 1.2 Let A ∈ A and let M > 1. Then the following statements hold:

(i) Let f ∈ H(Bn) be a normalized mapping. Then f ∈ Rlog M (idBn ,NA) if andonly if there exists an A-normalized univalent subordination chain f (z, t) suchthat f (·, 0) = f , f (·, log M) = eA log M idBn and {e−t A f (·, t)}t≥0 is a normalfamily on Bn.

(ii) If f ∈ Rlog M (idBn ,NA), then

b

apaπ (‖z‖) ≤ ‖ f (z)‖ ≤ a

bpb

0(‖z‖), z ∈ Bn,

where a = ek(A) log M and b = em(A) log M .(iii) The family Rlog M (idBn ,NA) is compact.

(iv) Let f ∈ ex Rlog M (idBn ,NA). Also let f (z, t) be an A-normalized uni-valent subordination chain as in the statement (i). Then e−t A f (·, t) ∈ex Rlog M−t (idBn ,NA) for 0 ≤ t ≤ log M.

(v) Let f ∈ supp Rlog M (idBn ,NA). Also let f (z, t) be an A-normalized univalentsubordination chain as in the statement (i). Then there exists a constant ε ∈(0, log M) such that e−t A f (·, t) ∈ supp Rlog M−t (idBn ,NA) for 0 ≤ t < ε.

(vi) Rlog M (idBn , ex NA) = Rlog M (idBn ,NA).

As a consequence of Theorem 1.2 (vi), we obtain the following density theorem:

Corollary 1.3 Let A ∈ A . If f ∈ S0A(Bn), then for any ε > 0 and for any compact

set K ⊂ Bn, there exist M > 1 and fε,K ∈ Rlog M (idBn , ex NA) such that ‖ f (z) −fε,K (z)‖ < ε on K .

Note that Theorem 1.1 is a generalization to higher dimensions of related resultsdue to Pell [43] and Kirwan [37], while Theorem 1.2 is a generalizations to C

n of wellknown results due to Prokhorov [52] and Roth [56]. On the other hand, Corollary 1.3may be considered as an n-dimensional version of a well known density result due toLoewner [40].

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2 Preliminaries

Let Cn denote the space of n complex variables z = (z1, . . . , zn) with the Euclidean

inner product 〈z, w〉 = ∑nj=1 z jw j and the Euclidean norm ‖z‖ = 〈z, z〉1/2. The

open ball {z ∈ Cn : ‖z‖ < ρ} is denoted by Bn

ρ and the unit ball Bn1 is denoted by

Bn . The closed ball {z ∈ Cn : ‖z‖ ≤ ρ} is denoted by B

nρ . In the case of one complex

variable, B1 is denoted by U .Let L(Cn, C

m) denote the space of linear operators from Cn into C

m with thestandard operator norm and let In be the identity in L(Cn), where L(Cn) = L(Cn, C

n).If � is a domain in C

n , let H(�) be the family of holomorphic mappings from � intoC

n with the compact-open topology. If f ∈ H(Bn), we say that f is normalized iff (0) = 0 and D f (0) = In . Let S(Bn) be the family of normalized biholomorphicmappings on Bn . In the case of one complex variable, the family S(U ) is denoted byS. Also let S∗(Bn) be the subfamily of S(Bn) consisting of starlike mappings on Bn .

If f ∈ H(Bn), we say that f is locally biholomorphic on Bn if J f (z) �= 0, z ∈ Bn ,where J f (z) = det D f (z) and D f (z) is the complex Jacobian matrix of f at z. LetL Sn be the family of normalized locally biholomorphic mappings on Bn .

We shall use the following notions related to an operator A ∈ L(Cn) (cf. [35]):

m(A) = min{�〈A(z), z〉 : ‖z‖ = 1},k(A) = max{�〈A(z), z〉 : ‖z‖ = 1},

k+(A) = max{�λ : λ ∈ σ(A)},

where σ(A) is the spectrum of A. Then k+(A) ≤ ‖A‖ and it is known ([12]; see also[16] and [54, p. 311]) that

k+(A) = limt→∞

log ‖et A‖t

.

Note that k+(A) is usually called the upper exponential index of A.

Remark 2.1 If A ∈ L(Cn), then

m(A) ≤ k+(A) ≤ k(A) ≤ ‖A‖. (2.1)

Proof Clearly, k(A) ≤ ‖A‖. If λ is an eigenvalue of A, then there exists z ∈ Cn\{0}

such that A(z) = λz. Then λ = 〈A(z), z‖z‖2 〉, and hence m(A) ≤ �λ ≤ k(A). Since

λ is arbitrary, the relation (2.1) follows, as desired. ��Let

A = {A ∈ L(Cn) : k+(A) < 2m(A)}.

Then In ∈A . Also, if A=diag(λ1, . . . , λn) with max j=1,...n �λ j <2 min j=1,...,n �λ j ,then A ∈ A . Note that if A ∈ A , then m(A) > 0 by (2.1).

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Remark 2.2 The authors [14, Lemma 2.1] (see also, [21, Lemma 1.1]) proved that ifA ∈ L(Cn) and u ∈ C

n with ‖u‖ = 1, then the following growth estimates hold:

e−k(A)t ≤ ‖e−t A(u)‖ ≤ e−m(A)t , t ≥ 0, (2.2)

and

em(A)t ≤ ‖et A(u)‖ ≤ ek(A)t , t ≥ 0. (2.3)

Definition 2.3 (see [31,59]) Let A ∈ L(Cn) be such that m(A) > 0. Also let � be adomain in C

n which contains the origin. We say that � is spirallike with respect to Aif e−t A(w) ∈ �, for all w ∈ � and t ≥ 0.

A mapping f ∈ S(Bn) is called spirallike with respect to A if f (Bn) is a spirallikedomain with respect to A. We denote by SA(Bn) the family of spirallike mappingswith respect to A.

Remark 2.4 Let A ∈ L(Cn) be such that m(A) > 0. It is well known (see [59]) thatif f ∈ L Sn , then f is spirallike with respect to A if and only if

�〈[D f (z)]−1 A f (z), z〉 > 0, z ∈ Bn\{0}.

If f, g ∈ H(Bn), we say that f is subordinate to g ( f ≺ g) if there exists a Schwarzmapping v (i.e. v ∈ H(Bn) and ‖v(z)‖ ≤ ‖z‖, z ∈ Bn) such that f = g ◦ v.

Definition 2.5 A mapping f : Bn ×[0,∞) → Cn is called a univalent subordination

chain if f (·, t) is biholomorphic on Bn , f (0, t) = 0 for t ≥ 0, and f (·, s) ≺ f (·, t),0 ≤ s ≤ t < ∞. A univalent subordination chain is said to be A-normalized ifD f (0, t) = et A for t ≥ 0, where A ∈ L(Cn) with m(A) > 0. We say that f (z, t)is a Loewner chain (or a normalized univalent subordination chain) if f (z, t) is In-normalized.

The above subordination condition is equivalent to the existence of a uniqueSchwarz mapping v = v(z, s, t), called the transition mapping associated with f (z, t),such that f (z, s) = f (v(z, s, t), t) for z ∈ Bn and t ≥ s ≥ 0.

We remark that the transition mapping v(z, s, t) satisfies the following semigroupproperty:

v(z, s, t) = v(v(z, s, u), u, t), z ∈ Bn, 0 ≤ s ≤ u ≤ t < ∞. (2.4)

The following families of holomorphic mappings on the unit ball Bn play the roleof the Carathéodory family in C

n :

N = {h ∈ H(Bn) : h(0) = 0, �〈h(z), z〉 > 0, z ∈ Bn\{0}},NA = {h ∈ N : Dh(0) = A}, where A ∈ L(Cn), m(A) > 0.

The family NIn is denoted by M . If n = 1, then f ∈ M if and only if f (z)/z ∈ P ,where

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P = {p ∈ H(U ) : p(0) = 1,�p(z) > 0, z ∈ U }.

These families play an essential role in the study of Loewner chains and the Loewnerdifferential equation in dimension n ≥ 2 (see [20,22,23,25,28,29,44,49]).

Definition 2.6 (see [22]) Let A ∈ L(Cn) be such that m(A) > 0. We say that anormalized mapping f ∈ H(Bn) has A-parametric representation if there exists amapping h : Bn × [0,∞) → C

n such that h(·, t) ∈ NA for t ∈ [0,∞), h(z, ·) ismeasurable on [0,∞) for z ∈ Bn , and

f (z) = limt→∞ et Av(z, t)

locally uniformly on Bn , where v = v(z, t) is the unique locally absolutely continuoussolution on [0,∞) of the initial value problem

∂v

∂t= −h(v, t) a.e. t ≥ 0, v(z, 0) = z,

for all z ∈ Bn .If A = In and f has In-parametric representation, then f has parametric represen-

tation in the usual sense (see [20,28]; cf. [50]).Let S0

A(Bn) (resp. S0(Bn)) be the family of mappings which have A-parametricrepresentation (resp. parametric representation).

Remark 2.7 Let A ∈ A . The condition in Definition 2.6 is equivalent to the fact thatthere exists an A-normalized univalent subordination chain f (z, t) such that f =f (·, 0) and {e−t A f (·, t)}t≥0 is a normal family on Bn (see [22, Corollary 2.9] and[14]; cf. [20,28,29,33,51]).

Remark 2.8 (i) If n = 1, S0(U ) = S (see [48]), but S0(Bn) � S(Bn) for n ≥ 2 (see[20,28]; cf. [50]). It is well known that S(Bn) is not compact in dimension n ≥ 2.On the other hand, S0

A(Bn) is a compact family for A ∈ A (see [22, Theorem2.15]).

(ii) Note that S∗(Bn) � S0(Bn) and SA(Bn) ⊆ S0A(Bn) for A ∈ A (see [22]).

Moreover, if A + A∗ = 2α In , where α > 0, then S0A(Bn) = S0(Bn) (see [22]).

Hence, in the case n = 1, S0a (U ) = S0(U ) = S, where a ∈ C, �a > 0.

(iii) It is not difficult to see that f ∈ SA(Bn) (resp. f ∈ S∗(Bn)) if and only if f (z, t) =et A f (z) is an A-normalized univalent subordination chain (resp. f (z, t) = et f (z)is a Loewner chain).

Definition 2.9 Let X be a locally convex linear space over C and let E ⊆ X .

(i) A point x ∈ E is called an extreme point of E provided x = t y + (1 − t)z, wheret ∈ (0, 1), y, z ∈ E , implies x = y = z. That is, x ∈ E is an extreme point of Eif x is not a proper convex combination of two points in E .

(ii) A point w ∈ E is called a support point of E if �L(w) = maxy∈E �L(y) forsome continuous linear functional L : X → C such that �L is nonconstant on E .

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Let ex E and supp E be the sets of extreme points of E and support points of Erespectively. By the Krein–Milman theorem (see e.g. [32, Chapter 4]), it is known thatif E is a nonempty compact subset of X then ex E is a nonempty subset of E . Also,it is known that if E is a compact subset of X which has at least two distinct points,then supp E is a nonempty subset of E . We shall consider X = H(Bn).

In this paper we are concerned with extreme points and support points for thefamilies S0

A(Bn) and Rlog M (idBn ,NA) (the normalized time-log M-reachable familygenerated by the Carathéodory mappings on Bn × [0, log M]), where A ∈ A andM > 1.

3 Extreme points and support points for the family S0A(Bn)

In this section, we shall consider extreme points and support points for the compactfamily S0

A(Bn), where A ∈ A (compare [30, Theorem 3.1]). In particular, Theorems3.1 and 3.5 apply for the family S0(Bn) (see [25]). These results were obtained byPell [43] and Kirwan [37] for n = 1. Note that the condition k+(A) < 2m(A) impliesthat m(A) > 0, in view of (2.1).

Theorem 3.1 Let A ∈ A . Also, let f ∈ ex S0A(Bn) and let f (z, t) be an A-normalized

univalent subordination chain such that f = f (·, 0) and {e−t A f (·, t)}t≥0 is a normalfamily on Bn. Then e−t A f (·, t) ∈ ex S0

A(Bn) for t ≥ 0.

