PARAMETRIC REPRESENTATION AND LINEAR FUNCTIONALS ASSOCIATED
WITH EXTENSION OPERATORS FOR BIHOLOMORPHIC MAPPINGS
IAN GRAHAM, GABRIELA KOHR and JOHN A. PFALTZGRAFF
We consider an operator introduced by Pfaltzgraff and Suffridge which provides a way of extending a locally biholomorphic mappingf ∈H(Bn) to a locally bi- holomorphic mappingF ∈H(Bn+1). Whenn= 1, this operator reduces to the well known Roper-Suffridge extension operator. In the first part of this paper we prove that iffhas parametric representation onBn then so doesF onBn+1. In particular, iff∈S∗(Bn) thenF∈S∗(Bn+1). We also prove that iffis convex on Bn, then the image ofFcontains the convex hull of the image of some egg domain contained inBn. In the second part of the paper we investigate some problems related to extreme points and support points for biholomorphic mappings onBn generated using the Roper-Suffridge extension operator. Given a parametric rep- resentation for an extreme point (respectively a support point) generated in this way, we consider whether the corresponding Loewner flow consists only of extreme points (respectively support points). This generalizes work of Pell.
AMS 2000 Subject Classification: Primary 32H02; Secondary 30C45.
Key words: convexity, starlikeness, Loewner chain, parametric representation, Roper-Suffridge extension operator, extreme point, support point.
1. INTRODUCTION AND PRELIMINARIES
Let Cn be the space of n complex variables z = (z1, . . . , zn) with the Euclidean inner product z, w = n
j=1zjwj and the Euclidean norm z = z, z1/2. Forn≥2, let z= (z2, . . . , zn)∈Cn−1 so that z= (z1,z) ∈Cn. The unit ball in Cn is denoted by Bn. In the case of one variable, B1 is denoted byU. The ball of radiusr >0 in Cnwith center at 0 will be denoted by Brn. Let L(Cn,Cm) denote the space of complex-linear mappings from Cn intoCm with the standard operator norm
A= sup{A(z): z= 1}
REV. ROUMAINE MATH. PURES APPL.,52(2007),1, 47–68
and let In be the identity inL(Cn,Cn). If Ω is a domain in Cn, letH(Ω) be the set of holomorphic mappings from Ω into Cn. Also, let H(Bn,C) be the set of holomorphic functions fromBnintoC. A mappingf ∈H(Bn) is called normalized if f(0) = 0 and Df(0) = In. We say that f ∈ H(Bn) is locally biholomorphic on Bn if the complex Jacobian matrix Df(z) is nonsingular at each z ∈ Bn. Let Jf(z) = detDf(z) for z ∈ Bn. Let LSn be the set of normalized locally biholomorphic mappings on Bn, and let S(Bn) denote the set of normalized biholomorphic mappings on Bn. In the case of one variable, the set S(B1) is denoted by S, and LS(B1) is denoted by LS. A mapping f ∈ S(Bn) is called starlike (respectively convex) if its image is a starlike domain with respect to the origin (respectively convex domain). The classes of normalized starlike (respectively convex) mappings on Bn will be denoted byS∗(Bn) (respectively K(Bn)). In the case of one variable,S∗(B1) (respectivelyK(B1)) is denoted byS∗ (respectivelyK).
Letf, g∈H(Bn). We say thatf is subordinate tog(and writef ≺g) if there is a Schwarz mappingv(i.e.,v∈H(Bn) andv(z) ≤ z,z∈Bn) such that f(z) =g(v(z)), z ∈Bn. If g is biholomorphic on Bn, this is equivalent to requiring thatf(0) =g(0) and f(Bn)⊆g(Bn).
We recall that a mapping f : Bn ×[0,∞) → Cn is called a Loewner chain iff(·, t) is biholomorphic on Bn,f(0, t) = 0, Df(0, t) = etIn for t≥0, and f(z, s) ≺ f(z, t) whenever 0 ≤ s ≤ t < ∞ and z ∈ Bn. We note that the requirement f(z, s) ≺ f(z, t) is equivalent to the condition that there is a unique biholomorphic 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∈Bn, t≥s≥0.
