Gabriela Kohr

Geometric and analytic aspects of Loewner chains 

in C n and complex Banach spaces 

Gabriela Kohr 

Faculty of Mathematics and Computer Science 

Babes¸-Bolyai University 

Cluj-Napoca, Romania 

Based on joint works with Ian Graham (Toronto), Hidetaka Hamada (Fukuoka), 

Mirela Kohr (Cluj-Napoca) 

Gabriela Kohr (UBB Cluj) Geometric and analytic aspects of Loewner chains 1 / 62

Non-normalized univalent subordination chains and the 

generalized Loewner differential equation on the unit ball in 

C n 

Solutions for the generalized Loewner differential equation in 

C n associated with non-normalized univalent subordination 

chains given by a time-dependent linear operator 

Univalent subordination chains and the generalized Loewner 

differential equation in reflexive complex Banach spaces 


1. Main contributions 

1. Contributions in SCV 

• J.A. Pfaltzgraff (1974-1975): 

-The first generalizations to higher dimensions of various results in the 

theory of Loewner chains. 

-Generalizations to higher dimensions of the one variable univalence 

and quasiconformal extension results due to J. Becker 

1. J.A. Pfaltzgraff, Subordination chains and univalence of holomorphic 

mappings in C n , Math. Ann., 210 (1974), 55-68. 

2. J.A. Pfaltzgraff, Subordination chains and quasiconformal extension 

of holomorphic maps in C n , Ann. Acad. Scie. Fenn. Ser. A I Math., 

1(1975), 13-25. 



• T. Poreda (1990): 

-The Loewner differential equation on the unit polydisc in C n 

-Applications of parametric representation to growth theorems and 

coefficient estimates on the unit polydisc 

-Extensions to complex Banach spaces 

1. T. Poreda, On generalized differential equations in Banach spaces, 

Dissertationes Mathematicae, 310 (1991), 1-50. 

• M. Chuaqui (1995): 

-Growth results for strongly starlike maps on the unit ball in C n by using 

the theory of Loewner chains 

-Explicit quasiconformal extensions to C n of quasiconformal strongly 

starlike mappings on B n 

1. M. Chuaqui, Applications of subordination chains to starlike 

mappings in C n , Pacif. J. Math., 168(1995), 33-48. 



• I. Graham, H. Hamada, G. K., M. Kohr, J.A. Pfaltzgraff, T.J. 

Suffridge, and P. Duren: 

-The Loewner differential equation on the unit ball in C n ; 

-Distortion results for mappings with parametric representation on B n 

-Quasiconformal extension results and Loewner chains 

-Geometric aspects of Loewner chains: asymptotic starlikeness and 

asymptotic spirallikeness; Extension operators and Loewner chains 

-Extreme points and support points associated to families of 

biholomorphic mappings with parametric representation 

1. I. Graham and G. Kohr, Geometric Function Theory in One and 

Higher Dimensions, Marcel Dekker, New York, 2003. 

2. I. Graham, H. Hamada, G. Kohr and M. Kohr, Parametric 

representation and asymptotic starlikeness in C n , Proc. AMS, 136 

(2008), 3963–3973. 

3. P. Duren, I. Graham, H. Hamada and G. Kohr, Solutions for the 

generalized Loewner differential equation in several complex variables, 

Math. Ann. 347 (2010) 411–435. 



• M. Elin, S. Reich and D. Shoikhet (2000): 

-Parametric representation for spirallike mappings in C n and complex 

Banach spaces, by using the nonlinear semigroups theory 

1. M. Elin, S. Reich, D. Shoikhet, Complex dynamical systems and the 

geometry of domains in Banach spaces, Dissertationes Math., 

427(2004), 1-62. 

2. S. Reich and D. Shoikhet, Nonlinear Semigroups, Fixed Points, and 

Geometry of Domains in Banach Spaces, Imperial College Press, 

London, 2005. 

• M. Elin; I. Graham, H. Hamada, and G.K.; J.R. Muir: 

1. M. Elin, Extension operators via semigroups, J. Math. Anal. Appl., 

377(2011), 239-250. 

2. I. Graham, H. Hamada and G.K., Extension operators and univalent 

subordination chains, J. Math. Anal. Appl., 386(2012), 278–289. 

3. J.R. Muir, A class of Loewner chain preserving extension operators, 

J. Math. Anal. Appl., 337(2008), 862-979. 



• F. Bracci, M.D. Contreras, S.Diaz-Madrigal (2008): 

-The study of a general version of the Loewner differential equation on 

the unit disc and complex hyperbolic manifolds, by using the iteration 

and the semigroups theory 

1. F. Bracci, M.D. Contreras and S.Diaz-Madrigal, Evolution families 

and the Loewner equation I: the unit disk, J. Reine Angew. Math., to 

appear. 

2. F. Bracci, M.D. Contreras and S.Diaz-Madrigal, Evolution families 

and the Loewner equation II: complex hyperbolic manifolds, Math. 

Ann., 344(2009), 947-962. 

• M.D. Contreras, S. Diaz-Madrigal and P. Gumenyuk (2009): a 

general version of the notion of Loewner chains and the 

correspondence with the evolution families. 

1. M.D. Contreras, S. Diaz-Madrigal and P. Gumenyuk, Loewner 

chains in the unit disk, Rev. Mat. Iberoamer., 26 (2010), 975–1012 

2. M.D. Contreras, S. Diaz-Madrigal and P. Gumenyuk, Geometry 

behind chordal Loewner chains, Complex Analysis and Operator 

Theory, 4 (2010), 541–587. 



• L. Arosio, F. Bracci, H. Hamada and G.K. (2010): 

-Abstract construction of Loewner chains on the unit ball and 

hyperbolic manifolds: one-to one correspondence between 

L d -Loewner chains and L d -evolution families. 

1. L. Arosio, F. Bracci, H. Hamada and G.Kohr, An abstract approach 

of Loewner’s chains, J. Anal. Math., to appear. 

2. H. Hamada, G. K., and J. Muir, Extensions of L d -Loewner chains to 

higher dimensions, submitted. 

• L. Arosio; M. Vodă (2010): resonances to construct solutions to the 

generalized Loewner differential equation on the unit ball in C n . 

1. L. Arosio, Resonances in Loewner equations, Advances Math., 

227(2011), 1413-1435. 

2. M. Vodă, Solution of a Loewner chain equation in several complex 

variables, J. Math. Anal. Appl., 375 (2011), 58-74. 


2. Univalent subordination chains in several complex variables 

3. Univalent subordination chains 

Subordination chains in C n 

Definition 

(a) If f , g ∈ H(B n ), we say that f is subordinate to g (written f ≺ g) if 

there is a Schwarz mapping v such that f = g ◦ v. 

• If g is biholomorphic, then f ≺ g iff f (0) = g(0) and f (B n ) ⊆ g(B n ). 

(b) f : B n × [0, ∞) → C n is an univalent subordination chain if: 

(i) f (·, t) is biholomorphic on B n , f (0, t) = 0, t ≥ 0; 

(ii) f (·, s) ≺ f (·, t) whenever 0 ≤ s ≤ t < ∞. 

• f (z, t) = e t 

0 A(τ)dτ z + · · · , where A : [0, ∞) → L(C n , C n ) is locally 

Lebesgue integrable, m(A(t)) > 0, t ≥ 0, and ∞ 

0 m(A(τ))dτ = ∞. 