Proof Let vs,t (z) = v(z, s, t) be the transition mapping associated with f (z, t). Also,let v(z, t) = v0,t (z) for z ∈ Bn and t ≥ 0. Fix t ≥ 0. Clearly, e−t A f (·, t) ∈S0

A(Bn). Indeed, if L(z, s) = e−t A f (z, t + s) for z ∈ Bn and s ≥ 0, then L(·, s) isbiholomorphic on Bn , L(0, s) = 0, DL(0, s) = es A, s ≥ 0, and

L(z, s) = L(w(z, s, τ ), τ ), z ∈ Bn, 0 ≤ s ≤ τ < ∞,

where w(z, s, τ ) = v(z, t + s, t + τ). Thus, L(z, s) is an A-normalized univalentsubordination chain, and it is easy to see that {e−s A L(·, s)}s≥0 is a normal family onBn . Since L(·, 0) = e−t A f (·, t), it follows that e−t A f (·, t) ∈ S0

A(Bn), as claimed.Next, suppose that

e−t A f (z, t) = λg(z) + (1 − λ)h(z), z ∈ Bn,

where λ ∈ (0, 1) and g, h ∈ S0A(Bn). Then

f (z) = f (v(z, t), t) = λet Ag(v(z, t)) + (1 − λ)et Ah(v(z, t)), z ∈ Bn .

We shall prove that the mappings et Ag(v(·, t)) and et Ah(v(·, t)) belong to S0A(Bn).

Since g ∈ S0A(Bn), there exists an A-normalized univalent subordination chain

g(z, s) such that {e−s Ag(·, s)}s≥0 is a normal family on Bn and g = g(·, 0). Then it isnot difficult to deduce that the mapping F = F(z, s) : Bn × [0,∞) → C

n given by

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I. Graham et al.

F(z, s) ={

et Ag(v(z, s, t)), z ∈ Bn, 0 ≤ s ≤ tet Ag(z, s − t), z ∈ Bn, s > t,

is an A-normalized univalent subordination chain such that et Ag(v(·, t)) = F(·, 0).Indeed, F(·, s) is biholomorphic on Bn , F(0, s) = 0 and DF(0, s) = es A for s ≥ 0.Let

ω(z, s, τ ) =⎧⎨

⎩

v(z, s, τ ), 0 ≤ s ≤ τ ≤ tv(v(z, s, t), 0, τ − t), 0 ≤ s ≤ t ≤ τ < ∞v(z, s − t, τ − t), τ ≥ s ≥ t,

where v(z, s, τ ) is the transition mapping associated with g(z, τ ). Then ω(·, s, τ ) is aunivalent Schwarz mapping and

F(z, s) = F(ω(z, s, τ ), τ ), z ∈ Bn, 0 ≤ s ≤ τ < ∞.

Thus, F(z, s) is an A-normalized univalent subordination chain. Since the family{e−s Ag(·, s)}s≥0 is a normal family on Bn , we deduce that {e−s A F(·, s)}s≥0 is also anormal family on Bn . Indeed, for each r ∈ (0, 1), there exists M = M(r, A) > 0 suchthat ‖e−s Ag(z, s)‖ ≤ M(r, A) for ‖z‖ ≤ r and s ≥ 0. Now, if 0 ≤ s ≤ t , we obtainthat

‖e−s A F(z, s)‖ = ‖e(t−s)Ag(v(z, s, t))‖ ≤ ‖e(t−s)A‖ · ‖g(v(z, s, t))‖≤ M(r, A)‖e(t−s)A‖ ≤ M(r, A)‖et A‖ = M∗

t (r, A), ‖z‖ ≤ r < 1.

Here we have used the fact that ‖v(z, s, t)‖ ≤ r , ‖z‖ ≤ r < 1, and ‖e−s A‖ ≤e−m(A)s ≤ 1, in view of the relations m(A) > 0 and (2.2). On the other hand, if s > t ,then

‖e−s A F(z, s)‖ = ‖e−(s−t)Ag(z, s − t)‖ ≤ M(r, A), ‖z‖ ≤ r < 1.

Letting M∗∗t (r, A) = max{M(r, A), M∗

t (r, A)}, we deduce that

‖e−s A F(z, s)‖ ≤ M∗∗t (r, A), ‖z‖ ≤ r < 1, s ≥ 0.

Thus {e−s A F(·, s)}s≥0 is a normal family on Bn .In view of the above arguments, we deduce that et Ag(v(·, t)) ∈ S0

A(Bn), asdesired. Similarly, et Ah(v(·, t)) ∈ S0

A(Bn). Since f ∈ ex S0A(Bn), we must have

et Ag(v(·, t)) ≡ et Ah(v(·, t)). By the identity theorem for holomorphic mappings, wededuce that g ≡ h. Thus e−t A f (·, t) ∈ ex S0

A(Bn), as desired. This completes theproof. ��

It is well known that no bounded mapping in S is an extreme or support point of S(see e.g. [13]). Indeed, if f ∈ ex S or f ∈ supp S, then f maps the unit disc U ontothe complement of a continuous arc tending to ∞ with increasing modulus (see e.g.[10,32,37,41]). In higher dimensions, the following results hold:

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Proposition 3.2 Let A ∈ A . Also, let f (z, t) be an A-normalized univalent subor-dination chain such that f = f (·, 0) and {e−t A f (·, t)}t≥0 is a normal family on Bn.Let vs,t (z) = v(z, s, t) be the transition mapping associated with f (z, t). Also, letvt (z) = v(z, t) = v0,t (z) for z ∈ Bn and t ≥ 0. Then the mapping et Av(·, t) is not anextreme point of S0

A(Bn) for any t ≥ 0. In particular, the identity mapping idBn is notan extreme point of S0

A(Bn).

Proof Let t ≥ 0 be fixed and let gt = et Avt . In view of the proof of Theorem 3.1 andthe fact that the identity mapping idBn belongs to S0

A(Bn), we deduce that gt ∈ S0A(Bn).

Then for any F ∈ SA(Bn), ϕ(z) = et A F(e−t Agt (z)) belongs to S0A(Bn), by the proof

of Theorem 3.1. Let Fε±(z) = z ±εz2ne1, where ε > 0. Since DFε±(z) = In ±2εG(zn)

with ‖G(zn)‖ ≤ 1, [DFε±(z)]−1 exists for 0 < ε < 1/2 and for z ∈ Bn. Since

�〈[DFε±(z)]−1 AFε±(z), z〉 → �〈Az, z〉

uniformly on Bn

as ε → +0, and �〈Az, z〉 ≥ m(A)‖z‖2 > 0 for z �= 0, there existsa number ε > 0 such that

�〈[DFε±(z)]−1 AFε±(z), z〉 ≥ 1

2m(A), ‖z‖ = 1.

For any fixed w with ‖w‖ = 1, by applying the minimum principle for harmonicfunctions on U to

p(ζ ) ={� 1

ζ〈[DFε±(ζw)]−1 AFε±(ζw),w〉, ζ �= 0

�〈Aw,w〉, ζ = 0,

we obtain that p(ζ ) > 0 on U . This implies that the mapping Fε± is spirallike withrespect to A. Let ϕε±(z) = et A Fε±(e−t Agt (z)) ∈ S0

A(Bn). Then we have ϕε± �= gt

and gt = (ϕε+ + ϕε−)/2. Thus gt = et Avt is not an extreme point of S0A(Bn). This

completes the proof. ��Before proving the following result, we recall the definition of a Runge pair in C

n .For details, see [53].

Definition 3.3 Let �1 ⊆ �2 ⊆ Cn be two domains. The pair (�1,�2) is a Runge

pair if O(�2) is dense in O(�1), where O(� j ) is the family of holomorphic functionson � j , j = 1, 2. A domain � ⊆ C

n is Runge if (�, Cn) is a Runge pair.

Proposition 3.4 below has been recently proved by Schleissinger [58], in the caseA = In . This gives an affirmative answer to [25, Conjecture 2.6]. In the case of supportpoints for S0

A(Bn), we have the following result:

Proposition 3.4 With notation as in the statement of Proposition 3.2, et Av(·, t) is nota support point of S0

A(Bn) for any t ≥ 0. In particular, the identity mapping idBn isnot a support point of S0

A(Bn).

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I. Graham et al.

Proof We shall use arguments similar to those in [58]. Fix t ≥ 0. Let gt (z) = et Avt (z)for z ∈ Bn . Taking into account the proof of Theorem 3.1, it follows that gt ∈ S0

A(Bn).Also, let Q : C

n → Cn be a polynomial mapping such that Q(0) = 0 and DQ(0) =

0n . First, we prove that there exists η > 0 such that gt +εet A Q(e−t Agt ) ∈ S0A(Bn) for

|ε| < η. Indeed, let qε(z) = z + εQ(z) for z ∈ Bn. Since qε is locally biholomorphic

on Bn

for ε small enough, it follows that

[Dqε(z)]−1 = [In + εDQ(z)]−1 = In + εSε(z), z ∈ Bn,

for some Sε(z) ∈ L(Cn) such that ‖Sε(z)‖ ≤ C1 for all z ∈ Bn

and small ε > 0,where C1 > 0 is a constant. Also, it is easy to see that

[Dqε(z)]−1 Aqε(z) = A(z) + εqε(z),

for some qε(z) ∈ Cn such that ‖qε(z)‖ ≤ C2 for all z ∈ B

nand small ε > 0, where

C2 > 0 is a constant, and thus

�〈[Dqε(z)]−1 Aqε(z), z〉 = �〈A(z), z〉 + �[ε〈qε(z), z〉] → �〈A(z), z〉

uniformly on Bn

as ε → 0. Since �〈A(z), z〉 > 0 for z �= 0, it follows by using anargument similar to that in the proof of Proposition 3.2 that there exists some η > 0such that

�〈[Dqε(z)]−1 Aqε(z), z〉 > 0, z ∈ Bn\{0},

for |ε| < η. Hence qε ∈ SA(Bn) for |ε| < η, and since k+(A) < 2m(A), it followsthat qε ∈ S0

A(Bn) (see [22]). Next, taking into account the proof of Theorem 3.1, wededuce that et Aqε(e−t Agt ) ∈ S0

A(Bn) for |ε| < η, i.e. gt + εet A Q(e−t Agt ) ∈ S0A(Bn)

for |ε| < η, as claimed.Now, suppose that gt ∈ supp S0

A(Bn). Then there exists a continuous linear func-tional L on H(Bn) such that �L|S0

A(Bn) �= constant and

�L(gt ) = maxF∈S0

A(Bn)

�L(F).

Let Q : Cn → C

n be a polynomial mapping such that Q(0) = 0 and DQ(0) = 0n .In view of the above arguments, we deduce that there exists some η > 0 such thatgt + εet A Q(e−t Agt ) ∈ S0

A(Bn) for |ε| < η. Since gt is a support point of S0A(Bn) and

�L(gt + εet A Q(e−t Agt )) = �L(gt ) + �L(εet A Q(e−t Agt )) for |ε| < η,

we deduce that

�L(et A Q(e−t Agt )) = �L(et A Q(vt )) = 0. (3.1)

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In view of [5, Proposition 5.1], we deduce that (vt (Bn), Bn) is a Runge pair andsince (Bn, C

n) is also a Runge pair, we easily deduce that (vt (Bn), Cn) is a Runge pair

too, i.e. vt (Bn) is a Runge domain in Cn . Hence, if p ∈ H(Bn) is such that p(0) = 0

and Dp(0) = 0n , then there exists a sequence of polynomial mappings {Qk}k∈N in Cn

such that Qk(0) = 0, DQk(0) = 0n and p ◦ v−1t = limk→∞ Qk locally uniformly on

vt (Bn). Since vt is biholomorphic on Bn , it follows that p = limk→∞ Qk(vt ) locallyuniformly on Bn . In view of (3.1) and the fact that Pt = et A Q is a polynomial mappingin C

n if and only if Q is a polynomial mapping in Cn , we deduce that �L(p) = 0,

for all p ∈ H(Bn) with p(0) = 0 and Dp(0) = 0n .Finally, if F ∈ S0

A(Bn), then F(z) = z + G(z), z ∈ Bn , where G ∈ H(Bn) is suchthat G(0) = 0 and DG(0) = 0n . Then

�L(F) = �L(idBn ) + �L(G) = �L(idBn ),

and since F is arbitrary, we conclude that �L|S0A(Bn) is constant. However, this is a

contradiction. Hence gt �∈ supp S0A(Bn), as desired. This completes the proof. ��

We next prove the following result related to support points for the family S0A(Bn)

(cf. [25, Theorem 2.5] and [30, Theorem 3.3], in the case A = In). In the case A = In ,this result was recently proved by Schleissinger [58]. In the case n = 1, this resultwas obtained by Kirwan [37] and Pell [43]; see also [46].

