Parametric Representation of Univalent Mappings in Several Complex Variables

Theorem 1.2 Let p ∈M and Pm = 1 m!Dmp(0) for m ≥1. Then the following assertions hold: https://doi.org/10.4153/CJM-2002-011-2 Published online by Cambridge University Press

Corollary 1.3 The set M is compact.

Theorem 2.1]) can now be improved by omitting the boundedness assumptions on https://doi.org/10.4153/CJM-2002-011-2 Published online by Cambridge University Press

Theorem 1.4 Let g : U →C satisfy the assumptions of Definition 1.1 and let ht(z) = h(z,t): B × [0, ∞) →Cn satisfy the following conditions: (i) for each t ≥0, ht(·) ∈Mg; (ii) for each z ∈B, h(z,t) is a measurable function of t ∈[0, ∞). Then the limit (1.2) lim t→∞etv(z, s,t) = f (z, s) exists locally uniformly on B for each s ≥0, where v = v(z, s,t) is the unique solution of the initial value problem (1.3) ∂v ∂t = −h(v,t), a.e. t ≥s, v(s) = z.

Lemma 1.6 Let f : B × [0, ∞) →Cn be such that f (· ,t) ∈H(B), f (0,t) = 0, D f (0,t) = etI, for each t ≥0 and f (z, ·) is a locally Lipschitz continuous function of t ∈[0, ∞) locally uniformly with respect to z ∈B. Let g : U →C satisfy the conditions of Definition 1.1 and let h: B × [0, ∞) →Cn satisfy the requirements of Theorem 1.4. Suppose that ∂f ∂t (z,t) = Df (z,t)h(z,t), a.e. t ≥0, for all z ∈B. Further, assume there exists an increasing sequence {tm} such that tm > 0, tm →∞and lim m→∞e−tm f

Theorem 1.9 (Vitali’s Theorem in Several Complex Variables) Let Ωbe a domain in Cn and let Q ⊂Ωbe a set of uniqueness for the holomorphic functions on Ω. Suppose that {Φk}k≥1 is a sequence of holomorphic functions on Ωwhich is locally bounded and which has the property that {Φk}k≥1 converges for all z ∈Q. Then there exists a holomorphic function Φ on Ωsuch that Φk →Φ locally uniformly on Ω. https://doi.org/10.4153/CJM-2002-011-2 Published online by Cambridge University Press

Theorem 1.10 Let f (z,t) be a Loewner chain which is locally Lipschitz in t locally uniformly with respect to z. Then there is a mapping h = h(z,t) such that h(z,t) ∈M for each t ≥0, h(z,t) is measurable in t for each z ∈B, and for a.e. t ≥0, (1.7) ∂f ∂t (z,t) = Df (z,t)h(z,t), ∀z ∈B. Moreover, if there is a sequence {tm} such that tm > 0, tm →∞and lim m→∞e−tm f (z,tm) = F(z) locally uniformly on B, then f (z) = f (z, 0) ∈S0(B).

Lemma 2.1 Let g and h satisfy the assumptions of Theorem 1.4. If v = v(z, s,t) is the solution of the initial value problem (1.3), then es∥z∥exp Z ∥z∥ ∥v(z,s,t)∥  1 max{g(x), g(−x)} −1  dx x (2.1) ≤et∥v(z, s,t)∥≤es∥z∥exp Z ∥z∥ ∥v(z,s,t)∥ 

Theorem 2.2 Let g : U →C satisfy the assumptions of Definition 1.1 and f ∈S0 g(B). Then ∥z∥exp Z ∥z∥ 0  1 max{g(x), g(−x)} −1  dx x (2.4) ≤∥f (z)∥≤∥z∥exp Z ∥z∥ 0

Corollary 2.3 Let g : U →C satisfy the assumptions of Definition 1.1 and f (z,t) be a g-Loewner chain. Then ∥z∥exp Z ∥z∥ 0  1 max{g(x), g(−x)} −1  dx x ≤∥e−t f (z,t)∥≤∥z∥exp Z ∥z∥ 0  1

Corollary 2.4 If f ∈S0(B) then ∥z∥ (1 + ∥z∥)2 ≤∥f (z)∥≤ ∥z∥ (1 −∥z∥)2 , z ∈B. Consequently, f (B) ⊇B1/4.