Here m(A(t)) = minz=1 Re 〈A(t)(z), z〉. 

• f (z, t) = etAz + · · · –A-normalized univalent subordination chain. 

• f (z, t) = etz + · · · –usual Loewner chain. 



Remark 

The first part of the talk refers to univalent subordination chains 

with the normalization Df (0, t) = e t 

0 A(τ)dτ for t ≥ 0. 

f (·, s) ≺ f (·, t) is equivalent to the existence of a Schwarz mapping 

v = v(z, s, t)-transition mapping associated with f (z, t), such that 

f (z, s) = f (v(z, s, t), t), z ∈ B n , 0 ≤ s ≤ t < ∞. 

(vs,t)-evolution family associated to f (z, t). 

Remark 

Let f (z, t) be a univalent subordination chain and let v(z, s, t) be the 

transition mapping associated with f (z, t). Then 

(i) v(·, s, t) is biholomorphic on B n and v(z, s, s) = z, z ∈ B n , s ≥ 0. 

(ii) v(z, s, ·) is decreasing on [s, ∞), for all z ∈ B n and s ≥ 0. 

(iii) Semigroup property: 

v(z, s, τ) = v(v(z, s, t), t, τ), z ∈ B n , 0 ≤ s ≤ t ≤ τ < ∞. 



S(B n ) = {h ∈ H(B n ) : h biholomorphic , h(0) = 0, Dh(0) = In}; S(B 1 ) = S 

• Every function f ∈ S can be embedded in a Loewner chain. 

• In C n , n ≥ 2, there exist mappings f ∈ S(B n ) that cannot be 

embedded in Loewner chains f (z, t) such that {e −t f (·, t)}t≥0 is a 

normal family on B n . 

• Every function f ∈ S has parametric representation, i.e. 

f (z) = limt→∞ e t v(z, t), where v = v(z, t) is the unique Lipschitz 

continuous solution on [0, ∞) of the initial value problem 

∂v 

= −vp(v, t) a.e. t ≥ 0, v(z, 0) = z, 

∂t 

for some choice of p = p(z, t) such that p(·, t) ∈ P for almost all 

t ∈ [0, ∞) and p(z, ·) is measurable on [0, ∞) for z ∈ U. 

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

• S 0 (B 1 ) = S; S 0 (B n ) S(B n ) for n ≥ 2, where S 0 (B n ) is the family of 

mappings which have parametric representation. 



• S 0 (B n ) is compact, but S(B n ) is not compact for n ≥ 2. 

• There exist mappings f ∈ S 0 (B n ) such that 1 

2 D2 f (0) > 2. 

• In dimension n ≥ 2, there exist Loewner chains f (z, t) such that 

{e −t f (·, t)}t≥0 is not a normal family. 

• Let p : U × [0, ∞) → C be a function such that p(·, t) ∈ P, t ≥ 0, and 

p(z, ·) is measurable on [0, ∞), z ∈ U. Then there exists a unique 

Loewner chain f (z, t) which satisfies the Loewner differential equation 

∂f 

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

• In dimension n ≥ 2, the Loewner differential equation 

∂f 

(z, t) = Df (z, t)h(z, t), a.e. t ≥ 0, ∀z ∈ U, 

∂t 

does not have a unique normalized univalent solution f (z, t). 



• Many classical results in the theory of holomorphic mappings in C n 

do not hold in infinite dimensional complex Banach spaces. 

• Montel’s theorem does not hold in the infinite dimensional setting 

(but surprisingly, Vitali’s theorem does). 

• On a domain in Cn , any univalent (holomorphic and injective) 

mapping into Cn is also biholomorphic. However, this result is no 

longer true in infinite dimensional complex Banach spaces. For 

example, if f : ℓ2 → ℓ2 is given by f (x) = (x 2 1 , x 3 1 , x 2 2 , x 3 2 , . . .), then f is 

univalent on the unit ball of ℓ2, but is not biholomorphic. 

• On a domain in C n any univalent mapping is open. Heath and 

Suffridge (1980): an example of a univalent mapping on the unit ball B 

of a complex Banach space which is not biholomorphic, f (B) contains 

an open set, but f (B) is not open. 



• There exist biholomorphic mappings on the unit ball B of an infinite 

dimensional complex Banach space X which are not bounded on the 

ball Br for r ∈ (0, 1). 

• Graham, Hamada, K (2012): Let f (x) = (0, f2(x1), f3(x2), . . .) for 

x = (x1, x2, . . .) ∈ ℓ2, where fn+1(xn) = 1 

n(n+1) (2xn) n+1 , n ≥ 1. Also, let 

F(x) = x + f (x). Then F is biholomorphic on ℓ2 and is not bounded on 

Br for r ∈ (2/3, 1). 

• L. Harris, S. Reich, D. Shoikhet (2000): Let X be a complex Banach 

space and let h : B → X be a holomorphic mapping. If L(h) is finite, 

then hs is bounded on B for each s ∈ (0, 1), where hs(z) = h(sz) and 

L(h) = lim 

s→1 sup Re V (hs). 

• It is natural to consider to what extent such phenomena require 

changes in the development of Loewner theory in complex Banach 

spaces. 



Example 

A mapping f ∈ H(B n ) is normalized if f (0) = 0 and Df (0) = In. 

LSn-the family of normalized locally biholomorphic maps on B n 

• If f ∈ LSn, then the following conditions are equivalent: 

(i) f ∈ S ∗ (B n ) (i.e. f is biholomorphic and e −t f (B n ) ⊆ f (B n ) for t ≥ 0); 

(ii) f (z, t) = e t f (z) is a Loewner chain. 

Example 

• Let A ∈ L(C n , C n ) be such that Re 〈A(z), z〉 > 0, z = 0. If f ∈ LSn, 

then the following conditions are equivalent: 

(i) f is spirallike with respect to A (i.e. f is biholomorphic on B n and 

e −tA f (B n ) ⊆ f (B n ) for t ≥ 0); 

(ii) f (z, t) = e tA f (z) is an A-univalent subordination chain; 



I. Graham, H. Hamada, G.K, M.K, 2008: 

Example 

Let A : [0, ∞) → L(Cn , Cn ) be a measurable mapping such that 

m(A(t)) > 0 for t ≥ 0 and ∞ 

0 m(A(t))dt = ∞. Moreover, assume that 

A(·) is uniformly bounded on [0, ∞). If f ∈ LSn, then the following 

conditions are equivalent: 

(i) f is (generalized) spirallike with respect to A (i.e. f is biholomorphic 

on B n and exp(− t 

0 A(τ)dτ)f (Bn ) ⊆ f (B n ) for t ≥ 0); 

(ii) f (z, t) = e t 

0 A(τ)dτ f (z) is a univalent subordination chain; 

(iii) ℜ〈[Df (z)] −1 A(t)f (z), z〉 > 0, a.e. t ≥ 0, ∀z ∈ B n \ {0}. 



• f (z, t) univalent subordination chain with Df (0, t) = U(t), t ≥ 0, 

where U(t) is the unique locally absolutely continuous solution of 

dU/dt = A(t)U a.e. t ≥ 0, U(0) = In. 