We also note that the normalization off(z, t) implies the normalization Dv(0, s, t) = es−tIn for 0≤s≤t <∞.
Certain subclasses of S(Bn) can be characterized in terms of Loewner chains. In particular,f is starlike if and only iff(z, t) = etf(z) is a Loewner chain (see [21]).
Definition 1.1 (see [9], [11], [12], [25], [26]). We say that a normalized mappingf ∈H(Bn) has parametric representation if there exists a mapping h:Bn×[0,∞)→Cn such that
(i) h(·, t) ∈ H(Bn), h(0, t) = 0, Dh(0, t) =In, t ≥0, Reh(z, t), z > 0, forz∈Bn\ {0}, t≥0;
(ii)h(z,·) is measurable on [0,∞) for z∈Bn,
and f(z) = limt→∞etv(z, t) locally uniformly onBn, where v=v(z, t) is the unique solution of the initial value problem
∂v
∂t =−h(v, t) a.e. t≥0, v(z,0) =z, for allz∈Bn.
The above condition is equivalent to the fact that there exists a Loewner chain f(z, t) such that {e−tf(·, t)}t≥0 is a normal family on Bn and f(z) = f(z,0), z∈Bn.
LetS0(Bn) be the set of mappings which have parametric representation onBn.
Let Aut(Bn) be the set of holomorphic automorphisms ofBn. Also, let U denote the set of unitary transformations in Cn. Then it is well known that
Aut(Bn) ={V ϕa: a∈Bn, V ∈ U}, where
(1.1) ϕa(z) =ϕ(z;a) =Ta
z−a 1− z, a
, z∈Bn,
(1.2) Ta= 1
a2{(1−sa)aa∗+saa2In} and
sa=
1− a2.
Note thatT0 =In andϕ0(z) =z,z∈Bn. The following conditions hold (see [29]):
(1.3) |Jϕa(0)|= (1− a2)n+12 ;
(1.4) ϕa(0) =−a, ϕa(a) = 0, ϕ−1a =ϕ−a. Moreover, ifφ∈Aut(Bn) then
(1.5) |Jφ(z)|=
1− φ(z)2 1− z2
n+12
, z∈Bn.
A key role in our discussion is played by the following Schwarz-type lemma for the Jacobian determinant of a holomorphic mapping fromBn into Bn. We have (cf. [29]; see also [13])
Lemma1.1. Let ψ∈H(Bn) be such that ψ(Bn)⊆Bn. Then
(1.6) |Jψ(z)| ≤
1− ψ(z)2 1− z2
n+12
, z∈Bn.
This inequality is sharp and equality at a given point z∈Bn holds if and only ifψ∈Aut(Bn).
Proof. Fixa∈Bn and letb=ψ(a). First, assume thata= 0 andb= 0.
Let f :Bn → Cn be given by f(z) = (ϕb ◦ψ◦ϕ−a)(z), z ∈ Bn. Then f is a holomorphic mapping onBn, f(0) = 0 andf(Bn)⊆Bn. Consequently, by
the Schwarz lemma for holomorphic mappings we deduce thatf(z) ≤ z, z∈Bn, andDf(0) ≤1. Further, since
|Jf(z)| ≤ Df(z)n, z∈Bn, we have|Jf(0)| ≤1. A simple computation yields
Df(0) =Dϕb(b)Dψ(a)Dϕ−a(0), and using the last equality in (1.4) we obtain
Jf(0) = Jϕ−a(0) Jϕ−b(0)Jψ(a).
Next, taking into account (1.3) and the above equation, we deduce that 1≥ |Jf(0)|=|Jψ(a)|
1− a2 1− b2
n+12 , hence (1.6) now follows.
If a= 0 and b = 0 then it suffices to consider the mapping g = ϕb ◦ψ and to use a similar argument as above. Similarly, ifa= 0 andb= 0, then we may consider the mappingh =ψ◦ϕ−a and apply the above argument. The casea=b= 0 is clear.
Ifψ∈Aut(Bn) then equality holds according to (1.5).
