Gabriela Kohr

Theorem
3. Parametric representation on the unit ball
Let A ∈ L(C n , C n ) be such that m(A) > 0 and k+(A) < 2m(A). Also let
f ∈ S(B n ). Then f is A-asymptotically spirallike if and only if f has
A-parametric representation.
Remark
The class of A-asymptotically spirallike mappings with k+(A) < 2m(A)
is compact, however the full class of asymptotically spirallike mappings
is not compact in dimension n ≥ 2.
Indeed, let n = 2 and f : B2 → C2 be given by f (z) = (z1, z2 + az2 1 ) for
z = (z1, z2) ∈ B2 . Also let A ∈ L(C2 , C2 ) be given by A(z) = (z1, 2z2).
Then f is spirallike with respect to A for all a ∈ C. Thus f is also
asymptotically spirallike. But, if z0 = (1/2, 0) then f (z0) = (1/2, a/4)
and f (z0) → ∞ as a → ∞.
• The class of asymptotically starlike mappings is compact.
Gabriela Kohr (UBB Cluj) Geometric and analytic aspects of Loewner chains 44 / 62