I. Graham, H. Hamada, G. K, M. K (2008-); L. Arosio; M. Vodă (2011). 

L. Arosio, F. Bracci, H.Hamada, G.K (2010): L d -Loewner chains. 

Definition 

Let X be a complex Banach space. For z ∈ X \ {0}, we define 

T (z) = {ℓz ∈ L(X, C) : ℓz(z) = z, ℓz = 1}. 

Then T (z) = ∅. If A ∈ L(X), let 

m(A) = inf{Re [ℓz(A(z))] : z = 1, ℓz ∈ T (z)} 

k(A) = sup{Re [ℓz(A(z))] : z = 1, ℓz ∈ T (z)} 

|V (A)| = sup{|ℓz(A(z))| : z = 1, ℓz ∈ T (z)}. 

V (A)-the numerical radius of the operator A. 



Definition 

Let k+(A) = max{Re λ : λ ∈ σ(A)} be the upper exponential index 

(Lyapunov) index of A, where σ(A) is the spectrum of A ∈ L(X). 

Let k−(A) = min{Re λ : λ ∈ σ(A)} be the lower exponential index 

(Lyapunov) index of A. 

Remark 

Let A ∈ L(X) be such that m(A) > 0. Then 

(i) k−(A) ≤ m(A) ≤ k+(A) ≤ k(A) ≤ |V (A)| ≤ A ≤ e|V (A)|; 

; 

(ii) For each ω > k+(A), there is δ = δ(ω) > 0 s.t. etA ≤ δeωt , t ≥ 0; 

(iii) 

e m(A)t ≤ e tA (u) ≤ e k(A)t , t ∈ [0, ∞), u = 1. 

k+(A) = limt→∞ ln etA t 

(iv) L. Harris, 1971: If Pm : X → X is a homogeneous polynomial 

mapping of degree m, then Pm ≤ km|V (Pm)|, where km = m m/(m−1) 

when m ≥ 2 and k1 = e (k1 = 2 for complex Hilbert spaces). 



The Carathéodory family N 

Definition 

N = {h ∈ H(B) : h(0) = 0, Re [ℓz(h(z))] > 0, z ∈ B \ {0}, ℓz ∈ T (z)}, 

N A = {h ∈ N : Dh(0) = A} and M = N I = {h ∈ N : Dh(0) = I}. 

• If n = 1, then f ∈ M iff 

f (z)/z ∈ P = {h ∈ H(U) : h(0) = 1, Re h(z) > 0, z ∈ U}. 

Remark 

L. Harris, S. Reich, D. Shoikhet, 2000: Bloch radii; distortion form of 

Schwarz’s lemma; T. Poreda, 2001; M. Elin, S. Reich, D. Shoikhet, 

2004: geometric aspects of biholomorphic mappings in complex 

Banach spaces; I. Graham, H. Hamada, G. K (2002): compactness of 

M when X = C n ; radius problems for compact subsets of S(B n ); 

applications to Loewner’s theory in C n (2008-). 



J.A. Pfaltzgraff (1974; X = C n ); K. Gurganus (1975): 

Lemma 

If h ∈ N A, where A ∈ L(X) with m(A) > 0, then 

1 − z 

Re [ℓz(A(z))] 

1 + z ≤ Re [ℓz(h(z))] 

1 + z 

≤ Re [ℓz(A(z))] 

1 − z , 

for all z ∈ B \ {0} and ℓz ∈ T (z). 

• H. Hamada, G. K (2002): Any h ∈ N is bounded on Br , r ∈ (0, 1). 

I. Graham, H. Hamada, G. K, 2002 (X = C n and Df (0) = In); (2008): 

Theorem 

Let X be a complex Banach space and A ∈ L(X) be s.t. m(A) > 0. 

If f (z) = Az + ∞ 

m=2 Pm(z) ∈ N A, then 

(i) |V (Pm)| ≤ 2|V (A)| and Pm ≤ 4m|V (A)| for m ≥ 2. 

(ii) m(A)r(1 − r)/(1 + r) ≤ f (z) ≤ 4|V (A)|r/(1 − r) 2 for z ≤ r < 1. 

(iii) N A is a compact family when X = C n . 



I. Graham, H. Hamada, G.K, M.K. (2008-2011), L. Arosio (2011); M. 

Voda (2011): non-normalized univalent subordination chains with 

normalization given by a time-dependent linear operator. 

Assumption 

Let A : [0, ∞) → L(Cn , Cn ) be a measurable mapping such that 

(a) m(A(t)) > 0 for t ≥ 0, and ∞ 

0 m(A(t))dt = ∞; 

(b) A(·) is uniformly bounded on [0, ∞); 

(c) t 

s A(τ)dτ ◦ s 

r A(τ)dτ = s 

r A(τ)dτ ◦ t 

s A(τ)dτ, t ≥ s ≥ r ≥ 0. 

(d) There exists δ > 0 such that k+(A(t)) ≤ 2m(A(t)) − δ for t ≥ 0; 

Remark 

(i) (c) holds if A(t) ≡ A ∈ L(C n , C n ) or if A(t) is diagonal, t ≥ 0. 

(ii) A-univalent subordination chains (I. Graham, H. Hamada, G. K, 

and M. K, 2008). 

(iii) Solutions to the Loewner differential equation generated by 

A-univalent subordination chains (P. Duren, I. Graham, H. Hamada, 

and G.K, 2010). 



J. Pfaltzgraff (1974); T. Poreda (1990); M. Elin, S. Reich, D. Shoikhet 

(2000); I. Graham, H. Hamada, G.K, M.K. (2008); L. Arosio; M. Voda 

(2011). 

Theorem 

Let A : [0, ∞) → L(C n , C n ) be such that (a)-(c) hold. Also let 

h : B n × [0, ∞) → C n satisfy the following conditions: 

(i) h(·, t) ∈ N and Dh(0, t) = A(t) for t ≥ 0; 

(ii) h(z, ·) is measurable on [0, ∞) for each z ∈ B n . 

Then for each z ∈ B n and s ≥ 0, the initial value problem 

(2.1) 

∂v 

∂t 

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

has a unique solution v = v(z, s, t) such that v(·, s, t) is a univalent 

Schwarz mapping, v(z, s, ·) is Lipschitz continuous on [s, ∞) locally 

uniformly with respect to z ∈ B n and Dv(0, s, t) = exp(− t 

s A(τ)dτ). 

h(z, t)-generating vector field (Herglotz vector field). 



Theorem 

T. Poreda, 1990; Graham, Hamada and K., 2001 (A = In); M. Elin, S. 

Reich, D. Shoikhet, 2000; I. Graham, H. Hamada, G.K, M.K., 2008: 

Let A : [0, ∞) → L(C n , C n ) be such that (a)-(d) hold. Also let 

vs,t(z) = v(z, s, t) be the solution of (2.1). Then the limit 

t 

(2.2) lim e 0 

t→∞ A(τ)dτ vs,t(z) = f (z, s) 

exists locally uniformly on B n for s ≥ 0. Moreover, f (z, t) is an univalent 

subordination chain and {e − t 

0 A(τ)dτ f (·, t)}t≥0 is a normal family on B n 

and 

t≥0 ft(B n ) = C n . In addition, f (z, ·) is locally Lipschitz continuous 

on [0, ∞) locally uniformly with respect to z ∈ B n , and 

(2.3) 

∂f 

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

The univalent subordination chain f (z, t) given by (2.2) is called 

the canonical solution of the Loewner differential equation (2.3). 