Theorem 3.5 Let A ∈ A . Also let f ∈ supp S0A(Bn) and let f (z, t) be an A-

normalized univalent subordination chain such that f = f (·, 0) and {e−t A f (·, t)}t≥0is a normal family on Bn. Then e−t A f (·, t) ∈ supp S0

A(Bn) for t ≥ 0.

Proof Note that e−t A f (·, t) ∈ S0A(Bn) for t ≥ 0, by the same argument as in the proof

of Theorem 3.1. Let v(z, s, t) denote the transition mapping associated to f (z, t). Also,let vt (z) = v(z, t) = v(z, 0, t), z ∈ Bn , t ≥ 0. Since f ∈ supp S0

A(Bn), there is acontinuous linear functional L on H(Bn) such that �L|S0

A(Bn) �= constant and

�L( f ) = maxg∈S0

A(Bn)

�L(g). (3.2)

Now fix t ≥ 0. Let Lt : H(Bn) → C be given by

Lt (g) = L(et Ag ◦ vt ), g ∈ H(Bn).

It is clear that Lt is a continuous linear functional on H(Bn) and

Lt (e−t A f (·, t)) = L( f (v(·, t), t)) = L( f ).

In view of the above arguments, we deduce that

�Lt (e−t A f (·, t)) = �L( f ) ≥ �L(et Ag ◦ vt ) = �Lt (g),

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I. Graham et al.

for all g ∈ S0A(Bn), i.e. �Lt (e−t A f (·, t)) = maxg∈S0

A(Bn) �Lt (g). Here we have used

the fact that et Ag ◦vt ∈ S0A(Bn) for g ∈ S0

A(Bn), by the same argument as in the proofof Theorem 3.1. From (3.2) and Proposition 3.4, we have

�Lt (idBn ) = �L(et Avt ) < �L( f ) = �Lt (e−t A f (·, t)),

and thus �Lt |S0A(Bn) is nonconstant, as desired. This completes the proof. ��

Conjecture 3.6 No bounded mapping in S0A(Bn) is a support point (resp. extreme

point) of S0A(Bn) for n ≥ 2 and A ∈ A .

As a consequence of the proof of Theorem 3.5, we obtain a generalization to then-dimensional case of an extremal principle (called “Basic Lemma”) of Kirwan andSchober [38] (see also [17,56]). We note that the functional λ need not be linear.

Theorem 3.7 Let A ∈ A . Also let λ : S0A(Bn) → R be a real-valued functional. If

f ∈ S0A(Bn) provides the maximum for λ over the family S0

A(Bn) and if f (z, t) is anA-normalized univalent subordination chain such that {e−t A f (·, t)}t≥0 is a normalfamily on Bn and f = f (·, 0), then for all t ≥ 0, e−t A f (·, t) ∈ S0

A(Bn) provides themaximum for the associated functional λt : S0

A(Bn) → R given by

λt (g) = λ(et Ag ◦ vt ), g ∈ S0A(Bn).

Here vt = v0,t and vs,t = v(·, s, t) is the transition mapping associated to f (z, t).Moreover, λ( f ) = λt (e−t A f (·, t)).

If A ∈ A is a diagonal matrix, we obtain the following partial growth theorem forthe family S0

A(Bn).

Theorem 3.8 Let A = diag(a1, . . . , an) ∈ A be a diagonal matrix. Assume that

0 < �a1 = · · · = �ap < �ap+1 ≤ · · · ≤ �an .

Let f ∈ S0A(Bn). Then

‖( f1(z), . . . , f p(z), 0, . . . , 0)‖ ≤ ‖z‖(1 − ‖z‖)2 , z ∈ Bn . (3.3)

This estimate is sharp.

Proof Let M = diag(m1, . . . , mn) be the diagonal matrix such that mi = ai for1 ≤ i ≤ p and mi = a1 for p + 1 ≤ i ≤ n. Since f ∈ S0

A(Bn), there exist mappingsh : Bn × [0,∞) → C

n and v : Bn × [0,∞) → Cn which satisfy the conditions of

Definition 2.6. Let z ∈ Bn\{0} be fixed. Then

limt→∞ et Mv(z, t) = lim

t→∞ et (M−A)et Av(z, t) = ( f1(z), . . . , f p(z), 0, . . . , 0).

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Since

‖et Mv(z, t)‖ = em(A)t‖v(z, t)‖ ≤ ‖z‖(1 − ‖z‖)2 ,

by [22, Theorem 2.1], the estimate (3.3) follows.Finally, we prove that (3.3) is sharp. For a given A = diag(a1, . . . , an) ∈ A which

satisfies the assumptions of the theorem, let A2 = diag(a2, . . . , an). Let

f (z) = (f1(z1), f2(z

′)), z = (z1, z′) ∈ Bn,

where

f1(z1) = z1

(1 − z1)2 and f2 ∈ S0A2

(Bn−1).

Since f1 ∈ S, f1 ∈ S0a (U ) by Remark 2.8 (ii), where a = a1. Therefore, f ∈ S0

A(Bn)

and the equality ‖ f (r, 0′)‖ = r/(1 − r)2 holds. Thus, the estimate (3.3) is sharp, asdesired. ��Remark 3.9 Let A = diag(a1, . . . , an) be a diagonal matrix such that �a1 = · · · =�an > 0. In [14, Corollary 2.12], the authors proved that if f ∈ S0

A(Bn), then

‖z‖(1 + ‖z‖)2 ≤ ‖ f (z)‖ ≤ ‖z‖

(1 − ‖z‖)2 , z ∈ Bn .

This result is sharp.

Example 3.10 Let A be a diagonal matrix such that

A =[

a 00 A2

]

∈ A ,

where m(A) = �a > 0. Let f : Bn → Cn be given by

f (z) = ( f1(z1), f2(z′)), z = (z1, z′) ∈ Bn,

where

f1(z1) = z1

(1 − z1)2 and f2 ∈ S0A2

(Bn−1)\SA2(Bn−1).

It is easy to see that f ∈ S0A(Bn)\SA(Bn). For fixed r ∈ (0, 1), let z0 = (r, 0′) and

L(g) = g1(z0), for g ∈ H(Bn).

Then L is a continuous linear functional on H(Bn). For g ∈ S0A(Bn), we have

�L(g) ≤ |g1(z0)| ≤ f1 (r) = �L( f )

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I. Graham et al.

by Theorem 3.8. We also have idBn ∈ S0A(Bn) and �L(idBn ) < f1 (r) = �L( f ).

Thus f ∈ supp S0A(Bn), and this gives a non-trivial example of a mapping satisfying

the hypotheses of Theorem 3.5.

The following remark provides a basic difference between the theory in one dimen-sional case and that in higher dimensions. Note that the Koebe function is an extremepoint and also a support point for the family S (see [13]).

Remark 3.11 Let f : Bn → Cn be given by

f (z) =(

z1

(1 − z1)2 , . . . ,zn

(1 − zn)2

)

, z = (z1, . . . , zn) ∈ Bn .

Then f ∈ supp S0(Bn)\ex S0(Bn) for n ≥ 2.

Proof It is clear that ex S0(Bn) ∩ S∗(Bn) ⊆ exS∗(Bn). Since f �∈ exS∗(Bn) forn ≥ 2, by [61, Proposition 2.3.2], it follows that f �∈ ex S0(Bn). On the other hand,by using an argument similar to that in Example 3.10, we have f ∈ supp S0(Bn). ��Remark 3.12 Let A = diag(a1, . . . , an) be a diagonal matrix such that max{�a j :j = 1, . . . , n} < 2 min{�a j : j = 1, . . . , n}. Then A ∈ A . Also, let f j ∈ S,j = 1, . . . , n, and f (z) = ( f1(z1), . . . , fn(zn)) for z = (z1, . . . , zn) ∈ Bn . SinceS = S0

a (U ) for a ∈ C, �a > 0, it is not difficult to deduce that f ∈ S0A(Bn). On

the other hand, it is easily seen that if f ∈ ex S0A(Bn), then f j ∈ ex S, j = 1, . . . , n.

However, the converse of this statement does not hold in dimension n ≥ 2, by Remark3.11.

4 Extremal problems for reachable families generated by the Carathéodoryfamily NA

In this section we study bounded mappings in S(Bn) which have A-parametric rep-resentation, where A ∈ A . To this end, we use ideas from control theory to obtainproperties of the time-log M-reachable family Rlog M (idBn ,NA), where M > 1. Weprove a growth theorem for this family and a characterization in terms of univalent sub-ordination chains. We also prove a density theorem using the Krein–Milman theoremand some technical lemmas.

Let α ∈ R and let κα : U → C be the Koebe function

κα(ζ ) = ζ

(1 − eiαζ )2 , ζ ∈ U.

It is well known that this function provides the maximum for various extremal problemsin the family S. Also let M ∈ [1,∞) and let pM

α : U → C be the Pick function givenby

pMα (ζ ) = Mκ−1

α

(1

Mκα(ζ )

)

, ζ ∈ U.

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Note that pMα (ζ ) maps the unit disc onto the disc UM = {ζ ∈ U : |ζ | < M} except

for a radial segment, and satisfies the relation

pMα (ζ )

(1 − eiα pMα (ζ )/M)2 = κα(ζ ), ζ ∈ U.

The Pick function pMα plays a basic role in the study of various extremal problems for

the family

S(M) ={

f ∈ S : | f (ζ )| < M, ζ ∈ U}

of bounded normalized univalent functions on the unit disc (see [52]).Let A ∈ A . Also, let M ∈ [1,∞) and

S0A(M, Bn) =

{f ∈ S0

A(Bn) : ‖ f (z)‖ < M, z ∈ Bn}.

The family S0In

(M, Bn) is denoted by S0(M, Bn). Clearly, S0(M, U ) = S(M). It isnot difficult to see that f : Bn → C

n given by

f (z) = (pMα (z1), . . . , pM

α (zn)), z = (z1, . . . , zn) ∈ Bn,

belongs to S0(M, Bn).

Definition 4.1 Let E ⊆ [0,∞) be an interval and let � ⊆ H(Bn) be a normal family.A mapping h = h(z, t) : Bn × E → C

n is called a Carathéodory mapping on E withvalues in � if the following conditions hold:

(i) h(·, t) ∈ � for t ∈ E .(ii) h(z, ·) is a measurable mapping on E for z ∈ Bn .

Let C (E,�) be the family of all Carathéodory mappings on E with values in �. Interms of control theory, the mapping h = h(z, t) may be called a control function andthe family C (E,�) may be called a control system in H(Bn). Also, the family � maybe called an input family (cf. [36,39,56]).

Definition 4.2 Let A ∈ A , T ∈ [0,∞) and let � ⊆ NA. Also, let h ∈ C ([0, T ],�)

and let v = v(z, t; h) be the unique Lipschitz continuous solution on [0, T ] of theinitial value problem

∂v

∂t(z, t) = −h(v(z, t), t) a.e. t ∈ [0, T ], v(z, 0) = z, (4.1)

for z ∈ Bn , such that v(·, t; h) is a univalent Schwarz mapping and Dv(0, t; h) = e−t A

for t ∈ [0, T ]. Also let

RT (idBn ,�) ={v(·, T ; h) : h ∈ C ([0, T ],�)

}

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I. Graham et al.

denote the family of all such solutions at t = T generated by all Carathéodorymappings on [0, T ] with values in �. The family RT (idBn ,�) is called the time-T -reachable family of (4.1) (cf. [25,56,57]). The set � is called the input set or inputfamily (cf. [56]). Let

RT (idBn ,�) = eT ART (idBn ,�) for T ∈ [0,∞)

and

R∞(idBn ,�) ={

limt→∞ et Av(·, t; h) : h ∈ C ([0,∞),�)

}.

