Gabriela Kohr

Definition
4. Subordination chains in complex Banach spaces
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.
Gabriela Kohr (UBB Cluj) Geometric and analytic aspects of Loewner chains 52 / 62