J. Pfaltzgraf (1975, A = In); I. Graham, H. Hamada, G.K., M.K., 2008. 

Theorem 

Let A : [0, ∞) → L(C n , C n ) be such that (a)-(d) hold. Let 

f = f (z, t) : B n × [0, ∞) → C n be such that f (·, t) ∈ H(B n ), f (0, t) = 0 

and Df (0, t) = e t 

0 A(τ)dτ for t ≥ 0, and f (z, ·) is locally Lipschitz 

continuous on [0, ∞) locally uniformly with respect to z ∈ B n . Assume 

that there exists a generating vector field h(z, t) = A(t)z + · · · such that 

∂f 


Then f (z, t) is a subordination chain. If {e − t 

0 A(τ)dτ f (·, t)}t≥0 is a 

normal family on B n , then f (z, t) is a univalent subordination chain 

which coincides with the canonical solution of (2.3). 

g(z, t)-standard solution of (2.3) if g(·, t) ∈ H(B n ), g(0, t) = 0 for 

t ≥ 0, g(z, ·) is locally Lipschitz continuous on [0, ∞) locally 

uniformly with respect to z ∈ B n , and g(z, t) is a solution of (2.3). 



• The above result does not hold if k+(A) = 2m(A). 

Example 

Let f : B 2 → C 2 be given by f (z) = (z1, z2 + az 2 1 ) for z = (z1, z2) ∈ B 2 , 

where a ∈ C \ {0}. Also let A ∈ L(C 2 , C 2 ) be given by A(z) = (z1, 2z2) 

and f (z, t) = e At f (z). Then k+(A) = 2m(A) and f is spirallike w.r.t. A. 

Also f (z, t) is a univalent subordination chain such that {e −At f (·, t)}t≥0 

is a normal family on B 2 and 

∂f 

∂t (z, t) = Df (z, t)h(z, t), t ≥ 0, z ∈ B2 , 

where h(z, t) = (z1, 2z2). Let v(z, s, t) = (e s−t z1, e 2(s−t) z2). Then 

v(z, s, t) is the unique solution of the initial value problem 

∂v 

∂t 

= −h(v, t), t ≥ s, v(z, s, s) = z. 

But e At v(z, s, t) = (e s z1, e 2s z2) f (z, s) = (e s z1, e 2s (z2 + az 2 1 )). 



Theorem 

Let A : [0, ∞) → L(Cn , Cn ) be such that (a)-(d) hold. Let f (z, t) be an 

univalent subordination chain such that Df (0, t) = exp( t 

0 A(τ)dτ). 

Then f (z, ·) is locally Lipschitz continuous on [0, ∞) locally uniformly 

with respect to z ∈ Bn . Also there is a generating vector field 

h = h(z, t) such that Dh(0, t) = A(t) and 

Theorem 

∂f 


I. Graham, H. Hamada, G. K., M. K., 2008: Assume the conditions 

(a)-(d) hold. Let f (z, t) = e t 

0 A(τ)dτ z + · · · be an univalent 

subordination chain. If {e− t 

0 A(τ)dτ f (·, t)}t≥0 is a normal family on Bn , 

then f (z, s) = limt→∞ e t 

0 A(τ)dτ v(z, s, t) locally uniformly on Bn for 

s ≥ 0, where v = v(z, s, t) is the transition mapping of f (z, t), and 

f (z, t) coincides with the canonical solution of (2.3). 



• T. Poreda, 1990 (starlikeness); M. Elin, S. Reich, D. Shoikhet, 2000 

(spirallikeness w.r.t. a linear operator). 

Corollary 

Assume k+(A) < 2m(A). If f (z, t) is an A-univalent subordination chain 

such that {e −tA f (·, t)}t≥0 is a normal family, then 

z 

(1 + z) 2 ≤ e−tA f (z, t) ≤ 

z 

(1 − z) 2 , z ∈ Bn , t ≥ 0. 



Corollary 

I. Graham, G.K, 2003: 

(i) If f (z, t) = e t z + · · · is a Loewner chain such that {e −t f (·, t)}t≥0 is a 

normal family on B n , then 

etz (1 + z) 2 ≤ f (z, t) ≤ etz (1 − z) 2 , z ∈ Bn , t ≥ 0. 

(ii) If, in addition, the generating vector field h(z, t) of (2.3) satisfies the 

condition 

1 

 

〈h(z, t), z〉 − 1 

z2 < 1, z ∈ Bn \ {0}, 

then 

etz 1 + z ≤ f (z, t) ≤ etz 1 − z , z ∈ Bn , t ≥ 0. 



Remark 

(a) (i) yields the growth result for S ∗ (B n ): R. Barnard, C. FitzGerald 

and S. Gong (1991); J.A. Pfaltzgraff (1991); M. Chuaqui (1995), T. Liu 

(1999). 

(b) (ii) yields the growth result for K (B n ) may be obtained from (ii): C. 

FitzGerald, C. Thomas, S. Gong (1995); P. Liczberski (1998), T. Liu 

(1999); H. Hamada, G.K. (2000). 

(c) There exist Loewner chains f (z, t) that do not satisfy the growth 

result (i) and {e −t f (·, t)}t≥0 is not a normal family on B n , for n ≥ 2. 

Example 

Let g(z, t) = 

 

et z1 

(1−z1) 2 e 

, 

t z2 

(1−z2) 2 

 

for z = (z1, z2) ∈ B2 , t ≥ 0. Then g(z, t) 

is a Loewner chain and if Φ(z) = (z1, z2 + z2 1 ) then Φ ∈ Aut(C2 ) and 

 

f (z, t) = Φ(g(z, t)) given by f (z, t) = 

is a 

 

et z1 

(1−z1) 2 e 

, 

t z2 

(1−z2) 2 + e2t z2 1 

(1−z1) 4 

Loewner chain such that f (r, 0) > r/(1 − r) 2 for r ∈ (0, 1) and 

{e−tf (·, t)}t≥0 is not a normal family on B2 . 



Solutions for the generalized Loewner differential equation (2.3): 

∂f 

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

where h(z, t) is a generating vector field such that Dh(0, t) = A(t) 

for t ≥ 0. 

n = 1: Ch. Pommerenke, 1965; J. Becker (1973-1980); 

n ≥ 2: I. Graham, G.K., J.A. Pfaltzgraff, 2005 (A = In); P. Duren, I. 

Graham, H. Hamada and G.K, 2010 (k+(A) < 2m(A)); L. Arosio 

(2010); M. Voda (2010); H. Hamada (2010): m(A) > 0; I. Graham, 

H. Hamada, G.K. (2010): Herglotz-vector fields with respect to a 

time-dependent linear operator. 

J. Becker: useful to construct the solutions of the Loewner 

differential equation which, for fixed t, are holomorphic on a 

punctured disc (rather than a disc) with center at zero. 

In several complex variables, point singularities of holomorphic 

mappings are removable. 



(i) If f (z, t) is a Loewner chain and Φ : C n → C n is a normalized 

biholomorphic mapping (i.e. Φ(0) = 0 and DΦ(0) = In), not the 

identity, then Φ(f (z, t)) is also a Loewner chain and satisfies the 

same Loewner differential equation as f (z, t). 

In C n , n ≥ 2, normalized univalent solutions of the generalized 

Loewner differential equation need not be unique. 

