Gabriela Kohr

2. Univalent subordination chains in several complex variables
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 .
Gabriela Kohr (UBB Cluj) Geometric and analytic aspects of Loewner chains 29 / 62