Gabriela Kohr

2. Univalent subordination chains in several complex variables
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;
Gabriela Kohr (UBB Cluj) Geometric and analytic aspects of Loewner chains 15 / 62