Spirallike mappings and univalent subordination chains in Cn

Theorem 1.2]).

Lemma 1.2. Let h : Bn →Cn be a mapping such that h ∈N , Dh(0) = A and m(A) > 0. Then ∥h(z)∥≤ 4r (1 −r)2 |V (A)|, ∥z∥≤r < 1. Definition 1.3. (cf. [12, 30]) Let A ∈L(Cn, Cn) be such that m(A) > 0. Also let
be a domain in Cn which contains the origin. We say that
is spirallike with respect to A if e−t A(w) ∈
, for all w ∈
and t ≥0. A mapping f ∈S(Bn) is called spirallike with respect to A if f (Bn) is a spirallike domain with respect to A.

Lemma 4.1] (in the case that h(·, t) = q ∈N for t ≥0) and [8, Theorem 2.1] (in the case that Dh(0, t) = A ∈L(Cn, Cn) for t ≥0, where m(A) > 0).

Theorem 2.1. Let A : [0, ∞) →L(Cn, Cn) be a measurable mapping which satisfies the assumptions of Definition 1.5. Also let h = h(z, t) : Bn×[0, ∞) →Cn be a mapping which satisfies the following conditions: (i) h(·, t) ∈N , Dh(0, t) = A(t) for t ≥0; (ii) h(z, ·) is measurable on [0, ∞) for z ∈Bn.

Theorem 2.3. Let h = h(z, t) : Bn × [0, ∞) →Cn and A : [0, ∞) →L(Cn, Cn) satisfy the assumptions of Theorem 2.1 and let v = v(z, s, t) be the Lipschitz con- tinuous solution on [s, ∞) of the initial value problem (2.1). Also, assume that the following condition holds: sup s≥0  ∞ s ∥e  t s [A(τ)−2m(A(τ))In]dτ∥dt < ∞. (2.6) Then the limit lim t→∞e

Lemma 4.2]. Fix s ≥0 and let u(z, s, t) = e  t 0 A(τ)dτv(z, s, t) for z ∈Bn and t ≥s. Also let g(z, t) = h(z, t) −A(t)(z) for z ∈Bn and t ≥0. Then g(·, t) ∈H(Bn), g(0, t) = 0 and Dg(0, t) = 0. We first prove that u(z, s, ·) is locally Lipschitz continuous on [s, ∞) locally uniformly with respect to z ∈Bn. Fix s ≥0, T > s and r ∈(0, 1). Also let t1, t2 ∈[s, T ]. Then ∥u(z, s, t1) −u(z, s, t2)∥ ≤e  t1 0 ∥A(τ)∥dτ∥v(z, s, t1) −v(z, s, t2)∥+ ∥v(z, s, t2)∥∥e  t1 0 A(τ)dτ −e

Corollary 2.4. Let A : [0, ∞) →L(Cn, Cn) be a measurable mapping which satisfies the assumptions of Definition 1.5 and the condition (2.6). Assume f : Bn →Cn has generalized parametric representation with respect to A. Then there exists a univalent subordination chain f (z, t) such that Df (0, t) = e  t 0 A(τ)dτ, t ≥0, {e−  t 0 A(τ)dτ f (·, t)}t≥0 is a normal family on Bn and f = f (·, 0).

Theorem 2.5. Let A : [0, ∞) →L(Cn, Cn) be a measurable mapping which satisfies the assumptions of Definition 1.5 and the condition (2.6). Also let f = f (z, t) : Bn ×[0, ∞) →Cn be a mapping such that f (·, t) ∈H(Bn), f (0, t) = 0, Df (0, t) = e  t 0 A(τ)dτ for t ≥0, and f (z, ·) is locally absolutely continuous on [0, ∞) locally uniformly with respect to z ∈Bn. Let h = h(z, t) : Bn × [0, ∞) → Cn satisfy the assumptions of Theorem 2.1. Assume f (z, t) satisfies the Loewner differential equation ∂f

Theorem 2.6. Let A : [0, ∞) →L(Cn, Cn) be a measurable mapping which satisfies the assumptions of Definition 1.5 and the relation (2.6). Also let f (z, t) be a univalent subordination chain such that Df (0, t) = e  t 0 A(τ)dτ for t ≥0,

Corollary 2.7. Let A : [0, ∞) →L(Cn, Cn) be a measurable mapping which satisfies the assumptions of Definition 1.5 and the condition (2.6). Also let f ∈ S(Bn). Then f has generalized parametric representation with respect to A if and only if there exists a univalent subordination chain f (z, t) such that Df (0, t) = e  t 0 A(τ)dτ, {e−  t 0 A(τ)dτ f (·, t)}t≥0 is a normal family on Bn and f = f (·, 0). In view of (2.2) and (2.3), we deduce the following growth result for map- pings in ˜S0 A(Bn),

Theorem 3.3. Let A : [0, ∞) →L(Cn, Cn) be a locally Lebesgue integrable mapping such that m(A(t)) > 0 for t ≥0 and the condition (1.1) holds. Also let f ∈H(Bn) be a normalized mapping. Then f is generalized spirallike with respect to A if and only if f (z, t) = e  t 0 A(τ)dτ f (z) is a univalent subordination chain.

Corollary 3.4. Let A : [0, ∞) →L(Cn, Cn) be a measurable mapping which satisfies the assumptions of Definition 1.5 and the relation (2.6). Also let f : Bn → Cn be a generalized spirallike mapping with respect to A. Then f has generalized parametric representation with respect to A.

Theorem 3.5. Let A : [0, ