Gabriela Kohr

2. Univalent subordination chains in several complex variables
The Carathéodory family N
Definition
N = {h ∈ H(B) : h(0) = 0, Re [ℓz(h(z))] > 0, z ∈ B \ {0}, ℓz ∈ T (z)},
N A = {h ∈ N : Dh(0) = A} and M = N I = {h ∈ N : Dh(0) = I}.
• If n = 1, then f ∈ M iff
f (z)/z ∈ P = {h ∈ H(U) : h(0) = 1, Re h(z) > 0, z ∈ U}.
Remark
L. Harris, S. Reich, D. Shoikhet, 2000: Bloch radii; distortion form of
Schwarz’s lemma; T. Poreda, 2001; M. Elin, S. Reich, D. Shoikhet,
2004: geometric aspects of biholomorphic mappings in complex
Banach spaces; I. Graham, H. Hamada, G. K (2002): compactness of
M when X = C n ; radius problems for compact subsets of S(B n );
applications to Loewner’s theory in C n (2008-).
Gabriela Kohr (UBB Cluj) Geometric and analytic aspects of Loewner chains 19 / 62