The family RT (idBn ,�) will be called the normalized time-T-reachable family of(4.1).

Remark 4.3 It is known that R∞(idU ,M ) = S (see [47,48]). Also, if M ∈ (1,∞),then Rlog M (idU ,M ) = S(M) (see [18] and [56, Theorem 1.48]). On the other hand,R∞(idBn ,NA) = S0

A(Bn) (see [22]). Moreover, Rlog M (idBn ,NA) ⊆ S0A(M∗, Bn),

where M∗ = ‖eA log M‖, by Theorem 4.5. Obviously, if T = 0 then RT (idBn ,NA) ={idBn }.

We next obtain some properties of the family Rlog M (idBn ,NA) for M ∈ (1,∞)

and A ∈ A . First, we give an example of a mapping in the normalized reachablefamily Rlog M (idBn ,NA). In the case A = In , see [25].

Example 4.4 Let A ∈ A . Also, let M > 1 and F ∈ SA(Bn). Let F MA : Bn → C

n begiven by

F MA (z) = eA log M F−1(e−A log M F(z)), z ∈ Bn . (4.2)

Then F MA ∈ Rlog M (idBn ,NA).

Proof Since F ∈ SA(Bn), it follows that the mapping F MA is well defined. Also,

since F ∈ SA(Bn), we deduce that F(z, t) = et A F(z) is an A-normalized univalentsubordination chain and F(z) = F(v(z, t), t) for z ∈ Bn and t ≥ 0, where v(z, t) =F−1(e−t A F(z)). Obviously,

eA log Mv(z, log M) = eA log M F−1(e−A log M F(z)),

i.e. F MA (z) = eA log Mv(z, log M), z ∈ Bn . Hence F M

A ∈ Rlog M (idBn ,NA), asdesired. ��Theorem 4.5 Let A ∈ A . Also, let M > 1 and let f ∈ H(Bn) be a normalized map-ping. Then f ∈ Rlog M (idBn ,NA) if and only if there exists an A-normalized univalentsubordination chain f (z, t) such that f (·, 0) = f , f (·, log M) = eA log M idBn and{e−t A f (·, t)}t≥0 is a normal family on Bn. Hence Rlog M (idBn ,NA) ⊆ S0

A(M∗, Bn),where M∗ = ‖eA log M‖.

123


Proof (i) First, assume that f ∈ Rlog M (idBn ,NA). Let h(z, t) and v(z, t) satisfy theassumptions of Definition 4.2. Also, let w = w(z, s, t) be the unique Lipschitzcontinuous solution of the initial value problem

∂w

∂t(z, s, t) = −h(w(z, s, t), t), a.e. s ≤ t ≤ log M, w(z, s, s) = z,

for z ∈ Bn and 0 ≤ s < log M . Then w(·, s, t) is a univalent Schwarz mappingand Dw(0, s, t) = e(s−t)A (see [22]). Also, w(z, 0, t) = v(z, t), and thus

f (z) = eA log Mv(z, log M) = eA log Mw(z, 0, log M).

Now, if F(z, s) = eA log Mw(z, s, log M), s ∈ [0, log M), then F(z, s) satisfiesthe subordination condition on Bn × [0, log M) needed to be the restriction of anA-normalized univalent subordination chain, in view of the semigroup property(2.4) of the Schwarz mapping w(z, s, t). Also, F(z, 0) = f (z) and F(z, log M) =eA log M z for z ∈ Bn . Next, let f (z, s) : Bn × [0,∞) → C

n be given by

f (·, s) ={

F(·, s), s ∈ [0, log M),

es AidBn , s ≥ log M.

Then f (z, s) is an A-normalized univalent subordination chain such that f (z, 0) =f (z) and f (z, log M) = eA log M z for z ∈ Bn . Moreover, {e−s A f (·, s)}s≥0 is anormal family on Bn . Indeed, if 0 ≤ s < log M , then

‖e−s A f (z, s)‖=‖eA(log M−s)w(z, s, log M)‖≤‖eA log M‖e−m(A)sr ≤‖eA log M‖r,

for ‖z‖ ≤ r < 1. If s ≥ log M , then

‖e−s A f (z, s)‖ ≤ r, ‖z‖ ≤ r < 1.

Letting K ∗M (r, A) = max{‖eA log M‖r, r}, we obtain that ‖e−s A f (z, s)‖ ≤

K ∗M (r, A) for ‖z‖ ≤ r and s ≥ 0. Thus, f ∈ S0

A(M∗, Bn), as desired.(ii) Conversely, assume that there is an A-normalized univalent subordination chain

f (z, t) such that f (·, 0) = f , f (·, log M) = eA log M idBn and {e−t A f (·, t)}t≥0 isa normal family on Bn . Let v(z, s, t) be the transition mapping associated withf (z, t) and let v(z, t) = v(z, 0, t). Then

f (z) = f (v(z, log M), log M) = eA log Mv(z, log M), z ∈ Bn,

and hence f = eA log Mv(·, log M). On the other hand, since f (z, t) is an A-normalized univalent subordination chain, there exists a mapping h = h(z, t) ∈C ([0,∞),NA) such that (see the proof of [22, Theorem 2.8]; cf. [29, Theorem2.2])

∂ f

∂t(z, t) = D f (z, t)h(z, t), a.e. t ≥ 0, ∀z ∈ Bn .

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I. Graham et al.

In view of [22, Theorem 2.6], we deduce that v(z, s, t) is the unique Lipschitzcontinuous solution on [s,∞) of the initial value problem

∂v

∂t= −h(v, t) a.e. t ≥ s, v(z, s, s) = z,

for all z ∈ Bn and s ≥ 0. Since f (z) = eA log Mv(z, log M) for z ∈ Bn , it followsthat f ∈ Rlog M (idBn ,NA), as desired. This completes the proof.

��

We next prove a growth result for the family Rlog M (idBn ,NA) (see [52, Theorem1], in the case n = 1; also see [25], in the case A = In). It would be interesting to seeif the growth result (4.4) remains valid for the family S0

A(M, Bn), n ≥ 2. Note thatthe estimates (4.4) are sharp in the case that A = In (see [25]).

Theorem 4.6 Let A ∈ A . Also, let M > 1 and let f ∈ Rlog M (idBn ,NA). Then

1

apaπ (‖z‖) ≤ ‖e−A log M f (z)‖ ≤ 1

bpb

0(‖z‖), z ∈ Bn, (4.3)

where a = ek(A) log M and b = em(A) log M . These estimates are sharp if A is a diagonalmatrix. Moreover,

b

apaπ (‖z‖) ≤ ‖ f (z)‖ ≤ a

bpb

0(‖z‖), z ∈ Bn . (4.4)

Proof Since f ∈ Rlog M (idBn ,NA), there is h = h(z, t) ∈ C ([0, log M],NA) suchthat f = eA log Mv(·, log M), where v = v(z, t) is the unique Lipschitz continuoussolution on [0, log M] of the initial value problem (4.1). Now, fix z ∈ Bn\{0}. Sincev(z, ·) is Lipschitz continuous on [0, log M] locally uniformly with respect to z ∈ Bn

(see [22]), it follows that ‖v(z, ·)‖ is differentiable a.e. on [0, log M]. After elementarycomputations we deduce that

∂‖v(z, t)‖∂t

= − 1

‖v(z, t)‖�〈h(v(z, t), t), v(z, t)〉, a.e. t ∈ [0, log M].

Since h(·, t) ∈ NA, we deduce that (see [31]; cf. [44])

m(A)‖z‖2 1 − ‖z‖1 + ‖z‖ ≤ �〈h(z, t), z〉 ≤ k(A)‖z‖2 1 + ‖z‖

1 − ‖z‖ , z ∈ Bn, t ∈ [0, log M].

In view of the above relations, we obtain that

−k(A)1 + ‖v(z, t)‖1 − ‖v(z, t)‖ ≤ 1

‖v(z, t)‖ · ∂‖v(z, t)‖∂t

≤ −m(A)1 − ‖v(z, t)‖1 + ‖v(z, t)‖ ,

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for a.e. t ∈ [0, log M]. It is not difficult to deduce that the above relation is equivalentto the following:

d log κπ(‖v(z, t)‖) ≥ −k(A)dt and d log κ0(‖v(z, t)‖) ≤ −m(A)dt.

Integrating both sides of the above inequalities on [0, log M] and using the fact that thefunctions κπ and κ0 are increasing on (0, 1) and ‖v(z, ·)‖ is decreasing on [0, log M],we obtain that

κπ

(‖e−A log M f (z)‖

)≥ 1

aκπ(‖z‖) and κ0

(‖e−A log M f (z)‖

)≤ 1

bκ0(‖z‖).

Hence, we have

κ−1π

(1

aκπ(‖z‖)

)

≤ ‖e−A log M f (z)‖ ≤ κ−10

(1

bκ0(‖z‖)

)

, z ∈ Bn,

i.e.

1

apaπ (‖z‖) ≤ ‖e−A log M f (z)‖ ≤ 1

bpb

0(‖z‖), z ∈ Bn .

Next, we show that the estimates (4.3) are sharp if A is a diagonal matrix. We give aproof for n = 2. Let

A =[

α 00 β

]

∈ A

with �α ≤ �β. Then m(A) = �α and k(A) = �β. Let

f (z) = (pb0(z1), pa

π (z2)), z = (z1, z2) ∈ B2.

Since pb0 ∈ S(b) and pa

π ∈ S(a), there exist Loewner chains f1 and f2 on U ×[0,∞)

such that

f1(·, 0) = pb0, f1(·, log b) = bidU , {e−t f1(·, t)}t≥0 is a normal family on U

and

f2(·, 0) = paπ , f2(·, log a) = aidU , {e−t f2(·, t)}t≥0 is a normal family on U

by Theorem 4.5. Let

f (z, t) = ( f1(ei(�α)t z1, (�α)t), f2(e

i(�β)t z2, (�β)t)).

Then f (z, t) is an A-normalized univalent subordination chain on B2 ×[0,∞) whichsatisfies f (·, 0) = f , f (·, log M) = eA log M idB2 and {e−t A f (·, t)}t≥0 is a normal

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I. Graham et al.

family on B2. Hence f ∈ Rlog M (idB2 ,NA) by Theorem 4.5. For z1 = (r, 0) andz2 = (0, r), we have

f (z1) = (pb0(r), 0), f (z2) = (0, pa

π (r)).

Therefore the equalities in (4.3) are attained by this mapping.Finally, using (2.2), (2.3) and (4.3), we obtain (4.4), as desired. This completes the

proof. ��Corollary 4.7 Let A ∈ A . Then the family Rlog M (idBn ,NA) is compact for M > 1.

Proof In view of Theorem 4.6, Rlog M (idBn ,NA) is locally uniformly bounded.Thus, it suffices to prove that Rlog M (idBn ,NA) is closed. To this end, let { fk}k∈N

be a sequence in Rlog M (idBn ,NA) which converges locally uniformly to a map-ping f . Clearly, f ∈ S(Bn) by Hurwitz theorem for biholomorphic mappings.Since fk ∈ Rlog M (idBn ,NA), we deduce from Theorem 4.5 that there existsan A-normalized univalent subordination chain fk(z, t) such that fk = fk(·, 0),fk(·, log M) = eA log M idBn . Also, for each r ∈ (0, 1) there exists some K ∗

M (r, A) > 0such that

‖e−t A fk(z, t)‖ ≤ K ∗M (r, A), ‖z‖ ≤ r, t ≥ 0, k ∈ N,

by the proof of Theorem 4.5. Using arguments similar to those in the proof of [29,Lemma 2.8] (see also, [22, Lemma 2.14]), we deduce that there exists a subse-quence { fkp (·, t)}p∈N of { fk(·, t)}k∈N which converges locally uniformly on Bn toan A-normalized univalent subordination chain f (z, t) such that {e−t A f (·, t)}t≥0 isa normal family on Bn . Clearly, f (·, 0) = f and f (·, log M) = eA log M idBn . Thusf ∈ Rlog M (idBn ,NA) by Theorem 4.5, as desired. This completes the proof. ��

Using arguments similar to those in the proofs of Theorems 3.1 and 3.5, we obtainthe following extremal results for mappings in the reachable family Rlog M (idBn ,NA)

(cf. [25] in the case A = In ; compare [56, Theorem 2.52], in the case n = 1).

