Gabriela Kohr

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2. Univalent subordination chains in several complex variables Theorem Let A : [0, ∞) → L(Cn , Cn ) be such that (a)-(d) hold. Let f (z, t) be an univalent subordination chain such that Df (0, t) = exp( t 0 A(τ)dτ). Then f (z, ·) is locally Lipschitz continuous on [0, ∞) locally uniformly with respect to z ∈ Bn . Also there is a generating vector field h = h(z, t) such that Dh(0, t) = A(t) and Theorem ∂f ∂t (z, t) = Df (z, t)h(z, t), a.e. t ≥ 0, ∀ z ∈ Bn . I. Graham, H. Hamada, G. K., M. K., 2008: Assume the conditions (a)-(d) hold. Let f (z, t) = e t 0 A(τ)dτ z + · · · be an univalent subordination chain. If {e− t 0 A(τ)dτ f (·, t)}t≥0 is a normal family on Bn , then f (z, s) = limt→∞ e t 0 A(τ)dτ v(z, s, t) locally uniformly on Bn for s ≥ 0, where v = v(z, s, t) is the transition mapping of f (z, t), and f (z, t) coincides with the canonical solution of (2.3). Gabriela Kohr (UBB Cluj) Geometric and analytic aspects of Loewner chains 26 / 62
2. Univalent subordination chains in several complex variables • T. Poreda, 1990 (starlikeness); M. Elin, S. Reich, D. Shoikhet, 2000 (spirallikeness w.r.t. a linear operator). Corollary Assume k+(A) < 2m(A). If f (z, t) is an A-univalent subordination chain such that {e −tA f (·, t)}t≥0 is a normal family, then z (1 + z) 2 ≤ e−tA f (z, t) ≤ z (1 − z) 2 , z ∈ Bn , t ≥ 0. Gabriela Kohr (UBB Cluj) Geometric and analytic aspects of Loewner chains 27 / 62
Page 1 and 2: Geometric and analytic aspects of L
Page 3 and 4: 1. Main contributions 1. Contributi
Page 5 and 6: 1. Main contributions • I. Graham
Page 7 and 8: 1. Main contributions • F. Bracci
Page 9 and 10: 2. Univalent subordination chains i
Page 25: 2. Univalent subordination chains i
Page 37 and 38: 3. Parametric representation on the
Page 43 and 44: Theorem 3. Parametric representatio
Page 45 and 46: Remark 3. Parametric representation
Page 47 and 48: 4. Subordination chains in complex
Page 49 and 50: Theorem 4. Subordination chains in
Page 51 and 52: Theorem 4. Subordination chains in
Page 53 and 54: Remark 4. Subordination chains in c
Page 55 and 56: 4. Subordination chains in complex
Page 57 and 58: 5. Open problems Conjecture 5. Open
Page 59 and 60: Definition 5. Open problems Let Ω
Page 61 and 62: 5. Open problems References 1. P. D

Gabriela Kohr

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