Loewner chains and quasiconformal extension

Lemma 2.2. For each r ∈(0, 1) there exists a constant M = M(r), independent of p, such that ∥p(z)∥≤M(r) for ∥z∥≤r, p ∈M. Using Lemma 2.2 and [Por3, Theorem 6], the authors of [Gr-Ha-Ko] obtained the following lemma.

Lemma 2.3. Let ht(z) = h(z, t) : B × [0, ∞) →Cn satisfy the following conditions: (i) for each t ≥0, ht(·) ∈M; (ii) for each z ∈B, h(z, t) is a measurable function of t ∈[0, ∞). Let ft(z) = f(z, t) : B×[0, ∞) →Cn be such that f(·, t) ∈H(B), f(0, t) = 0, Df(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. Suppose that ∂f ∂t (z, t) = Df(z, t)h(z, t) a.e. t ≥0, for all z ∈B. Further, assume that there exists an inc

Lemma 3.1. Let f(z, t) be a Loewner chain which satisfies the assump- tions of Lemma 2.3. If there exists a constant c1 > 0 such that c1∥z∥2 ≤ℜ⟨h(z, t), z⟩ for z ∈B \ {0}, t ≥0, then there exists a constant d such that ∥f(z, t)∥≤det∥z∥ for z ∈B, t ≥0.

Theorem 3.2. Let f(z, t) be a Loewner chain which satisfies the as- sumptions of Lemma 2.3. Assume that the following conditions are satisfied: (i) ∥Df(z, t)∥≤ M1(t) (1 −∥z∥)α, z ∈B, t ≥0, where M1(t) is locally bounded with respect to t and α is a constant with 0 ≤α < 1; (ii) there exists a constant c1 > 0 such that c1∥z∥2 ≤ℜ⟨h(z, t), z⟩ for z ∈B \ {0}, t ≥0; (iii) there exists a constant c2 > 0 such that ∥h(z, t)∥≤c2 for z ∈B, t ≥0; (iv) f(z, t) is K1-quasiconformal for each t.

Theorem 3.2 cannot be omitted. Let B be the Euclidean unit ball in C2. Let f(z) = (z1 + az2 2, z2)′. Then f is starlike if and only if |a| ≤3 √ 3/2 by Example 3 in [Su3]. Put a = 3 √ 3/2. Since f is starlike, f(z, t) = etf(z) is a Loewner chain which satisfies the assumptions of Lemma 2.3. Since Df(z, t) = etDf(z) and f is a polynomial, condition (i) is satisfied. As h(z, t) = [Df(z)]−1f(z) = (z1 −az2 2, z2)′, condition (iii) is satisfied. Because ∥Df(z)∥is uniformly bounded in B2, det Df(z) = 1 an

Theorem 4.1. Let f : B →Cn be a normalized holomorphic mapping on B and let G(z) be a nonsingular n×n matrix, holomorphic as a function of z ∈B. Suppose that G(0) = I and the following assumptions hold: (i) ∥[G(z)]−1Df(z) −I∥≤c, z ∈B; (ii) ∥∥z∥2[[G(z)]−1Df(z) −I] + (1 −∥z∥2)[G(z)]−1DG(z)(z, ·)∥≤c, z ∈B, where 0 ≤c < 1; (iii) there exists a K ≥1 such that ∥G(z)∥n ≤K|det G(z)|, z ∈B. Then f is univalent and quasiregular on B and extends to a quasiconformal homeomorphism of R2n onto itself.

Corollary 4.2. Let f : B →Cn be a normalized holomorphic mapping on B and let a : B →C be a holomorphic function such that a(z) ̸= 0, z ∈B, and a(0) = 1. Suppose that the following assumptions hold:

Theorem 4.3. Let f be a quasiconformal, strongly spirallike mapping of type α with ∥[Df(z)]−1f(z)∥uniformly bounded on B. Then f(z) has a continuous extension to B (again denoted by f) and F(z) =  f(z), z ∈B, ∥z∥1−iaf(z/∥z∥1−ia), z ̸∈B, is a quasiconformal homeomorphism of R2n onto itself.

Corollary 4.4. Let f be a quasiconformal, strongly starlike mapping with ∥[Df(z)−1f(z)∥uniformly bounded on B. Then f(z) has a continuous extension to B (again denoted by f ) and F(z) =  f(z), z ∈B, ∥z∥f(z/∥z∥), z ̸∈B, is a quasiconformal homeomorphism of R2n onto itself. We remark that the mapping in Example 3.4 shows that the assumption of strong starlikeness in the above corollary cannot be omitted.

Definition 2.1. f is said to be strongly spirallike of type α if φz(U, 0) is contained in a compact subset of the right half-plane independent of z ∈∂B. Or, equivalently, there exists a c with 0 < c < 1 such that |σz(ζ, 0)| ≤c uniformly for z ∈∂B, ζ ∈U. When α = 0, the above definition coincides with the definition of strongly starlike mappings due to Chuaqui [Ch] (cf. [Ha]).