Theorem 4.8 Let A ∈ A . Also, let M > 1 and let f ∈ ex Rlog M (idBn ,NA). Also letf (z, t) be an A-normalized univalent subordination chain which satisfies the conclu-sion of Theorem 4.5. Then e−t A f (·, t) ∈ ex Rlog M−t (idBn ,NA) for 0 ≤ t ≤ log M.

Proof Note that e−t A f (·, t) ∈ Rlog M−t (idBn ,NA) for 0 ≤ t ≤ log M . Indeed, let

F(·, s) = e−t A f (·, t + s), s ≥ 0.

Then F(·, s) is an A-normalized univalent subordination chain such that F(·, log M −t) = eA(log M−t)idBn and {e−s A F(·, s)}s≥0 is a normal family on Bn .

Let w(z, s, t) be the transition mapping of f (z, t). Since f (·, log M) =eA log M idBn , it follows that f (·, t) = eA log Mw(·, t, log M) for 0 ≤ t ≤ log M .Suppose that

e−t A f (·, t) = λg + (1 − λ)h,

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where λ ∈ (0, 1) and g, h ∈ Rlog M−t (idBn ,NA). Then

f = f (w(·, t), t) = λet Ag(w(·, t)) + (1 − λ)et Ah(w(·, t)),

where w(·, t) = w(·, 0, t). Let qt (z) = et Ag(w(z, t)) and rt (z) = et Ah(w(z, t))for z ∈ Bn . Then, in view of the proof of Theorem 3.1, we obtain that qt , rt ∈Rlog M (idBn ,NA). Finally, since f ∈ ex Rlog M (idBn ,NA), we deduce that qt ≡ rt ,and thus g ≡ h, by the identity theorem for holomorphic mappings. Hence we haveproved that e−t A f (·, t) ∈ ex Rlog M−t (idBn ,NA), as desired. This completes theproof. ��Theorem 4.9 Let A ∈ A . Also, let M > 1 and let f ∈ supp Rlog M (idBn ,NA).Also let f (z, t) be an A-normalized univalent subordination chain that satisfies theconclusion of Theorem 4.5. Then there exists a constant ε ∈ (0, log M) such thate−t A f (·, t) ∈ supp Rlog M−t (idBn ,NA) for 0 ≤ t < ε.

Proof Note that e−t A f (·, t) ∈ Rlog M−t (idBn ,NA) for 0 ≤ t ≤ log M , by the proof ofTheorem 4.8. Let v(z, s, t) denote the transition mapping associated to f (z, t). Also,let vt (z) = v(z, t) = v(z, 0, t), z ∈ Bn , t ≥ 0. Since f ∈ supp Rlog M (idBn ,NA),there is a continuous linear functional L on H(Bn) such that �L|Rlog M (idBn ,NA)

�=constant and

�L( f ) = maxg∈Rlog M (idBn ,NA)

�L(g).

Now fix t ≥ 0. Let Lt : H(Bn) → C be given by

Lt (g) = L(et Ag ◦ vt ), g ∈ H(Bn).

It is clear that Lt is a continuous linear functional on H(Bn) and

Lt (e−t A f (·, t)) = L( f (v(·, t), t)) = L( f ).

In view of the above arguments, we deduce that

�Lt (e−t A f (·, t)) = �L( f ) ≥ �L(et Ag ◦ vt ) = �Lt (g),

for all g ∈ Rlog M−t (idBn ,NA), i.e.

�Lt (e−t A f (·, t)) = max

g∈Rlog M−t (idBn ,NA)

�Lt (g).

Here we used the fact that et Ag◦vt ∈ Rlog M (idBn ,NA) for g ∈ Rlog M−t (idBn ,NA),as noted in the proof of Theorem 4.8.

Finally, we will show that there exists ε ∈ (0, log M) such that �Lt |Rlog M−t (idBn ,NA)

is nonconstant for 0 ≤ t < ε. There exists F ∈ Rlog M (idBn ,NA) such that

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I. Graham et al.

�L(F) < �L( f ). Let F(z, t) be an A-normalized univalent subordination chainthat satisfies the conclusion of Theorem 4.5 with F(·, 0) = F(·). Then e−t A F(·, t) ∈Rlog M−t (idBn ,NA) and

�Lt (e−t A F(·, t)) = �L(F(vt , t))

for t ∈ [0, log M]. Since L(F(vt , t)) → L(F) as t → 0+, there exists ε ∈ (0, log M)

such that

�Lt (e−t A F(·, t)) < �L( f ) = �Lt (e

−t A f (·, t)), 0 ≤ t < ε

and thus �Lt |Rlog M−t (idBn ,NA)is nonconstant, as desired. This completes the proof.

��Example 4.10 Let

A =[

α 00 A2

]

∈ A ,

where m(A) = α > 0. Let M > 1 and let

f (z) = (pb0(z1), f2(z

′)), z = (z1, z′) ∈ Bn,

where b=em(A) log M and f2 ∈Rlog M (idBn−1 ,NA2). Then f ∈supp Rlog M (idBn ,NA).

Proof First, we prove that f ∈ Rlog M (idBn ,NA). Indeed, in view of the proof ofTheorem 4.6, there exists an α-normalized univalent subordination chain f1(z1, t) onU × [0,∞) such that f1(z1, 0) = pb

0(z1) and f1(z1, log M) = eα log M z1, |z1| < 1and {e−tα f1(·, t)}t≥0 is a normal family on U . Since f2 ∈ Rlog M (idBn−1 ,NA2), thereexists an A2-normalized univalent subordination chain f2(z′, t) on Bn−1×[0,∞) suchthat f2(·, 0) = f2, f2(·, log M) = eA2 log M idBn−1 and {e−t A2 f2(·, t)}t≥0 is a normalfamily on Bn−1. Then f (z, t) = ( f1(z1, t), f2(z′, t)) is an A-normalized univalentsubordination chain on Bn × [0,∞) such that f = f (·, 0). It is easy to see thatf (·, log M) = eA log M idBn and {e−t A f (·, t)}t≥0 is a normal family on Bn . Hence,f ∈ Rlog M (idBn ,NA), as desired.

Next, let z0 = (1/2, 0′) and

L(g) = e−α log M g1(z0), for g ∈ H(Bn).

Then L is a continuous linear functional on H(Bn). For g ∈ Rlog M (idBn ,NA), wehave

�L(g) ≤ ‖e−A log M g(z0)‖ ≤ 1

bpb

0

(1

2

)

= �L( f ),

by Theorem 4.6. We also have idBn ∈ Rlog M (idBn ,NA) and �L(idBn ) < 1b pb

0

( 12

) =�L( f ). Thus, f ∈ supp Rlog M (idBn ,NA). ��

123


Taking into account Example 4.4, we obtain an extremal result in the familyRlog M (idBn ,NA) involving spirallike mappings. This result yields that if F ∈ SA(Bn)

is an extremal mapping for a functional λ over S0A(Bn), then it leads to the solution of

an extremal problem over the family Rlog M (idBn ,NA), where A ∈ A . The followingresult is analogous to Theorem 3.7 in the case of reachable families. We have (see [56,Theorem 2.65] for n = 1, and [25] for A = In).

Theorem 4.11 Let A ∈ A . Assume that F ∈ SA(Bn) provides the maximum fora functional λ : S0

A(Bn) → R on S0A(Bn). Then for each M > 1, the mapping

F MA ∈ Rlog M (idBn ,NA) given by (4.2) provides the maximum on Rlog M (idBn ,NA)

for the associated functional λMA : Rlog M (idBn ,NA) → R, defined by

λMA (g) = λ(eA log M F(e−A log M g(·))), g ∈ Rlog M (idBn ,NA).

In addition, λMA (F M

A ) = λ(F).

Proof Since F ∈ SA(Bn), we deduce that F MA ∈ Rlog M (idBn ,NA) by Example

4.4. On the other hand, if g ∈ Rlog M (idBn ,NA), then it is not difficult to deducethat the mapping eA log M F(e−A log M g(·)) belongs to S0

A(Bn). Indeed, let w(z, s, t),s ≤ t ≤ log M , be as in the proof of Theorem 4.5 such that g = eA log Mw(·, 0, log M).Also let

L(z, t) ={

eA log M F(w(z, t, log M)), z ∈ Bn, 0 ≤ t ≤ log Met A F(z), z ∈ Bn, t ≥ log M.

Let

ω(z, s, t) =⎧⎨

⎩

w(z, s, t), 0 ≤ s ≤ t ≤ log MF−1(eA(log M−t)F(w(z, s, log M)), 0 ≤ s ≤ log M ≤ tF−1(e(s−t)A F(z)), log M ≤ s ≤ t.

Then ω(·, s, t) is a univalent Schwarz mapping and L(z, s) = L(ω(z, s, t), t) forz ∈ Bn and 0 ≤ s ≤ t < ∞. Since L(·, t) is biholomorphic on Bn , L(0, t) = 0and DL(0, t) = et A, t ≥ 0, it follows that L(z, t) is an A-normalized univalentsubordination chain. It is not difficult to see that {e−t A L(·, t)}t≥0 is a normal familyon Bn and L(·, 0) = eA log M F(e−A log M g(·)). Hence, eA log M F(e−A log M g(·)) ∈S0

A(Bn), as claimed.In view of the above arguments, we obtain that

λMA (g) = λ(eA log M F(e−A log M g(·))) ≤ λ(F) = λM

A (F MA ), g ∈ Rlog M (idBn ,NA),

as desired. This completes the proof. ��We next prove a density result related to the family Rlog M (idBn ,NA), which

involves the subfamily of NA consisting of extreme points (for n = 1, see [56];cf. [40,47]). It is known that Rlog M (idU , ex M ) is dense in S(M) for M ∈ (1,∞),

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I. Graham et al.

and R∞(idU , ex M ) is dense in S (see [40,56]; see also [47]). Note that in the casen = 1, this result may be used to prove the equality Rlog M (idU ,M ) = S(M) (seee.g. [56]).

For the proof of Theorem 4.14, we need the following lemmas. Some ideas for theproof come from [56]. We introduce a metric on H(Bn) in the usual way. Let {Kμ}μ∈N

be a sequence of compact sets Kμ ⊂ Bn such that Kμ ⊂ Kμ+1, ∪μ∈NKμ = Bn andsuch that for each compact set K ⊂ Bn there exists a μ ∈ N with K ⊂ Kμ. We defineNμ( f ) = maxz∈Kμ ‖ f (z)‖, μ ∈ N and

d( f ) =∞∑

μ=1

1

2μmin{1, Nμ( f )},

for f ∈ H(Bn). Then d( fk − f ) → 0 as k → ∞ if and only if the sequence { fk}k∈N ⊂H(Bn) converges in H(Bn) to f ∈ H(Bn). Obviously, min{1, Nμ( f )} ≤ 2μd( f )

for each μ ∈ N.

Lemma 4.12 Let A ∈ A and M ∈ (1,∞). Let �1 ⊂ NA and �2 = co �1, whereco �1 is the convex hull of �1. Then for every h = h(z, t) ∈ C ([0, log M],�2), thereexists a sequence {hk} ⊆ C ([0, log M], co �1) such that

t∫

0

hk(v(z, s; h), s)ds →t∫

0

h(v(z, s; h), s)ds

locally uniformly on Bn × [0, log M] as k → ∞, where v = v(z, t; h) is the uniquesolution on [0, log M] of the initial value problem (4.1). Moreover, let vk = vk(·, t; hk)

be the unique Lipschitz continuous solution on [0, log M] of the initial value problem

∂vk

∂t(z, t) = −hk(vk(z, t), t) a.e. t ∈ [0, log M], vk(z, 0) = z, (4.5)

for each z ∈ Bn and k ∈ N. Then, vk(·, t; hk) → v(·, t; h) locally uniformly on Bn

for t ∈ [0, log M].Proof Let h = h(z, t) ∈ C ([0, log M],�2). For fixed k ∈ N, we define the mappinghk ∈ C ([0, log M], co �1) as follows. We divide the interval I = [0, log M] into ksubintervals I j = [τ j−1, τ j ), τ j − τ j−1 = log M/k, j = 1, 2, . . . , k. Let

r j (z) = k

log M

τ j∫

τ j−1

h(z, s)ds.

Then r j ∈ co �2 = co �1, and there exist g j ∈ co �1 such that d(g j − r j ) < 2−k foreach j = 1, . . . , k. Hence, Nμ(g j − r j ) ≤ 2μ−k for j = 1, . . . , k, and for all k ≥ μ.

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Now, for given 0 < δ < min{2−k, log M/2k}, we define

hk(z, s) =

⎧⎪⎪⎨

⎪⎪⎩

g1(z), s ∈ [0, τ1 − δ),

g j (z), s ∈ [τ j−1 + δ, τ j − δ),τ j−1+δ−s

2δg j−1(z) + −τ j−1+δ+s

2δg j (z), s ∈ [τ j−1 − δ, τ j−1+δ), 2≤ j ≤k

gk(z), s ∈ [τk−1 + δ, log M].

Then hk ∈ C ([0, log M], co �1) is jointly continuous in (z, s) ∈ Bn × I .Since NA is a normal family by [22, Lemma1.2], for each r ∈ (0, 1), there exists

a constant M(r) > 0 such that

max

{

‖h(z, s)‖, max1≤ j≤k

‖g j (z)‖, ‖hk(z, s)‖}

≤ M(r) z ∈ Bnr , s ∈ I, k ∈ N. (4.6)

Let R = (1 + r)/2. By using the Cauchy estimates, we have

‖hk(z, s) − hk(w, s)‖ ≤ 2M(R)

1 − r‖z − w‖, z, w ∈ B

nr , s ∈ I, k ∈ N. (4.7)

Let r ∈ (0, 1) be fixed. Choose μ ∈ N in such a way that Bnr ⊂ Kμ. By using (4.6),

we obtain∥∥∥∥∥∥∥

τ j∫

τ j−1

[hk(z, s) − h(z, s)]ds

∥∥∥∥∥∥∥

≤

∥∥∥∥∥∥∥

τ j∫

τ j−1

[hk(z, s)−g j (z)]ds

∥∥∥∥∥∥∥

+ log M

k

∥∥g j (z)−r j (z)

∥∥

≤ 2M(r)2δ + log M

k‖g j (z) − r j (z)‖

≤ 4M(r)δ + log M

kNμ(g j − r j )

≤ 4M(r)1

2k+ 2μ−k log M

k, (4.8)

for all z ∈ Bnr , 1 ≤ j ≤ k and k ≥ μ.

Now, we show that

t∫

0

hk(v(z, s; h), s)ds →t∫

0

h(v(z, s; h), s)ds

uniformly on Bnr × [0, log M] as k → ∞. The mappings h and hk have the Taylor

expansions

h(z, s) =∞∑

m=0

1

m! Dmh(0, s)(zm), for z ∈ Bn, s ∈ I

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I. Graham et al.

and

hk(z, s) =∞∑

m=0

1

m! Dmhk(0, s)(zm), for z ∈ Bn, s ∈ I

Since ‖h(z, s)‖ ≤ M(R) and ‖hk(z, s)‖ ≤ M(R) for z ∈ BnR , s ∈ I and k ∈ N,

where R = (1 + r)/2, the above convergence is uniform for z ∈ Bnr , s ∈ I and k ∈ N

by the Cauchy estimates. This implies that

supz∈B

nr

supt∈I

supk∈N

t∫

0

∥∥∥∥∥

∞∑

m=N

1

m! [Dmhk(0, s) − Dmh(0, s)](v(z, s)m)

∥∥∥∥∥

ds

can be arbitrarily small if we take N sufficiently large. So, it suffices to show that

t∫

0

[Dmhk(0, s) − Dmh(0, s)](v(z, s)m)ds → 0

uniformly for z ∈ Bnr and t ∈ I as k → ∞ for every m. There exists a constant

M(r, m) such that

‖v(z, s)m − v(z, t)m‖ ≤ M(r, m)|s − t | (4.9)

for z ∈ Bnr and s, t ∈ I . For fixed t ∈ (0, log M), there exists an l ∈ {1, . . . , k} such

that t ∈ [τl−1, τl). We calculate

t∫

0

[Dmhk(0, s) − Dmh(0, s)](v(z, s)m)ds

=l−1∑

j=1

τ j∫

τ j−1

[Dmhk(0, s) − Dmh(0, s)](v(z, s)m − v(z, τ j−1)m)ds

+l−1∑

j=1

τ j∫

τ j−1

[Dmhk(0, s) − Dmh(0, s)](v(z, τ j−1)m)ds

+t∫

τl−1

[Dmhk(0, s) − Dmh(0, s)](v(z, s)m)ds.

In view of the Cauchy estimates and (4.6), we obtain the following relations:

‖Dmhk(0, s)(v(z, s)m − v(z, τ j−1)m)‖ ≤ m!M(r)

rm‖v(z, s)m − v(z, τ j−1)

m‖,

123


and

‖Dmh(0, s)(v(z, s)m − v(z, τ j−1)m)‖ ≤ m!M(r)

rm‖v(z, s)m − v(z, τ j−1)

m‖.

Then, by applying the relation (4.9) to the above inequalities, we obtain that

‖Dmhk(0, s)(v(z, s)m − v(z, τ j−1)m)‖ ≤ m!M(r)M(r, m)

rm(s − τ j−1),

and

‖Dmh(0, s)(v(z, s)m − v(z, τ j−1)m)‖ ≤ m!M(r)M(r, m)

rm(s − τ j−1).

Hence, in view of the above relations, we obtain that

∥∥∥∥∥∥∥

l−1∑

j=1

τ j∫

τ j−1

[Dmhk(0, s) − Dmh(0, s)](v(z, s)m − v(z, τ j−1)m)ds

∥∥∥∥∥∥∥

≤l−1∑

j=1

2m!M(r)M(r, m)

rm

τ j∫

τ j−1

(s − τ j−1)ds.

Combining the above arguments, and taking into account the Cauchy integral formula,the Cauchy estimates, the relations (4.6) and (4.8), we deduce that

∥∥∥∥∥∥

t∫

0

[Dmhk(0, s) − Dmh(0, s)](v(z, s)m)ds

∥∥∥∥∥∥

≤l−1∑

j=1

2m!M(r)M(r, m)

rm

τ j∫

τ j−1

(s − τ j−1)ds

+l−1∑

j=1

∥∥∥∥∥∥∥

m!2π

√−1

∫

|ζ |=1/2

1

ζm+1

τ j∫

τ j−1

[hk(ζv(z, τ j−1), s)−h(ζv(z, τ j−1), s)]dsdζ

∥∥∥∥∥∥∥

+2M(r)m! log M

k

≤ m!M(r)M(r, m)(log M)2

rmk+ 2m+2m!M(r)

k

2k+ 2mm!2μ−k log M

+2M(r)m! log M

k

for all z ∈ Bnr , t ∈ I and all k ≥ μ, and this expression tends to 0 as k → ∞.

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I. Graham et al.

Next, we show that vk(·, t; hk) → v(·, t; h) locally uniformly on Bn , for all t ∈[0, log M]. From (4.7), we obtain

‖vk(z, t) − v(z, t)‖ ≤∥∥∥∥∥∥

t∫

0

[hk(vk(z, s), s) − hk(v(z, s), s)]ds

∥∥∥∥∥∥

+∥∥∥∥∥∥

t∫

0

[hk(v(z, s), s) − h(v(z, s), s)]ds

∥∥∥∥∥∥

≤ M

t∫

0

‖vk(z, s) − v(z, s)‖ds + εk,

for all z ∈ Bnr and all t ∈ [0, log M] and all k ∈ N, where

M = 2M(R)

1 − r

and {εk} is a sequence of positive numbers with limit 0. It follows by applying Gron-wall’s lemma that vk(z, t) → v(z, t) uniformly in Bn

r for each t ∈ I . This completesthe proof. ��Lemma 4.13 Let A ∈ A , M ∈ (1,∞), and let �1 ⊆ NA. Then

Rlog M (idBn , co �1) = Rlog M (idBn ,�1).

Proof It suffices to show that

Rlog M (idBn , co �1) ⊆ Rlog M (idBn ,�1).

Let f ∈ Rlog M (idBn , co �1). Then there exists a mapping h ∈ C ([0, log M], co �1)

such that f = v(·, log M; h), where v = v(z, t; h) is the unique solution on [0, log M]of the initial value problem (4.1). Then, by Lemma 4.12 and its proof, there exists asequence {hi } such that hi ∈ C ([0, log M], co �1) is jointly continuous in (z, s) ∈Bn × [0, log M] and fi = vi (·, log M; hi ) converges locally uniformly to f on Bn

as i → ∞, where vi = vi (z, t; hi ) is the unique solution on [0, log M] of the initialvalue problem (4.5). Thus, we can suppose that the mapping h(z, t) is continuous withrespect to (z, t) ∈ Bn × [0, log M].

For each k ∈ N, we define the mapping hk as follows. We divide the interval I =[0, log M] into subintervals I j = [τ j−1, τ j ), τ j − τ j−1 = log M/k. By assumption,we have

h(z, τ j−1) =m j∑

p=1

λjpg j

p(z),

123


with specific g jp ∈ �1 and non-negative numbers λ

jp ∈ [0, 1] satisfying λ

j1 + · · · +

λjm j = 1 for all j = 1, . . . , k. Let I p

j = [t jp−1, t j

p), p = 1, . . . , m j be subintervals ofI j such that

t jp − t j

p−1 = log M

kλ

jp.

Let

hk(z, t) = g jp(z), if t ∈ I p

j .

Then hk ∈ C ([0, log M],�1). For fixed k ∈ N, there exists some l ∈ {1, . . . , k} suchthat t ∈ [τl−1, τl) and we calculate

t∫

0

hk(v(z, ρ), ρ)dρ =l−1∑

j=1

τ j∫

τ j−1

[hk(v(z, ρ), ρ) − hk(v(z, τ j−1), ρ)]dρ

+l−1∑

j=1

τ j∫

τ j−1

hk(v(z, τ j−1), ρ)dρ +t∫

τl−1

hk(v(z, ρ), ρ)dρ

=l−1∑

j=1

τ j∫

τ j−1


+l−1∑

j=1

m j∑

p=1

t jp∫

t jp−1

g jp(v(z, τ j−1))dρ +

t∫

τl−1

hk(v(z, ρ), ρ)dρ

=l−1∑

j=1

τ j∫

τ j−1


+l−1∑

j=1

log M

kh(v(z, τ j−1), τ j−1) +

t∫

τl−1

hk(v(z, ρ), ρ)dρ.

Let r ∈ (0, 1) be fixed. Since NA is a normal family, there exists a constantM = M(r) such that

‖hk(z, s) − hk(w, s)‖ ≤ M, z, w ∈ Bnr , s ∈ I, k ∈ N

and

‖v(z, s) − v(z, t)‖ ≤ M |s − t |, z ∈ Bnr , s, t ∈ I

123

I. Graham et al.

as in the proof of Lemma 4.12. Since

∥∥∥∥∥∥∥

l−1∑

j=1

τ j∫

τ j−1

[hk(v(z, ρ), ρ) − hk(v(z, τ j−1), ρ)]dρ +t∫

τl−1

hk(v(z, ρ), ρ)dρ

∥∥∥∥∥∥∥

≤ Ml−1∑

j=1

τ j∫

τ j−1

‖v(z, ρ) − v(z, τ j−1)‖dρ + M log M

k

≤ M2l−1∑

j=1

τ j∫

τ j−1

(ρ − τ j−1)dρ + M log M

k≤ M2 (log M)2

2k+ M log M

k, z ∈ B

nr ,

limk→∞

t∫

0

hk(v(z, ρ), ρ)dρ = limk→∞

l−1∑

j=1

log M

kh(v(z, τ j−1), τ j−1)=

t∫

0

h(v(z, ρ), ρ)dρ

pointwise in Bn , where the last equality follows from the continuity of h(v(z, ρ), ρ)

in ρ ∈ [0, t]. Since �1 is a normal family, the above convergence is locally uniformon Bn by Vitali’s theorem in several complex variables.

As in the proof of the preceding lemma, we obtain that vk(·, t; hk) → v(·, t; h)

locally uniformly on Bn for t ∈ [0, log M], where vk = vk(·, t; hk) is the uniqueLipschitz continuous solution on [0, log M] of the initial value problem (4.5). Con-sequently, vk(·, log M; hk) → f locally uniformly on Bn . This completes the proof.

��Theorem 4.14 Let A ∈ A and let M ∈ (1,∞). Then

Rlog M (idBn , ex NA) = Rlog M (idBn ,NA). (4.10)

Proof First, note that NA is a compact family. Indeed, in view of [22, Lemma 1.2])(cf. [20]), NA is a locally uniformly bounded family, and since NA is also closed, it iscompact. Hence, ex NA �= ∅. Let �1 = ex NA and �2 = NA. Then �1 is a normalfamily in H(Bn). Since �2 is a nonempty, convex and compact family, we deduce inview of the Krein–Milman theorem that co �1 = �2, where co �1 is the convex hullof �1. Next, we deduce that

Rlog M (idBn ,�1) = Rlog M (idBn ,�2). (4.11)

Indeed, since �1 ⊆ �2, it is not difficult to deduce that

Rlog M (idBn ,�1) ⊆ Rlog M (idBn ,�2). (4.12)

123


Now, let f ∈ Rlog M (idBn ,�2). Then there exists h = h(z, t) ∈ C ([0, log M],�2)

such that f = v(·, log M; h), where v = v(z, t; h) is the unique solution on [0, log M]of the initial value problem (4.1). Since �2 = co �1, there exists a sequence {hμ} ⊆C ([0, log M], co �1) such that

t∫

0

hμ(v(z, s; h), s)ds →t∫

0

h(v(z, s; h), s)ds

locally uniformly on Bn × [0, log M] as μ → ∞, by Lemma 4.12. Let vμ =vμ(·, t; hμ) be the unique Lipschitz continuous solution on [0, log M] of the initialvalue problem (4.5). Then, by Lemma 4.12, we deduce that vμ(·, t; hμ) → v(·, t; h)

locally uniformly on Bn for t ∈ [0, log M]. Hence vμ(·, log M; hμ) → f locallyuniformly on Bn , and thus f ∈ Rlog M (idBn , co �1). Since f is arbitrary, we deducethat Rlog M (idBn ,�2) ⊆ Rlog M (idBn , co �1), and thus

Rlog M (idBn ,�2) ⊆ Rlog M (idBn , co �1). (4.13)

On the other hand, by Lemma 4.13, it follows that

Rlog M (idBn , co �1) = Rlog M (idBn ,�1).

Hence, in view of (4.13) and the above relation, we obtain that Rlog M (idBn ,�2) ⊆Rlog M (idBn ,�1), and thus Rlog M (idBn ,�2) ⊆ Rlog M (idBn ,�1). Combining theabove relation with (4.12), we deduce that

Rlog M (idBn ,�1) = Rlog M (idBn ,�2),

as claimed. On the other hand, since Rlog M (idBn ,�2) is a compact family by Theorem4.7, we conclude in view of the above relation that the relation (4.10) holds, as desired.This completes the proof. ��For any f ∈ S0

A(Bn), where A ∈ A , we have f (z) = limt→∞ et Av(z, t; h) locallyuniformly on Bn by [22, Theorem 2.3], where v = v(z, t; h) is the unique solutionon [0,∞) of the initial value problem (4.1). Then, from Theorem 4.14 and the above,we obtain the following consequence:

Corollary 4.15 Let A ∈ A and let f ∈ S0A(Bn). Then for any ε > 0 and for any

compact set K ⊂ Bn, there exist M > 1 and fε,K ∈ Rlog M (idBn , ex NA) such that‖ f (z) − fε,K (z)‖ < ε on K .

In view of Theorem 4.14 and the above result, it is natural to consider the followingconjecture. Note that it is true in the case n = 1 [40].

Conjecture 4.16 Let A ∈ A . Then the family R∞(idBn , ex NA) is dense in S0A(Bn).

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I. Graham et al.

In view of Theorems 4.5 and 4.14, it would be interesting to give an answer to thefollowing questions. A discussion of these questions in the case n = 1 may be foundin [56].

Question 4.17 Is Rlog M (idBn ,NA) = Rlog M (idBn , ex NA) = S0A(M∗, Bn) for

M > 1, n ≥ 2 and A ∈ A , where M∗ = ‖eA log M‖?

Question 4.18 Is R∞(idBn , ex NA) = S0A(Bn) for n ≥ 2 and A ∈ A ?

Taking into account Theorem 4.14, we propose the following conjecture. In thecase of one complex variable, see [57, Theorem 1].

Conjecture 4.19 Let A ∈ A and let M ∈ (1,∞]. Also, let � ⊆ NA be a compactand convex family. Then Rlog M (idBn ,�) is compact and

Rlog M (idBn , ex �) = Rlog M (idBn ,�).

Finally, we consider the dependence of the reachable family Rlog M (idBn ,NA) onM . To this end, we recall the Hausdorff metric ρ on the space of compact subsets ofa metric space (X, d):

ρ(V, W ) = max

{

maxv∈V

d(v, W ), maxw∈W

d(w, V )

}

,

for all compact subsets V, W of X . Then ρ is a metric on the space of compact subsetsof (X, d) (cf. [56]). Since H(Bn) is a metric space and each family Rlog M (idBn ,NA)

is compact for M ≥ 1 and A ∈ A , we obtain the following result (see [56, Theorem1.45] for n = 1):

Proposition 4.20 Let A ∈ A . Then M �→ Rlog M (idBn ,NA) is a continuous map-ping on [1,∞).

Proof Suppose that there exists M0 ∈ [1,∞) such that M �→ Rlog M (idBn ,NA) isnot continuous at M = M0. Then there exist ε > 0 and a sequence {Mk} ⊂ [1,∞)

such that Mk → M0 and

ρ(Rlog Mk (idBn ,NA),Rlog M0(idBn ,NA)) ≥ ε, k ∈ N.

Hence, for each k ∈ N, there exist a solution vk(·, t) of the initial value problem(4.1) on [0, T ] with T = max{log Mk, log M0} and a compact set K ⊂ Bn such thatvk(z, log Mk)− vk(z, log M0) does not converge uniformly in K to 0. However, sincethere exists a constant M = M(K ) > 0 such that ‖vk(z, s) − vk(z, t)‖ ≤ M |s − t |for z ∈ K and s, t ∈ [0,∞), this is a contradiction. This completes the proof. ��Remark 4.21 The results in the third and fourth sections may be generalized to thecase of an arbitrary norm in C

n .

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5 Extremal problems associated with extension operators

In this section we are concerned with extreme points and support points related tocompact families generated by extension operators. We also prove that if � is anextension operator that preserves A-normalized univalent subordination chains, wheremax{k+(A), 1} < 2 min{m(A), 1}, then it preserves the notions of A-parametric rep-resentation and spirallikeness with respect to the operator A (see also [25], in the caseA = In).

The notions of continuous extension operator � : L S1 → L Sn and Loewner chainpreserving extension operator were recently introduced by Muir [42]. We generalizethese notions for continuous operators � : L Sn → L Sn+p.

For n, p ≥ 1, set z′ = (z1, . . . , zn) ∈ Cn and z = (zn+1, . . . , zn+p) ∈ C

p so thatz = (z′, z) ∈ C

n+p.

Definition 5.1 The mapping � : L Sn → L Sn+p is called an extension operator if �

is continuous (with respect to the compact open topologies of H(Bn) and H(Bn+p))and

�( f )(z′, 0) = ( f (z′), 0), ∀ f ∈ L Sn, z′ ∈ Bn .

If A ∈ L(Cn) with m(A) > 0 and if � is an extension operator, we say that �

preserves A-normalized univalent subordination chains provided that, whenever f =f (z′, t) : Bn × [0,∞) → C

n is an A-normalized univalent subordination chain, themapping F = F(z, t) : Bn+p × [0,∞) → C

n+p given by

F(·, t) = et A�(e−t A f (·, t)), t ≥ 0, (5.1)

is an A-normalized univalent subordination chain, where

A =[

A 00 Ip

]

. (5.2)

If A = In , we say that the operator � preserves Loewner chains ([25]; cf. [42]).

Remark 5.2 A detailed discussion and examples of extension operators may be foundin [15,21,25,28,42]. The well known Roper-Suffridge extension operator (see [55])is an example of an extension operator that preserves Loewner chains.

Muir [42] has recently proved that if � : L S1 → L Sn is an extension operator thatpreserves Loewner chains, then �(S) ⊆ S0(Bn) and �(S∗) ⊆ S∗(Bn). In the caseof extension operators that preserve A-normalized univalent subordination chains, wehave (see [25] for A = In)

Theorem 5.3 (i) Let A ∈ L(Cn) be such that max{k+(A), 1} < 2 min{m(A), 1}.If � : L Sn → L Sn+p is an extension operator that preserves A-normalizedunivalent subordination chains, then �(S0

A(Bn)) ⊆ S0A(Bn+p) and �(SA(Bn)) ⊆

SA(Bn+p), where A is given by (5.2).

123

I. Graham et al.

(ii) In addition, if M > 1 and �(idBn ) = idBn+p , then

�(Rlog M (idBn ,NA)) ⊆ Rlog M (idBn+p ,N A).

Proof First, we prove the statement (i). Let F ∈ �(S0A(Bn)) and f ∈ S0

A(Bn) besuch that F = �( f ). Since k+(A) < 2m(A), there exists a univalent subordinationchain f (z′, t) such that D f (0, t) = et A for t ≥ 0, {e−t A f (·, t)}t≥0 is a normalfamily on Bn and f = f (·, 0). Let F(z, t) be the univalent subordination chain givenby (5.1). Since {e−t A f (·, t)}t≥0 is a normal family on Bn and � is continuous, it

is not difficult to see that {e−t A F(·, t)}t≥0 is also a normal family on Bn+p. Sincek+( A) < 2m( A), in view of the fact that max{k+(A), 1} < 2 min{m(A), 1}, and sinceF = �( f ) = �( f (·, 0)) = F(·, 0), we deduce that F ∈ S0

A(Bn+p), as desired.

Next, if F = �( f ) ∈ �(SA(Bn)), then f (z′, t) = et A f (z′) is an A-normalizedunivalent subordination chain. Hence F(·, t) = et A�(e−t A f (·, t)) = et A F is alsoan A-normalized univalent subordination chain, i.e. F = F(·, 0) ∈ SA(Bn+p), asdesired.

(ii) Now, we assume that F = �( f ) ∈ �(Rlog M (idBn ,NA)). In view of The-orem 4.5, there exists an A-normalized univalent subordination chain f (z′, t) suchthat {e−t A f (·, t)}t≥0 is a normal family on Bn , f (·, 0) = f and f (·, log M) =eA log M idBn . Let F(z, t) be the mapping given by (5.1). Then F(z, t) is an A-normalized univalent subordination chain such that {e−t A F(·, t)}t≥0 is a normal fam-ily on Bn+p. Also, F = F(·, 0) and since �(idBn ) = idBn+p , it is easily seen thatF(·, log M) = eA log M idBn+p . Taking into account Theorem 4.5, we deduce thatF ∈ Rlog M (idBn+p ,N A). This completes the proof. ��Remark 5.4 Recently, the authors [25] proved that if � : L Sn → L Sn+p is anextension operator and F ⊆ L Sn is a nonempty compact set, then

�(ex F ) ⊆ ex �(F ) and �(supp F ) ⊆ supp �(F ).

From Remark 5.4, we obtain the following consequences (see [25] for A = In):

Corollary 5.5 Let A ∈ A . If � : L Sn → L Sn+p is an extension operator, then�(ex S0

A(Bn)) ⊆ ex �(S0A(Bn)) and �(supp S0

A(Bn)) ⊆ supp �(S0A(Bn)). In addi-

tion, if M > 1, then

�(ex Rlog M (idBn ,NA)) ⊆ ex �(Rlog M (idBn ,NA))

and

�(supp Rlog M (idBn ,NA)) ⊆ supp �(Rlog M (idBn ,NA)).

Proof It suffices to use the fact that Rlog M (idBn ,NA) is compact in view of Corollary4.7, and S0

A(Bn) is also compact for A ∈ A . ��

123


Corollary 5.6 Let A ∈ A . Also, let f ∈ ex S0A(Bn) (resp. f ∈ supp S0

A(Bn)) andlet f (z′, t) be an A-normalized univalent subordination chain such that f = f (·, 0)

and {e−t A f (·, t)}t≥0 is a normal family on Bn. Also let � : L Sn → L Sn+p be anextension operator that preserves A-normalized univalent subordination chains and

let F(z, t) be given by (5.1). Then e−t A F(·, t) ∈ ex �(S0A(Bn)) (resp. e−t A F(·, t) ∈

supp �(S0A(Bn)) for t ≥ 0.

Proof Fix t ≥ 0. From Theorem 3.1 (resp. Theorem 3.5), we obtain that e−t A f (·, t) ∈ex S0

A(Bn) (resp. e−t A f (·, t) ∈ supp S0A(Bn)). Then it suffices to apply Corollary 5.5.

This completes the proof. ��Acknowledgments The authors would like to thank the referee for a very careful reading of the paper andfor valuable suggestions that improved the manuscript. Also, the authors are grateful to S. Schleissinger forproviding the proof of Proposition 3.4 in the case A = In . Some of the research for this paper was done inAugust, 2012, and August, 2013, while Gabriela and Mirela Kohr visited the Department of Mathematicsof the University of Toronto. They are grateful to the members of this department for the hospitality duringthese visits.

References

1. Abate, M., Bracci, F., Contreras, M.D., Díaz-Madrigal, S.: The evolution of Loewner’s differentialequations. Eur. Math. Soc. Newsl. 78, 31–38 (2010)

2. Arosio, L.: Resonances in Loewner equations. Adv. Math. 227, 1413–1435 (2011)3. Arosio, L., Bracci, F.: Infinitesimal generators and the Loewner equation on complete hyperbolic

manifolds. Anal. Math. Phys. 1, 337–350 (2011)4. Arosio, L., Bracci, F., Hamada, H., Kohr, G.: An abstract approach to Loewner chains. J. Anal. Math.

119, 89–114 (2013)5. Arosio, L., Bracci, F., Wold, F.E.: Solving the Loewner PDE in complete hyperbolic starlike domains

of Cn . Adv. Math. 242, 209–216 (2013)

6. Becker, J.: Über die Lösungsstruktur einer Differentialgleichung in der konformen Abbildung. J. ReineAngew. Math. 285, 66–74 (1976)

7. Bracci, F., Contreras, M.D., Díaz-Madrigal, S.: Evolution families and the Loewner equation II: com-plex hyperbolic manifolds. Math. Ann. 344, 947–962 (2009)

8. Bracci, F., Contreras, M.D., Díaz-Madrigal, S.: Semigroups versus evolution families in the Loewnertheory. J. Anal. Math. 115, 273–292 (2011)

9. Bracci, F., Contreras, M.D., Díaz-Madrigal, S.: Evolution families and the Loewner equation I: theunit disk. J. Reine Angew. Math. 672, 1–37 (2012)

10. Brickman, L., Wilken, D.R.: Support points of the set of univalent functions. Proc. Am. Math. Soc.42, 523–528 (1974)

11. Cartan, H.: Sur la possibilité d’étendre aux fonctions de plusieurs variables complexes la théorie desfonctions univalentes, 129–155. Leçons sur les fonctions univalentes ou multivalentes, Gauthier-Villars,Paris, Note added to P. Montel (1933)

12. Daleckii, Yu.L., Krein, M.G.: Stability of solutions of differential equations in Banach space. Transl.Math. Monogr., vol. 43. American Mathematical Society, Providence (1974)

13. Duren, P.: Univalent Functions. Springer, New York (1983)14. Duren, P., Graham, I., Hamada, H., Kohr, G.: Solutions for the generalized Loewner differential equa-

tion in several complex variables. Math. Ann. 347, 411–435 (2010)15. Elin, M.: Extension operators via semigroups. J. Math. Anal. Appl. 377, 239–250 (2011)16. Elin, M., Reich, S., Shoikhet, D.: Complex dynamical systems and the geometry of domains in Banach

spaces. Diss. Math. 427, 1–62 (2004)17. Friedland, S., Schiffer, M.: On coefficient regions of univalent functions. J. Anal. Math. 31, 125–168

(1977)18. Goodman, G.S.: Univalent Functions and Optimal Control, Ph.D. Thesis, Stanford Univ. (1968)19. Gong, S.: Convex and Starlike Mappings in Several Complex Variables. Kluwer, Dordrecht (1998)

123

I. Graham et al.

20. Graham, I., Hamada, H., Kohr, G.: Parametric representation of univalent mappings in several complexvariables. Can. J. Math. 54, 324–351 (2002)

21. Graham, I., Hamada, H., Kohr, G.: Extension operators and subordination chains. J. Math. Anal. Appl.386, 278–289 (2012)

22. Graham, I., Hamada, H., Kohr, G., Kohr, M.: Asymptotically spirallike mappings in several complexvariables. J. Anal. Math. 105, 267–302 (2008)

23. Graham, I., Hamada, H., Kohr, G., Kohr, M.: Parametric representation and asymptotic starlikeness inC

n . Proc. Am. Math. Soc. 136, 3963–3973 (2008)24. Graham, I., Hamada, H., Kohr, G., Kohr, M.: Spirallike mappings and univalent subordination chains

in Cn . Ann. Scuola Norm. Sup. Pisa-Cl. Sci. 7, 717–740 (2008)

25. Graham, I., Hamada, H., Kohr, G., Kohr, M.: Extreme points, support points and the Loewner variationin several complex variables. Sci. China Math. 55, 1353–1366 (2012)

26. Graham, I., Hamada, H., Kohr, G., Kohr, M.: Univalent subordination chains in reflexive complexBanach spaces. Contemp. Math. (AMS) 591, 83–111 (2013)

27. Graham, I., Hamada, H., Kohr, G., Kohr, M.: Asymptotically spirallike mappings in reflexive complexBanach spaces. Complex Anal. Oper. Theory 7, 1909–1927 (2013)

28. Graham, I., Kohr, G.: Geometric Function Theory in One and Higher Dimensions. Marcel Dekker Inc,New York (2003)

29. Graham, I., Kohr, G., Kohr, M.: Loewner chains and parametric representation in several complexvariables. J. Math. Anal. Appl. 281, 425–438 (2003)

30. Graham, I., Kohr, G., Pfaltzgraff, J.A.: Parametric representation and linear functionals associated withextension operators for biholomorphic mappings. Rev. Roumaine Math. Pures Appl. 52, 47–68 (2007)

31. Gurganus, K.: �-like holomorphic functions in Cn and Banach spaces. Trans. Am. Math. Soc. 205,

389–406 (1975)32. Hallenbeck, D.J., MacGregor, T.H.: Linear Problems and Convexity Techniques in Geometric Function

Theory. Pitman, Boston (1984)33. Hamada, H.: Polynomially bounded solutions to the Loewner differential equation in several complex

variables. J. Math. Anal. Appl. 381, 179–186 (2011)34. Hamada, H., Kohr, G., Muir Jr, J.R.: Extensions of Ld -Loewner chains to higher dimensions. J. Anal.

Appl. 120, 357–392 (2013)35. Harris, L.: The numerical range of holomorphic functions in Banach spaces. Am. J. Math. 93, 1005–

1019 (1971)36. Jurdjevic, V.: Geometric Control Theory. Cambridge Univ Press, Cambridge (1997)37. Kirwan, W.E.: Extremal properties of slit conformal mappings. In: Brannan, D., Clunie, J. (eds.)

Aspects of Contemporary Complex Analysis, pp. 439–449. Academic Press, London (1980)38. Kirwan, W.E., Schober, G.: New inequalities from old ones. Math. Z. 180, 19–40 (1982)39. Lee, E., Markus, L.: Foundations of Optimal Control Theory. Wiley, New York (1967)40. Loewner, K.: Untersuchungen über schlichte konforme Abbildungen des Einheitskreises. I. Math. Ann.

89, 103–121 (1923)41. MacGregor, T.H., Wilken, D.R.: Extreme points and support points. In: Kühnau, R. (ed.) Handbook

of Complex Analysis: Geometric Function Theory, vol. I, pp. 371–392. Elsevier, Amsterdam (2002)42. Muir, J.R.: A class of Loewner chain preserving extension operators. J. Math. Anal. Appl. 337, 862–879

(2008)43. Pell, R.: Support point functions and the Loewner variation. Pacific J. Math. 86, 561–564 (1980)44. Pfaltzgraff, J.A.: Subordination chains and univalence of holomorphic mappings in C

n . Math. Ann.210, 55–68 (1974)

45. Pfaltzgraff, J.A.: Subordination chains and quasiconformal extension of holomorphic maps in Cn .

Ann. Acad. Scie. Fenn. Ser. A I Math. 1, 13–25 (1975)46. Pfluger, A.: Lineare extremal probleme bei schlichten Funktionen. Ann. Acad. Sci. Fenn. Ser. A I.,

489 (1971)47. Pommerenke, C.: Über die subordination analytischer funktionen. J. Reine Angew. Math. 218, 159–173

(1965)48. Pommerenke, C.: Univalent Functions. Vandenhoeck & Ruprecht, Göttingen (1975)49. Poreda, T.: On Generalized Differential Equations in Banach Spaces. Diss. Math. 310, 1–50 (1991)50. Poreda, T.: On the univalent holomorphic maps of the unit polydisc in C

n which have the parametricrepresentation, I-the geometrical properties. Ann. Univ. Mariae Curie Skl. Sect. A. 41, 105–113 (1987)

123


51. Poreda, T.: On the univalent holomorphic maps of the unit polydisc in Cn which have the parametric

representation, II-the necessary conditions and the sufficient conditions. Ann. Univ. Mariae Curie Skl.Sect. A. 41, 115–121 (1987)

52. Prokhorov, D.V.: Bounded univalent functions. In: Kühnau, R. (ed.) Handbook of Complex Analysis:Geometric Function Theory, vol. I, pp. 207–228, Elsevier, Amsterdam (2002)

53. Range, M.: Holomorphic Functions and Integral Representations in Several Complex Variables.Springer, New York (1986)

54. Reich, S., Shoikhet, D.: Nonlinear Semigroups, Fixed Points, and Geometry of Domains in BanachSpaces. Imperial College Press, London (2005)

55. Roper, K., Suffridge, T.J.: Convex mappings on the unit ball of Cn . J. Anal. Math. 65, 333–347 (1995)

56. Roth, O.: Control theory in H(D). Diss. Bayerischen Univ, Wuerzburg (1998)57. Roth, O.: A remark on the Loewner differential equation, Comput. Methods and Function Theory 1997

(Nicosia), Ser. Approx. Decompos, vol. 11, pp. 461–469. World Sci. Publ. River Edge (1999)58. Schleissinger, S.: On support points of the class S0(Bn). Proc. Am. Math. Soc. (2013, to appear)59. Suffridge, T.J.: Starlikeness, convexity and other geometric properties of holomorphic maps in higher

dimensions. In: Lecture Notes Math., vol. 599, pp. 146–159. Springer, Berlin (1977)60. Voda, M.: Solution of a Loewner chain equation in several complex variables. J. Math. Anal. Appl.

375, 58–74 (2011)61. Voda, M.: Loewner theory in several complex variables and related problems, Ph.D thesis, Univ.

Toronto (2011)

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extremal properties associated with univalent subordination chains in $$\mathbb {c}^n$$ c n

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