Conversely, if equality holds at a given pointa∈Bn, then reversing the steps in the above proof, we deduce that|Jf(0)|= 1 wheref =ϕb◦ψ◦ϕ−aand b=ψ(a). Hencef ∈Aut(Bn) by [16, Theorem 11.3.1], and thusψ∈Aut(Bn) too. This completes the proof.
Forn≥1, setz = (z1, . . . , zn)∈Cnand z= (z, zn+1)∈Cn+1.
Definition 1.2 ([22]). The extension operator Φn : LSn → LSn+1 is defined by
Φn(f)(z) =F(z) =
f(z), zn+1[Jf(z)]n+11 , z= (z, zn+1)∈Bn+1. We choose the branch of the power function such that
[Jf(z)]1/(n+1)
z=0= 1.
Then F = Φn(f) ∈ LSn+1 whenever f ∈ LSn. It is easy to see that if f ∈S(Bn) thenF ∈S(Bn+1).
If n = 1 then Φ1 reduces to the well-known Roper-Suffridge extension operator. For generaln≥2 we have
Definition1.3 ([27]). The Roper-Suffridge extension operator Ψn:LS → LSn is defined by
Ψn(f)(z) =
f(z1),z
f(z1) , z= (z1,z) ∈Bn. We choose the branch of the power function such that
f(z1)
z1=0= 1.
Roper and Suffridge [27] proved that if f is convex on U then Ψn(f) is also convex onBn. Graham and Kohr [8] proved that iff is starlike onU then so is Ψn(f) on Bn, and in [10] (see also [9] and [7]) it is shown that if f has parametric representation on the unit disc, then Ψn(f) has the same property on Bn. Moreover, if one begins with a complex valued function f(z1), then the extension to B2 under Φ1 = Ψ2 is (f(z1), z2
f(z1)). If this mapping is then extended to B3 then to B4, etc. up to Bn by successive applications of Φk, k = 1, . . . , n−1, one obtains the mapping Ψn(f)(z) = (f(z1),z
f(z1)), i.e. we obtain the Roper-Suffridge extension operator Ψn.
In this paper we prove that iff ∈S(Bn) can be imbedded in a Loewner chain f(z, t), then F = Φn(f) can also be imbedded in a Loewner chain F(z, t). Further, iff ∈S0(Bn) thenF = Φn(f)∈S0(Bn+1). In particular, we give a simplified proof of the theorem of Graham, Kohr and Kohr [10] that the Roper-Suffridge extension operator preserves the parametric representation property. Moreover, we prove that iff ∈S∗(Bn) thenF = Φn(f)∈S∗(Bn+1), and iff ∈K(Bn) then the image ofF contains the convex hull of the image of some egg domain contained inBn. It would be interesting to give a complete answer to the conjecture of Pfaltzgraff and Suffridge [22, Conjecture 1] that Φn preserves convexity, but so far we have not been able to do so. We also discuss the behaviour of Φn with respect to starlikeness and convexity on the unit polydiscPnin Cn.
In the last section we investigate some problems involving extreme points and support points for families of biholomorphic mappings on Bn generated with the Roper-Suffridge extension operator. We consider the following ques- tion: given a parametric representation for an extreme point (respectively a support point) of Ψn(S), must the corresponding Loewner flow e−tF(·, t) consist of extreme points (respectively support points)? The analogous one- variable questions were treated by Pell [19] (see also Kirwan [15]).
2. STARLIKENESS AND CONVEXITY PROPERTIES AND THE EXTENSION OPERATOR Φn
We begin this section with the following main result. In the case n= 1, we obtain a simplified proof of [10, Theorem 2.1].
Theorem2.1. Assume f ∈S(Bn) can be imbedded in a Loewner chain f(z, t). Then F = Φn(f) can also be imbedded in a Loewner chain F(z, t).
Proof. Since f ∈ S(Bn), it is easiy to see that F ∈ S(Bn+1). Let v = v(z, s, t) be the transition mapping associated tof(z, t). Then
(2.1) f(z, s) =f(v(z, s, t), t), z ∈Bn, 0≤s≤t <∞.
