Gabriela Kohr

2. Univalent subordination chains in several complex variables
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).
Gabriela Kohr (UBB Cluj) Geometric and analytic aspects of Loewner chains 21 / 62