Theorem 

I. Graham, G.K., J.A. Pfaltzgraff, 2005; P. Duren, I. Graham, H. 

Hamada and G.K, 2010 (A(t) ≡ A); Graham, Hamada, K (2010): 

Let A : [0, ∞) → L(C n , C n ) be such that (a)-(d) hold. Let 

f (z, t) = e t 

0 A(τ)dτ z + · · · be the canonical solution of (2.3) and let 

g(z, t) be a standard solution of (2.3). Then 

t≥0 f (Bn , t) = C n and 

there exists a mapping Φ ∈ H(C n ) such that 

g(z, t) = Φ(f (z, t)), z ∈ B n , t ≥ 0. 

In addition, g(·, t) is univalent on B n iff Φ is biholomorphic in C n . 



Theorem 

P. Duren, I. Graham, H. Hamada and G.K, 2010 (A(t) ≡ A); Graham, 

Hamada, K (2010): Let A : [0, ∞) → L(C n , C n ) be such that (a)-(d) 

hold. Let f (z, t) = e t 

0 A(τ)dτ z + · · · be the canonical solution of (2.3) 

and let g(z, t) be a standard solution of (2.3). Assume that 

{e − t 

0 A(τ)dτ g(·, t)}t≥0 is a normal family on B n . Then there exists 

Ψ ∈ L(C n , C n ) such that g(z, t) = Ψ(f (z, t)) for z ∈ B n and t ≥ 0. 

Corollary 

Let A : [0, ∞) → L(C n , C n ) be such that (a)-(d) hold. Let g(z, t) be a 

standard solution of the Loewner differential equation (2.3) such that 

{e − t 

0 A(τ)dτ g(·, t)}t≥0 is a normal family on B n and Dg(0, 0) = In. Then 

g(z, s) ≡ f (z, s), where f (z, s) is the canonical solution of (2.3). 



Corollary 

Let A : [0, ∞) → L(C n , C n ) be such that (a)-(d) hold. The canonical 

solution f (z, t) = e t 

0 A(τ)dτ z + · · · of the Loewner differential equation 

(2.3) is the unique normalized univalent subordination chain solution 

such that {e− t 

0 A(τ)dτ f (·, t)}t≥0 is a normal family on Bn 

. In addition, 

t≥0 f (Bn , t) = Cn . 

Corollary 

I. Graham, G.K., J.A. Pfaltzgraff, 2005: Any Loewner chain 

g(z, t) = e t z + · · · has the form g(z, t) = Φ(f (z, t)), z ∈ B n , t ≥ 0, 

where f (z, t) is a Loewner chain of the distinguished class and 

Φ : C n → C n is a normalized biholomorphic mapping (i.e. an 

automorphism of C n or a normalized Fatou-Bieberbach map). 



Example 

Let n = 2 and let f : B2 → C2 be given by f (z) = (z1, z2 + az2 1 ) for 

z = (z1, z2) ∈ B2 , where a ∈ C \ {0}. Also let A ∈ L(C2 , C2 ) be given 

by A(z) = (z1, 2z2) and f (z, t) = etAf (z). Then m(A) = 1 and 

k+(A) = 2. Also f is spirallike with respect to A and f (z, t) = etAf (z) is 

a univalent subordination chain such that 

∂f 

∂t (z, t) = Df (z, t)h(z), t ≥ 0, z ∈ B2 , 

where h(z) = A(z). Next, let g(z, t) = e tA (z) for z ∈ B 2 and t ≥ 0. 

Then g(z, t) is a univalent subordination chain and Dg(0, t) = e tA for 

t ≥ 0, {e −tA g(·, t)}t≥0 is a normal family on B 2 and g(z, t) satisfies the 

same Loewner differential equation as f (z, t). But, f (z, t) ≡ g(z, t). 



Corollary 

T. Poreda, 1991; M. Elin, S. Reich, D. Shoikhet, 2000: Let h ∈ M. 

Then the differential equation 

Df (z)h(z) = f (z), z ∈ B n , 

has a unique solution f ∈ H(B n ) such that f (0) = 0 and Df (0) = In. 

This solution is starlike. 

• T. Poreda, 1991 (A < 2m(A)); M. Elin, S. Reich, D. Shoikhet, 2004 

(k+(A) < 2k−(A)); Duren, Graham, Hamada, K., 2008. 

Corollary 

Let h ∈ N A be such that k+(A) < 2m(A). Then the differential equation 

Df (z)h(z) = f (z), z ∈ B n , 

has a unique solution f ∈ H(B n ) such that f (0) = 0 and Df (0) = In. 

This solution is spirallike w.r.t. A. 


3. Parametric representation on the unit ball 


Lemma 

Poreda-Szadkowska, 1990: Let f ∈ S∗ (Bn ) and h(z) = [Df (z)] −1f (z). 

Then f (z) = lim e 

t→∞ t v(z, t) locally uniformly on Bn , where v(z, t) is the 

unique solution of the initial value problem 

∂v 

∂t 

= −h(v), ∀ t ≥ 0, v(z, 0) = z, 

i.e. f has parametric representation. In addition, 

z 

≤ f (z) ≤ 

(1 + z) 2 

z 

(1 − z) 2 , z ∈ Bn . 

• J.A. Pfaltzgraff (1991); R. Barnard, C. FitzGerald and S. Gong 

(1991); M. Elin, S. Reich, D. Shoikhet, 2000. 

• If f ∈ S then f has parametric representation, i.e. there exists a 

Loewner chain f (z, t) such that f = f (·, 0). 



T. Poreda, 1991 (A < 2m(A)); M. Elin, S. Reich, D. Shoikhet, 

2004 (k+(A) < 2k−(A)); Duren, Graham, Hamada, K., 2008. 

Theorem 

Let A ∈ L(C n , C n ) be an operator such that k+(A) < 2m(A). Also let 

f : B n → C n be a spirallike mapping with respect to A and let 

h(z) = [Df (z)] −1 Af (z) for z ∈ B n . Then f (z, t) = e tA f (z) is a univalent 

subordination chain and 

lim 

t→∞ eAtv(z, s, t) = f (z, s) 

locally uniformly on B n , where v(z, s, t) = f −1 (e (s−t)A f (z)). Also 

∂f 

∂t (z, t) = Df (z, t)h(z), t ≥ 0, z ∈ Bn . 

Spirallike mappings w.r.t. A with k+(A) < 2m(A) have a 

”generalized” parametric representation. 



• In dimension n ≥ 2, there exist mappings f ∈ S(B n ) which cannot be 

imbedded in Loewner chains f (z, t) such that {e −t f (·, t)}t≥0 is a 

normal family on B n (for example, spirallike mappings). 

Definition 

Let f ∈ H(B n ) be such that f (0) = 0, Df (0) = In. Also let A ∈ L(C n , C n ) 

such that k+(A) < 2m(A); f has A-parametric representation (and 

denote by f ∈ S 0 A (Bn )) if there exists a generating vector field h(z, t) 

such that Dh(0, t) = A for t ≥ 0, and 

f (z) = lim 

t→∞ e tA v(z, t) 

locally uniformly on B n , where v = v(z, t) is the unique Lipschitz 

continuous solution on [0, ∞) of the initial value problem 

∂v 

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

• A = In-parametric representation; S 0 In (Bn ) := S 0 (B n ). 



I. Graham, H. Hamada, G.K., 2002 (A = In); I. Graham, H. Hamada, 

G.K., M.K., 2008 (k+(A) < 2m(A)); cf. T. Poreda (A = In). 