Let ft(z) = f(z, t) for z ∈Bn and t ≥0, and let vs,t(z) = v(z, s, t), z ∈Bn,t≥s≥0. Also, let F :Bn+1×[0,∞)→Cn+1 be given by
(2.2) F(z, t) =
f(z, t), zn+1en+1t [Jft(z)]n+11
for z = (z, zn+1) ∈ Bn+1 and t ≥ 0. We choose the branch of the power function such that
[Jft(z)]n+11
z=0 = ent/(n+1).
Let us prove that F(z, t) is a Loewner chain. Indeed, since f(·, t) is biholomorphic onBn,f(0, t) = 0 and Df(0, t) = etIn, it is not difficult to see thatF(·, t) is biholomorphic onBn+1,F(0, t) = 0 and DF(0, t) = etIn+1.
LetVs,t:Bn+1→Cn+1 be given by Vs,t(z) =V(z, s, t), where (2.3) V(z, s, t) =
v(z, s, t), zn+1en+1s−t[Jvs,t(z)]n+11
forz= (z, zn+1)∈Bn+1 and t≥s≥0. We choose the branch of the power function such that [Jvs,t(z)]n+11
z=0 = en(s−t)n+1 . ThenVs,t is biholomorphic on Bn,Vs,t(0) = 0, DVs,t(0) = es−tIn+1, andVs,t(z)<1,z∈Bn+1. Indeed, by Lemma 1.1 we obtain that
Vs,t(z)2=vs,t(z)2+ e2(s−t)n+1 |zn+1|2|Jvs,t(z)|n+12 ≤
≤ vs,t(z)2+ |zn+1|2
1− z2(1− vs,t(z)2)<
<vs,t(z)2+ 1− vs,t(z)2 = 1, z= (z, zn+1)∈Bn+1. HenceVs,t(z)<1 for z∈Bn+1, as claimed.
Further, taking into account (2.1), we easily deduce that F(z, s) = F(V(z, s, t), t) forz∈Bn+1,t≥s≥0. Indeed,
F(V(z, s, t), t) = (f(v(z, s, t), t), zn+1en+1s [Jvs,t(z)]n+11 [Jft(vs,t(z))]n+11 =
=
f(z, s), zn+1en+1s [Jfs(z)]n+11 =F(z, s),
for allz∈Bn+1 and t≥s≥0. Here we have used (2.1) and the fact that Jfs(z) =Jft(vs,t(z))Jvs,t(z), z ∈Bn, t≥s≥0.
This completes the proof.
Corollary2.1. Assume f ∈S0(Bn). Then F = Φn(f)∈S0(Bn+1).
Proof. Since f ∈ S0(Bn), there is a Loewner chain f(z, t) such that f(z,0) =f(z),z ∈Bn, and {e−tf(·, t)}t≥0 is a normal family. Then
(2.4) r
(1 +r)2 ≤ e−tf(z, t) ≤ r
(1−r)2, z=r <1, t≥0,
by [11, Corollary 2.6] (see also [6]). Applying the Cauchy integral formula for vector valued holomorphic functions, it is easy to see that for each r ∈(0,1) there isK=K(r)≥0 such that
e−tDf(z, t) ≤K(r), z ≤r, t≥0.
Moreover, since
|Jft(z)| ≤ Dft(z)n, z ∈Bn, we deduce that there is someK∗ =K∗(r)≥0 such that (2.5) |Jft(z)|n+11 ≤en+1nt K∗(r), z ≤r, t≥0.
Let F : Bn+1 ×[0,∞) → Cn+1 be the Loewner chain given by (2.2).
Taking into account (2.4) and (2.5) we now easily deduce that for each r ∈ (0,1) there is some L = L(r) ≥ 0 such that e−tF(z, t) ≤ L(r), z ≤ r, t≥0. Consequently, {e−tF(·, t)}t≥0 is a locally uniformly bounded family on Bn+1, and thus is normal. Hence F = F(·,0) ∈ S0(Bn+1). This completes the proof.
