Gabriela Kohr

2. Univalent subordination chains in several complex variables
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 .
Gabriela Kohr (UBB Cluj) Geometric and analytic aspects of Loewner chains 20 / 62