Theorem 

Let f ∈ S(B n ) and k+(A) < 2m(A). Then f ∈ S 0 A (Bn ) if and only if there 

exists an A-univalent subordination chain f (z, t) such that 

{e −tA f (·, t)}t≥0 is a normal family on B n and f = f (·, 0). 

Remark 

• Ch. Pommerenke (1965): S 0 (B 1 ) = S. 

• If n ≥ 2 then S 0 (B n ) S(B n ). 

• The most important compact subfamilies of S(B n ): starlikeness, 

convexity, close-to-starlikeness, are also subfamilies of S 0 (B n ). 

Theorem 

S 0 A (Bn ) is a compact family for k+(A) < 2m(A). 


Theorem 


I. Graham, H. Hamada, G.K. 2003: Let f ∈ S 0 (B n ). Then 

z 

≤ f (z) ≤ 

(1 + z) 2 

These estimates are sharp. Also 

Moreover, 1 

2 D2 f (0) ≤ 8. 

Corollary 

S 0 (B n ) is compact. 

Remark 

z 

(1 − z) 2 , z ∈ Bn . 

 

 

1 

2 〈D2 

 

f (0)(v, v), v〉 ≤ 2, v = 1. 

There exist mappings f ∈ S 0 (B n ), n ≥ 2, such that (1/2)D 2 f (0) > 2. 



• Asymptotic spirallikeness, a natural generalization of spirallikeness. 

Definition 

Let Ω ⊆ C n be a domain which contains the origin and let 

A ∈ L(C n , C n ) be such that m(A) > 0. We say that Ω is 

A-asymptotically spirallike if there exists a mapping 

Q = Q(z, t) : Ω × [0, ∞) → C n which satisfies the following conditions: 

(i) Q(·, t) is a holomorphic mapping on Ω, Q(0, t) = 0, DQ(0, t) = A, 

t ≥ 0, and the family {Q(·, t)}t≥0 is locally uniformly bounded on Ω. 

(ii) Q(z, ·) is measurable on [0, ∞) for all z ∈ Ω; 

(iii) The initial value problem 

∂w 

∂t 

= −Q(w, t) a.e. t ≥ s, w(z, s, s) = z, 

has a unique solution w = w(z, s, t) for each z ∈ Ω and s ≥ 0, such 

that w(·, s, t) is a holomorphic mapping of Ω into Ω for t ≥ s, w(z, s, ·) 

is locally absolutely continuous on [s, ∞) locally uniformly with respect 

to z ∈ Ω for s ≥ 0, and lim 

t→∞ e At w(z, 0, t) = z locally uniformly on Ω. 



• A domain Ω ⊆ C n which contains the origin is called asymptotically 

spirallike if there exists an operator A ∈ L(C n , C n ) with m(A) > 0 such 

that Ω is A-asymptotically spirallike. 

Definition 

Let f ∈ S(B n ) and let A ∈ L(C n , C n ) be such that m(A) > 0. We say 

that f is A-asymptotically spirallike if f (B n ) is an A-asymptotically 

spirallike domain. 

Remark 

If Ω is a spirallike domain with respect to A with m(A) > 0, then Ω is 

A-asymptotically spirallike. 

• Ω ⊆ C n is In-asymptotically spirallike if and only if Ω is asymptotically 

starlike. 

Problem Does there exist a connection between A-asymptotic 

spirallikeness and A-parametric representation? 


Theorem 


Let A ∈ L(C n , C n ) be an operator with m(A) > 0 and k+(A) < 2m(A). 

Also let f ∈ S(B n ). Then f has A-parametric representation iff there is 

a univalent subordination chain f (z, t) such that Df (0, t) = e At , t ≥ 0, 

{e −At f (·, t)}t≥0 is a normal family on B n and f = f (·, 0). 

Corollary 

Let f : Bn → Cn be a spirallike mapping with respect to A ∈ L(Cn , Cn ) 

such that k+(A) < 2m(A). Then f has A-parametric representation. In 

addition, there is a univalent subordination chain f (z, t) such that 

f (0, t) = 0, Df (0, t) = etA , {e−tAf (·, t)}t≥0 is a normal family on Bn and 

f = f (·, 0). Let v(z, s, t) be the transition mapping of f (z, t). Then 

f (z) = lim e 

t→∞ tA v(z, t) locally uniformly on Bn , where v(z, t) = v(z, 0, t). 

• Particular cases of this result were obtained by T. Poreda and A. 

Szadkowska, 1990; M. Elin, S. Reich, D. Shoikhet, 2004. 


Theorem 


Let A ∈ L(C n , C n ) be such that m(A) > 0 and k+(A) < 2m(A). Also let 

f ∈ S(B n ). Then f is A-asymptotically spirallike if and only if f has 

A-parametric representation. 

Remark 

The class of A-asymptotically spirallike mappings with k+(A) < 2m(A) 

is compact, however the full class of asymptotically spirallike mappings 

is not compact in dimension n ≥ 2. 

Indeed, let n = 2 and f : B2 → C2 be given by f (z) = (z1, z2 + az2 1 ) for 

z = (z1, z2) ∈ B2 . Also let A ∈ L(C2 , C2 ) be given by A(z) = (z1, 2z2). 

Then f is spirallike with respect to A for all a ∈ C. Thus f is also 

asymptotically spirallike. But, if z0 = (1/2, 0) then f (z0) = (1/2, a/4) 

and f (z0) → ∞ as a → ∞. 

• The class of asymptotically starlike mappings is compact. 


Remark 


Let f : B n → C n be an A-asymptotically spirallike mapping, where A is 

a diagonal matrix whose eigenvalues λ1, . . . , λn satisfy the conditions 

ℜλ j = a > 0 for j = 1, . . . , n. Then 

Theorem 

z 

≤ f (z) ≤ 

(1 + z) 2 

z 

(1 − z) 2 , z ∈ Bn . 

Let f ∈ S(B n ) and let A ∈ L(C n , C n ) be such that A + A ∗ = 2aIn, where 

a > 0 and A ∗ is the adjoint of A. Then f is asymptotically spirallike if 

and only if f is asymptotically starlike. 

Theorem 

Let f : B n → C n be a spirallike mapping with respect to an operator 

A ∈ L(C n , C n ) such that A + A ∗ = 2aIn, where a > 0. Then f ∈ S 0 (B n ). 


Examples 

Example 


Let f1, . . . , fn ∈ S be such that f1 is not spirallike and let 

f (z) = (f1(z1), . . . , fn(zn)) for z = (z1, . . . , zn) ∈ B n . Then f is 

asymptotically starlike, and hence asymptotically spirallike, but is not 

spirallike. 

Example 

Let f1 ∈ S be a function which has a1-parametric representation and 

assume f : B n−1 → C n−1 has A-parametric representation. Then 

F : B n → C n given by F(z) = (f1(z1), f (z ′ )), z = (z1, z ′ ) ∈ B n , has 

Ã-parametric representation where Ã = 

a1 0 

0 A 

(3.1) max{ℜa1, k+(A)} < 2 min{ℜa1, m(A)}, 

 

. In addition, if 

then F is Ã-asymptotically spirallike. If f1 is not spirallike then F is also 

not spirallike. 


4. Subordination chains in complex Banach spaces 


I. Graham, H. Hamada, G.K, M.K, 2012; 

• Let X be a reflexive complex Banach space. 

X = C n : J. Pfaltzgraff, 1974 (A = In); T. Poreda, 1990; M. Elin, S. 

Reich, D. Shoikhet, 2004; Graham, H. Hamada, G.K, M.K. (2008): 

Theorem 

Let h(z, t) : B × [0, ∞) → X be a generating vector field: 

(i) h(·, t) ∈ N , Dh(0, t) = A ∈ L(X), t ≥ 0, where m(A) > 0. 

(ii) h(z, ·) is strongly measurable on [0, ∞) for z ∈ B. 

Then for each s ≥ 0 and z ∈ B, the initial value problem 

(4.1) 

∂v 

∂t 

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

has a unique solution v = v(z, s, t) such that v(·, s, t) is a univalent 

Schwarz mapping, v(z, s, ·) is Lipschitz continuous on [s, ∞) uniformly 

with respect to z ∈ Br , r ∈ (0, 1), Dv(0, s, t) = exp(−A(t − s)). 



• J. Pfaltzgraff (1975; X = C n ); T. Poreda (1990): h(z, t) generating 

vector field in (4.1) such that: 

h(z, t) ≤ M(r), z ≤ r < 1, t ≥ 0. 

• T. Poreda: h = h(z, t) is continuous on B × [0, ∞) 

Theorem 

Under the assumptions of the previous result, the following estimates 

hold: 

and 

v(z, s, t) 

(1 − v(z, s, t)) 

e −k(A)(t−s) 

z 

2 ≤ e−m(A)(t−s) 

z 

≤ 

(1 + z) 2 

, z ∈ B, t ≥ s ≥ 0, 

(1 − z) 2 

v(z, s, t) 

, z ∈ B, t ≥ s ≥ 0. 

(1 + v(z, s, t)) 2 


Theorem 


Let h = h(z, t) : B × [0, ∞) → X be a generating vector field such that 

Dh(0, t) = A ∈ L(X) for t ≥ 0, and k+(A) < 2m(A). Also, let 

v = v(z, s, t) be the unique Lipschitz continuous solution on [s, ∞) of 

the initial value problem (4.1). Then the limit 

(4.2) lim 

t→∞ e tA v(z, s, t) = f (z, s) 

exists uniformly on each closed ball Br for r ∈ (0, 1) and s ≥ 0. 

Moreover, f (z, t) is an A-normalized univalent subordination chain. 

Theorem 

Let A ∈ L(X) be s.t. k+(A) < 2m(A). Also, let f (z, t) be an 

A-normalized univalent subordination chain. Assume that for each 

r ∈ (0, 1), there is M = M(r, A) > 0 such that e −tA f (z, t) ≤ M, 

z ≤ r, t ≥ 0. If v(z, s, t) is the transition mapping of f (z, t), then 

f (z, s) = limt→∞ e tA v(z, s, t) uniformly on Br , r ∈ (0, 1). 


Corollary 


Let f (z, t) = e t z + · · · be a normalized univalent subordination chain. 

Assume that for each r ∈ (0, 1), there is M = M(r) > 0 such that 

e −t f (z, t) ≤ M, z ≤ r, t ≥ 0. Then 

Corollary 

r/(1 + r) 2 ≤ e −t f (z, t) ≤ r/(1 − r) 2 , z = r < 1, t ≥ 0. 

Assume X is separable. Let h = h(z, t) : B × [0, ∞) → X be a 

generating vector field such that Dh(0, t) = A ∈ L(X) for t ≥ 0, and the 

condition k+(A) < 2m(A) holds. Also, let f (z, t) = e tA z + · · · be the 

A-normalized univalent subordination chain given by (4.2). Assume 

that ∂f 

∂t (z, t) exists for t ∈ [0, ∞) \ E and z ∈ Bδ, for some δ ∈ (0, 1), 

where E ⊂ [0, ∞) (independent of z) has measure zero. Then 

∂f 

(z, t) = Df (z, t)h(z, t), t ∈ [0, ∞) \ E, ∀ z ∈ B. 

∂t 


Theorem 


(i) Assume that f (z, t) = e tA z + · · · is a standard solution of the 

generalized Loewner differential equation 

(4.3) 

∂f 

(z, t) = Df (z, t)h(z, t), t ∈ [0, ∞) \ E, ∀z ∈ B, 

∂t 

where E is a subset of [0, ∞) of measure zero. Also, assume that for 

each r ∈ (0, 1), there is M = M(r, A) > 0 such that 

e −tA f (z, t) ≤ M(r, A), z ≤ r, t ≥ 0. 

Then f (z, t) is an A-univalent subordination chain and (4.2) holds. 

(ii) Assume that g(z, t) is another standard solution of (4.3). If for each 

r ∈ (0, 1), there is K = K (r, A) > 0 such that e −tA g(z, t) ≤ K , 

z ≤ r and t ≥ 0, then g(z, t) is a subordination chain and there 

exists Ψ ∈ L(X) such that g(z, t) ≡ Ψ(f (z, t)). If, in addition, 

Dg(0, 0) = I, then g(z, t) ≡ f (z, t). 


Definition 


Let A ∈ L(X) be such that k+(A) < 2m(A) and let f ∈ H(B) be a 

normalized mapping. We say that f has A-parametric representation 

(f ∈ S0 A (B)) if there exists a generating vector field 

h = h(z, t) : B × [0, ∞) → Cn such that Dh(0, t) = A for t ≥ 0, and 

f (z) = lim e 

t→∞ tA v(z, t) uniformly on each closed ball Br for r ∈ (0, 1), 

where v = v(z, t) is the unique Lipschitz continuous solution on [0, ∞) 

of the initial value problem 

∂v 

∂t 

= −h(v, t) a.e. t ≥ 0, v(z, 0) = z, 

for each z ∈ B. 

If A = I and f has I-parametric representation, then we say that f has 

parametric representation in the usual sense. 

• If f ∈ S0 A (B), then f is univalent. 


Remark 


(i) Let S be the family of normalized univalent functions on the unit 

disc U in C. It is well known that f ∈ S if and only if there exists a 

Loewner chain f (z, t) such that f = f (·, 0). 

(ii) In C n , n ≥ 2, such result does not hold for the full family S(B n ) of 

normalized biholomorphic mappings on the unit ball B n in C n . 

(iii) However, if A ∈ L(C n , C n ) is such that k+(A) < 2m(A), then 

f ∈ S(B n ) has A-parametric representation if and only if there exists an 

A-normalized univalent subordination chain f (z, t) such that 

{e −tA f (·, t)}t≥0 is a normal family on B n and f = f (·, 0). 


Corollary 


Let A ∈ L(X) satisfy the condition k+(A) < 2m(A). Also, let f : B → X 

be a normalized holomorphic mapping. Then the following assertions 

hold: 

(i) If f has A-parametric representation, then there exists an 

A-normalized univalent subordination chain f (z, t) such that f = f (·, 0), 

and for each r ∈ (0, 1), there is M = M(r) > 0 such that 

(4.4) e −tA f (z, t) ≤ M(r), z ≤ r, t ≥ 0. 

(ii) Conversely, assume that f (z, t) = e tA z + · · · is a standard solution 

of the generalized Loewner differential equation (4.3), which satisfies 

the condition (4.4), and such that f = f (·, 0). Then f has A-parametric 

representation. 