From Corollary 2.1 and [11, Corollary 2.6] we obtain the following dis- tortion result of independent interest for mappings inS0(Bn). In particular, this result also holds for mappings inS∗(Bn), since any starlike mapping has parametric representation onBn (see e.g. [9]).
Corollary2.2. Assume f ∈S0(Bn) and r∈[0,1). Then r2
(1 +r)4 ≤ f(z)2+|zn+1|2|Jf(z)|n+12 ≤ r2 (1−r)4, z= (z, zn+1)∈∂Bn+1r .
Proof. Sincef ∈S0(Bn), we have F = Φn(f)∈S0(Bn+1). Then r
(1 +r)2 ≤ F(z) ≤ r
(1−r)2, z=r, by [11, Corollary 2.6]. The result now follows.
From Corollary 2.1 we can deduce the compactness of the set Φn(S0(Bn)).
We have
Corollary2.3. The set Φn(S0(Bn))is compact.
Proof. Since Φn(S0(Bn)) is a subset ofS0(Bn+1), Φn(S0(Bn)) is locally uniformly bounded by [11, Corollary 2.6]. We prove that Φn(S0(Bn)) is closed.
To this end, let{Fk}k∈Nbe a sequence in Φn(S0(Bn)) which converges locally uniformly onBn+1 to a mappingF ask→ ∞. Also let{fk}k∈Nbe a sequence in S0(Bn) be such that Fk = Φn(fk), k ∈ N. Since {fk}k∈N is a locally uniformly bounded sequence onBn, there is a subsequence{fkp}p∈Nof{fk}k∈N which converges locally uniformly on Bn to a mapping f. Since S0(Bn) is a compact set, by [11, Theorem 2.9], we deduce that f ∈ S0(Bn). Also it is easy to see that the subsequence{Φn(fkp)}p∈Nconverges locally uniformly on Bn+1 to Φn(f), and thus we must have F = Φn(f). Hence F ∈Φn(S0(Bn)).
This completes the proof.
Example 2.1. (i) Let fj ∈S, j = 1, . . . , n. Then f :Bn→ Cn given by f(z) = (f1(z1), . . . , fn(zn)) belongs to S0(Bn). Indeed, since fj ∈S, there is a Loewner chain fj(zj, t) such thatfj(zj) =fj(zj,0),j = 1, . . . , n. Moreover, {e−tfj(zj, t)}t≥0 is a normal family on U since e−tfj(zj, t) ∈ S. Next, let f(z, t) = (f1(z1, t), . . . , fn(zn, t)) forz= (z1, . . . , zn)∈Bn andt≥0. Then it is easy to see thatf(z, t) is a Loewner chain and {e−tf(z, t)}t≥0 is a normal family onBn. The desired conclusion now follows.
Further, by Corollary 2.1, we deduce thatF :Bn+1 →Cn+1 given by F(z) =
f1(z1), . . . , fn(zn), zn+1 n j=1
[fj(zj)]n+11 , z= (z, zn+1)∈Bn+1, belongs toS0(Bn+1).
(ii) Letf ∈ LSn be such that
(1− z2)[Df(z)]−1D2f(z)(z,·) ≤1, z ∈Bn. ThenF = Φn(f) belongs to S0(Bn+1).
Indeed, by [20, Theorem 2.4], f is biholomorphic on Bn and can be imbedded as the first element of the chain
f(z, t) =f(e−tz) + (et−e−t)Df(e−tz)(z), z∈Bn, t≥0.
Moreover, since lim
t→∞e−tf(z, t) =z locally uniformly on Bn, {e−tf(z, t)}t≥0 is a normal family on Bn. Hence f ∈ S0(Bn). By Corollary 2.1, we have F = Φn(f)∈S0(Bn+1).
Another consequence of Theorem 2.1 is given in the following result, which provides a positive answer to the question of Pfaltzgraff and Suffridge regarding the preservation of starlikeness under the operator Φn (see [22]).
Corollary2.4. Assume f ∈S∗(Bn). Then F = Φn(f)∈S∗(Bn+1).