Corollary 2.6 S0(B) is a normal family. Thus in Cn, n ≥2, S(B) is a larger set than S0(B). Actually we believe that S0(B) is a compact set. 2.2 Examples of Mappings in S0 g(B) The following result, obtained recently in [Gr-Ha-Ko-Su], gives many examples of mappings in S0(B) when B is the unit Euclidean ball of Cn. Properties of the operator Ψn,0,β have been recently studied in [Gr-Ko-Ko].

Theorem 2.7 Let α ∈[0, 1] and β ∈[0, 1/2] be such that α + β ≤1. If f ∈S then Ψn,α,β( f ) ∈S0(B), where Ψn,α,β( f )(z) =  f (z1), z2  f (z1) z1  αf ′(z1)
β, . . . , zn  f (z1) z1  αf ′(z1)
β ′ , for z = (z1, . . . , zn)′ ∈B. The branches of the power functions are chosen such that

Theorem 2.8 Let P: B →C be a holomorphic function on B such that P(0) = 1 and let F(z) = P(z)z, z ∈B. Also let g : U →C satisfy the requirements of Definition 1.1. Then F ∈S∗ g (B) if and only if 1 + DP(z)z P(z) ∈1 g (U), z ∈B.

Corollary 2.9 For each j = 1, 2, . . . , n, let fj(ζ) be a normalized holomorphic function on U such that fj(ζ) ζ f ′ j (ζ) ≺g(ζ), for ζ ∈U, where g : U →C satisfy the requirements https://doi.org/10.4153/CJM-2002-011-2 Published online by Cambridge University Press

Corollary 2.10 For each j = 1, 2, . . . , n, let f j be a normalized starlike function of order 1/2 on U and let F be defined by (2.7). Then F ∈S∗ g (B), where g(ζ) = 1 + ζ. We have seen that the class of spirallike mappings of type α, |α| < π/2, is a sub- class of S0(B). However, in general a spirallike mapping relative to a linear operator need not belong to S0(B). In other words, in higher dimensions there exist mappings in S(B) \ S0(B), which do not have parametric representation. We have the

Theorem 2.2 (see [Su4], [Fi-Th], [Ro-Su2], [Ha-Ko3], [Ko2], [Ha1]).

Corollary 2.13 Let g(ζ) = 1 + ζ, ζ ∈U, and f ∈S0 g(B). Then ∥z∥ 1 + ∥z∥≤∥f (z)∥≤ ∥z∥ 1 −∥z∥, z ∈B. Consequently, f (B) ⊇B1/2. Note that the above growth result is sharp in the case of the unit ball B(p) with respect to a p-norm, 1 ≤p ≤∞. To see this, let f (z) =  z1 1 −z1 , . . . ,

Theorem 2.14 Let g : U →C satisfy the assumptions of Definition 1.1 and let f ∈ S0 g(B). Then (2.8) 1 2!lw D2 f (0)(w, w)
 ≤|g ′(0)|, w ∈Cn, ∥w∥= 1, lw ∈T(w).

Corollary 2.15 Let g : U →C satisfy the assumptions of Definition 1.1 and f ∈S0 g(B). Then 1 2!D2 f (0)(z, z) ≤4|g ′(0)|, ∥z∥= 1.

Corollary 2.16 If f ∈S0(B) then (2.13) 1 2!lw D2 f (0)(w, w)
 ≤2, ∥w∥= 1, lw ∈T(w). Moreover 1 2!D2 f (0)(z, z) ≤8, ∥z∥= 1. It would be interesting to see if the mappings in S1(B) satisfy the above bound. Note that using the growth result in Corollary 2.4, one may prove that if f ∈S0(B) then

Corollary 2.19 If f ∈S0 g(B) with g(ζ) = 1 + ζ, ζ ∈U, then (2.14) 1 2!lw D2 f (0)(w, w)
 ≤1, ∥w∥= 1, lw ∈T(w). Moreover, 1 2!D2 f (0)(w, w) ≤4, ∥w∥= 1, and (2.15)

Theorem 2.22 Let f ∈S0 c(B) with c ∈(0, 1). Then ∥z∥ (1 + c∥z∥)2 ≤∥f (z)∥≤ ∥z∥ (1 −c∥z∥)2 , z ∈B. Moreover, 1 2lw D2 f (0)(w, w)
 ≤2c, ∥w∥= 1, lw ∈T(w), and 1