• Applications to spirallikeness and starlikeness 

Definition 

Let A ∈ L(X) be such that m(A) > 0. Also let Ω be a domain in X 

which contains the origin. We say that Ω is spirallike with respect to A if 

e −tA (w) ∈ Ω, for all w ∈ Ω and t ≥ 0. 

A mapping f ∈ S(B) is said to be spirallike with respect to A if f (B) is a 

spirallike domain with respect to A. 

Note that if A = I in the above definition, we obtain the usual notion of 

starlikeness. 

Remark 

A normalized locally biholomorphic mapping f on B is spirallike with 

respect to A ∈ L(X), m(A) > 0, if and only if (T. Suffridge; K. 

Gurganus, 1975) 

ℜ[ℓz([Df (z)] −1 Af (z))] > 0, z ∈ B \ {0}, ℓz ∈ T (z). 


Corollary 


Let A ∈ L(X) be such that m(A) > 0. Also, let f : B → X be a 

normalized locally biholomorphic mapping. Then f is spirallike with 

respect to A if and only if f (z, t) = e tA f (z) is an A-normalized univalent 

subordination chain. 

• T. Poreda (1999); M. Elin, S. Reich, D. Shoikhet (2004); Graham, 

Hamada, K.K. (2008); P. Duren, I. Graham, H. Hamada, G.K. (2010): 

Corollary 

Let A ∈ L(X) be such that k+(A) < 2m(A) and let f be a spirallike 

mapping with respect to A. Then f has A-parametric representation. 

Corollary 

Let A ∈ L(X) be s.t. the condition k+(A) < 2m(A) holds, and let h ∈ N 

be s.t. Dh(0) = A. Then the differential equation Df (z)h(z) = Af (z), 

z ∈ B, has a unique solution f ∈ H(B) with f (0) = 0 and Df (0) = I. 


5. Open problems 

Conjecture 


Let A ∈ L(X) be such that k+(A) < 2m(A). Also, let f (z, t) be an 

A-normalized univalent subordination chain f (z, t) such that the 

condition (4.4) holds. Then f = f (·, 0) has A-parametric representation. 

• In the case X = C n , every Loewner chain satisfies the generalized 

Loewner differential equation (Graham, Hamada, K, 2002). 

Problem 

Let A ∈ L(X) be such that k+(A) < 2m(A) and let f (z, t) be an 

A-normalized univalent subordination chain. Does there exist a 

generating vector field h(z, t) = Az + · · · such that 

∂f 

(z, t) = Df (z, t)h(z, t), a.e. t ≥ 0, ∀z ∈ B? 

∂t 


Conjecture 


Let A ∈ L(X) be such that k+(A) < 2m(A). If f (z, t) is the A-normalized 

univalent subordination chain given by (4.2), then f (·, t) is 

biholomorphic on B for t ≥ 0. 

• P. Duren, I. Graham, H. Hamada, G.K (2010): positive answer to the 

following problem when X = C n . 

Problem 

Let A ∈ L(X) be such that k+(A) < 2m(A) and let ft(z) = f (z, t) be the 

A-normalized univalent subordination chain given by (4.2). Also, let 

Ω = 

t≥0 ft(B). Is it true that Ω = X? 


Definition 


Let Ω ⊆ X be a domain which contains the origin and let A ∈ L(X) be 

such that m(A) > 0. We say that Ω is A-asymptotically spirallike if 

there exists a mapping Q = Q(z, t) : Ω × [0, ∞) → X such that: 

(i) Q(·, t) ∈ H(Ω), Q(0, t) = 0, DQ(0, t) = A, t ≥ 0, and for each closed 

ball B(z0, r) contained in Ω, there exists some M = M(r, z0) > 0 such 

that Q(z, t) ≤ M(r, z0) for z ∈ B(z0, r). 

(ii) Q(z, ·) is strongly measurable on [0, ∞), for all z ∈ Ω; 

(iii) The initial value problem 

∂w 

∂t 

= −Q(w, t) a.e. t ≥ s, w(z, s, s) = z, 

has a unique solution w = w(z, s, t) for each z ∈ Ω and s ≥ 0, such 

that w(·, s, t) is a holomorphic mapping of Ω into Ω for t ≥ s, w(z, s, ·) 

is strongly locally absolutely continuous on [s, ∞), for all z ∈ Ω and 

s ≥ 0, and lim e 

t→∞ tA w(z, 0, t) = z uniformly on each closed ball B(z0, r) 

contained in Ω. 


Definition 


A mapping f ∈ S(B) is called A-asymptotically spirallike 

(asymptotically starlike) if f (B) is an A-asymptotically spirallike 

(I-asymptotically spirallike) domain. 

Remark 

Note that any spirallike mapping with respect to an operator A ∈ L(X) 

is A-asymptotically spirallike. Also, any normalized starlike mapping is 

asymptotically starlike. 

It would be interesting to give an answer to the following question. In 

the case X = C n , positive answer: Graham, Hamada, K.K. (2008). 

Problem 

Does there exist a connection between A-normalized univalent 

subordination chains and A-asymptotically spirallike mappings in the 

case of reflexive complex Banach spaces? 



References 

1. P. Duren, I. Graham, H. Hamada, G. Kohr: Solutions for the 

generalized Loewner differential equation in several complex variables, 

Math. Ann., 347, 411-435 (2010). 

2. Graham, I., Hamada, H., Kohr, G.: Parametric representation of 

univalent mappings in several complex variables, Canadian J. Math., 

54, 324-351 (2002). 

3. Graham, I., Hamada, H., Kohr, G., Kohr, M.: Parametric 

representation and asymptotic starlikeness in C n , Proc. Amer. Math. 

Soc., 136, 3963-3973 (2008). 

4. Graham, I., Hamada, H., Kohr, G., Kohr, M.: Asymptotically 

spirallike mappings in several complex variables, J. Anal. Math., 105, 

267-302 (2008). 

5. Graham, I., Hamada, H., Kohr, G., Kohr, M.: Spirallike mappings 

and univalent subordination chains in C n , Ann. Scuola Norm. Sup. 

Pisa-Cl. Sci., 7 (2008), 717-740. 



6. Graham, I., Kohr, G.: Geometric Function Theory in One and Higher 

Dimensions, Marcel Dekker, New York (2003). 

7. Graham, I., Kohr, G., Kohr, M.: Loewner chains and parametric 

representation in several complex variables, J. Math. Anal. Appl., 281, 

425-438 (2003). 

8. Graham, I., Kohr, G., Pfaltzgraff, J.A.: The general solution of the 

Loewner differential equation on the unit ball in C n , Contemporary 

Mathematics, 382, 191-203 (2005). 

9. Hamada, H., Kohr, G.: Subordination chains and the growth 

theorem of spirallike mappings, Mathematica (Cluj), 42 (65), 153-161 

(2000). 

10. Hamada, H., Kohr, G.: An estimate of the growth of spirallike 

mappings relative to a diagonal matrix, Ann. Univ. Mariae Curie 

Sklodowska, Sect. A, 55, 53-59 (2001). 

11. L. Arosio, F. Bracci, H. Hamada, G. Kohr, An abstract approach to 

Loewner’s chains, submitted, 2010. 

12. I. Graham, H. Hamada, G. Kohr, Extreme points, support points 

and the Loewner variation in several complex variables, submitted.

Gabriela Kohr

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