Proof. The fact that f is starlike on Bn is equivalent to the statement that f(z, t) = etf(z) is a Loewner chain. With this choice of f(z, t), we
deduce thatF(z, t) given by (2.2) is a Loewner chain. On the other hand, a simple computation yields thatF(z, t) = etF(z) forz∈Bn+1andt≥0. Thus F =F(·,0)∈S∗(Bn+1), as claimed. This completes the proof.
Example 2.2. (i) Let fj ∈ S∗, j = 1, . . . , n. Then f : Bn → Cn given byf(z) = (f1(z1), . . . , fn(zn)) is starlike onBn. By Corollary 2.4, we deduce thatF :Bn+1→Cn+1 given by
F(z) =
f1(z1), . . . , fn(zn), zn+1 n j=1
[fj(zj)]n+11 , z= (z, zn+1)∈Bn+1, is starlike onBn+1. For example, the mapping
F(z) =
z1
(1−z1)2, . . . , zn
(1−zn)2, zn+1 n j=1
1 +zj (1−zj)3
1
n+1
is starlike onBn+1.
(ii) Letabe a complex number. Then the mappingF :B3 →C3given by F(z) =
z1+az1z2, z2, z3(1 +az2)1/3 is starlike if and only if|a| ≤1.
Indeed, if |a| ≤1 then f :B2 → C2 given by f(z) = (z1+az1z2, z2) is starlike onB2 (see [32], [28]). HenceF is also starlike on B3 by Corollary 2.4.
Conversely, if F ∈ S∗(B3) then F(z,0) is starlike on B2, and thus we must have|a| ≤1 by [32], [28].
We next discuss the case of convex mappings associated with the operator Φn. Pfaltzgraff and Suffridge [22] conjectured that iff ∈K(Bn) then Φn(f)∈ K(Bn+1).
Fora∈(0,1], let Ωa,n=
z= (z, zn+1)∈Cn+1: |zn+1|2 < an+12n (1− z2)
.
Then Ωa,n ⊆Bn+1 and Ω1,n = Bn+1. We have the following convexity result.
Theorem2.2. Let f ∈K(Bn) andα1, α2 >0 be such that α1+α2 ≤1.
Also letF = Φn(f). Then
(1−λ)F(z) +λF(w)∈F(Ωα1+α2,n), z∈Ωα1,n, w∈Ωα2,n, λ∈[0,1].
Proof. Our argument combines an idea of Gong and Liu [4] (see also [5]) with the estimates for the Jacobian determinant of a holomorphic mapping fromBnto itself (Lemma 1.1). Fixλ∈[0,1] and letz∈Ωα1,nandw∈Ωα2,n. Since f ∈ K(Bn), F is biholomorphic on Bn+1. We want to find a point u= (u, un+1)∈Ωα1+α2,n such that
(1−λ)F(z) +λF(w) =F(u),
i.e.,
f(u), un+1[Jf(u)]n+11 =
=
(1−λ)f(z) +λf(w),(1−λ)zn+1[Jf(z)]n+11 +λwn+1[Jf(w)]n+11 . Since z, w ∈ Bn and f ∈ K(Bn), there is a unique point u ∈ Bn such that
f(u) = (1−λ)f(z) +λf(w),
i.e.,u =f−1((1−λ)f(z) +λf(w)). Ifλ= 0 thenu =z, and if λ= 1 then u =w. Hence, we may assume that λ ∈(0,1). Then u =u(z, w) can be viewed as a mapping fromBn×Bn intoBn. Let
un+1 = (1−λ)zn+1
Jf(z) Jf(u)
1
n+1
+λwn+1
Jf(w) Jf(u)
1
n+1
. We prove thatu= (u, un+1)∈Ωα1+α2,n. It is obvious that
Ju
z(z, w) = (1−λ)nJf(z)
Jf(u) and Ju
w(z, w) =λnJf(w) Jf(u),
whereuz and uw denote the Fr´echet derivatives of u with respect to z and w, respectively. Hence
un+1 = (1−λ)n+11 zn+1[Ju
z(z, w)]n+11 +λn+11 wn+1[Ju
w(z, w)]n+11 . Next, using Lemma 1.1 and H¨older’s inequality in the previous equation, we obtain
|un+1| ≤(1−λ)n+11 |zn+1|
1−u(z, w)2 1−z2
1/2
+λn+11 |wn+1|
1−u(z, w)2 1−w2
1/2
≤
1− u2(1−λ+λ)n+11
|zn+1|2 1− z2
n+12n +
|wn+1|2 1− w2
n+12n n+1n
<
1− u2(α1+α2)n+1n .
Therefore, we have proved that |un+1|2 < (α1+α2)n+12n (1− u2), i.e., u = (u, un+1)∈Ωα1+α2,n. This completes the proof.
Corollary 2.5. Let f ∈K(Bn) and F = Φn(f). Then (1−λ)F(z) +λF(w)∈F(Bn+1), z, w∈Ω1/2,n, λ∈[0,1].
It is natural to investigate the situation concerning starlikeness and con- vexity on the unit polydiscPn=
z∈Cn: max1≤j≤n|zj|<1 .
Remark2.1.The operator Φndoes not preserve convexity onPn. Indeed, letf :Pn→Cnbe a normalized convex mapping and letF = Φn(f). By [31,
Theorem 2] there exist normalized convex functionsfk(zk) on the unit discU, k= 1, . . . , n, such that
f(z) = (f1(z1), . . . , fn(zn)), z = (z1, . . . , zn)∈Pn. ThenF = Φn(f) is given by
(2.6) F(z) =
f1(z1), . . . , fn(zn), zn+1 n k=1
[fk(zk)]n+11
, z= (z, zn+1)∈Pn+1. Here we choose the branch of the power function such that [fk(zk)]1/(n+1)|zk=0= 1 fork= 1, . . . , n. It is clear that if there is someksuch thatfk(zk)≡1, then F does not satisfy the necessary and sufficient condition for convexity in the polydisc given by Suffridge.
Remark 2.2.The operator Φn does not preserve starlikeness on the unit polydisc Pn either. Indeed, let fj(zj) be a function in S∗ for j = 1, . . . , n.
Then it is easy to see that the mappingf(z) = (f1(z1), . . . , fn(zn)) is starlike on Pn while F given by (2.6) is not starlike on Pn+1. Indeed, the necessary and sufficient condition for starlikeness ofF is (see [31, Theorem 1])
(2.7) Re
wj(z) zj
>0, j= 1, . . . , n+ 1, z∞ =|zj|>0, where
w(z) = [DF(z)]−1F(z) =
=
f1(z1)
f1(z1), . . . ,fn(zn) fn(zn), zn+1
1− 1 n+ 1
n k=1
fk(zk)fk(zk) (fk(zk))2
, for allz= (z, zn+1)∈Pn+1.
It is clear that the firstninequalities in (2.7) are satisfied, but ifj=n+1 then (2.7) becomes
0<Re
wn+1(z) zn+1
= 1− 1 n+ 1
n k=1
Re
fk(zk)fk(zk) (fk(zk))2
, |zn+1|=z∞>0, i.e.,
1− 1 n+ 1
n k=1
Re
fk(zk)fk(zk) (fk(zk))2
>0, |zk|<1, k= 1, . . . , n.
It suffices to choose n = 2 and fj(zj) = zj/(1−zj)2 for j = 1,2. Also let z1 =z2∈U. Then the above inequality is equivalent to
Re
(1−z1)(z1+ 3) (1 +z1)2
>0, |z1|<1.
However, it is elementary to check that this relation is not satisfied everywhere on the unit discU, and thus F is not starlike.
3. EXTREME POINTS AND SUPPORT POINTS ASSOCIATED WITH THE ROPER-SUFFRIDGE EXTENSION OPERATOR In this section we restrict our discussion to the Roper-Suffridge extension operator Ψn. A key role is played by the result established in Theorem 2.1.
We shall study extreme points and support points for families of univalent mappings on Bn constructed using the Roper-Suffridge operator, and their behaviour under the Loewner variation. To this end, we recall the definitions of extreme points and support points in the general setting of locally convex linear spaces. For applications of linear methods to the study of extremal problems in geometric function theory of one variable, the reader may consult [14], [1], [2], [3], [24], [30], and the recent survey [17]. In the case of several variables, see [18].
Definition 3.1. Let X be a locally convex linear space and let E be a subset ofX.
(i) A pointx∈Eis called anextreme pointofEprovidedx=ty+(1−t)z, where t∈ (0,1), y, z ∈ E, implies x =y =z. That is, x ∈E is an extreme point ofE ifx is not a proper convex combination of two points inE.
(ii) A point w∈E is called a support pointof E if ReL(w) = max
y∈E ReL(y)
for some continuous linear functionalL:X→Csuch that ReLis nonconstant onE.
Let exEand suppEbe the sets of extreme points ofEand support points ofE. From the general theory of locally convex linear spaces, in particular by the Krein-Milman theorem (see e.g. [14, Chapter 4]), it is known that ifE is a nonempty compact subset ofXthen exE and suppE are nonempty subsets ofE.
In the case of the classS, it is known that iff ∈exS orf ∈suppS, then f maps the unit disc U onto the complement of a continuous arc tending to
∞with increasing modulus (see e.g. [17]). In particular, a bounded univalent function cannot be a support point ofS.
We recall in view of the proof of Theorem 2.1 that iff(z1, t) is a Loewner chain thenF :Bn×[0,∞)→Cn given by
(3.1) F(z, t) =
f(z1, t),ze t/2(f(z1, t))1/2
, z= (z1,z) ∈Bn, t≥0,
is also a Loewner chain. We choose the branch of the power function such that (f(z1, t))1/2|z1=0 = et/2 for t ≥ 0. If t = 0 then F = F(·,0) is the Roper-Suffridge extension operator Ψn(f).
We also recall that Ψn(S) is a compact set by Corollary 2.3, hence ex Ψn(S)=∅ and supp Ψn(S)=∅.
We begin this section with some auxiliary results.
Lemma3.1. Ψn(exS)⊆ex Ψn(S).
Proof. LetF ∈Ψn(exS) andf ∈exS be such thatF = Ψn(f). Suppose F = sG+ (1−s)H where s ∈ (0,1) and G, H ∈ Ψn(S). Then there exist functionsg, h∈S such thatG= Ψn(g), H= Ψn(h) and
Ψn(f)(z) =sΨn(g)(z) + (1−s)Ψn(h)(z), z∈Bn. Hence
f(z1) =sg(z1) + (1−s)h(z1), z1 ∈U,
and since f ∈ exS, we must have g ≡ h. Therefore G ≡ H, too. This completes the proof.
Example 3.1.It is known that the rotations of the Koebe function, given by f(z1) = z1/(1 −λz1)2, where |λ| = 1, are extreme points of S. Then F = Ψn(f)∈ex Ψn(S) by Lemma 3.1, i.e., the mappingFλ given by
(3.2) Fλ(z) =
z1 (1−λz1)2,z
1 +λz1 (1−λz1)3
1/2
, z= (z1,z) ∈Bn, is an extreme point of Ψn(S) for |λ|= 1.
Lemma 3.2. Let f ∈exS and let f(z1, t) be a Loewner chain such that f(z1) =f(z1,0), z1 ∈U. Also let F(z, t) be given by(3.1). Then e−tF(·, t)∈ ex Ψn(S) for t≥0.
Proof. Sincef is an extreme point ofS, e−tf(·, t) also is an extreme point ofS by a result of Pell [19]. Then Ψn(e−tf(·, t))∈Ψn(exS) for t≥0. Hence Ψn(e−tf(·, t))∈ex Ψn(S) by Lemma 3.1. On the other hand, since
Ψn(e−tf(·, t)) = e−tF(·, t), t≥0, the conclusion follows.
We now prove one of the main results of this paper.
Theorem3.1. Letf ∈S andF = Ψn(f). Also letF(z, t)be the Loewner chain given by (3.1). If F ∈ex Ψn(S) then e−tF(·, t)∈ex Ψn(S) for t≥0.
Proof. Fixt≥0. Suppose that
e−tF(z, t) =λG(z) + (1−λ)H(z), z∈Bn,
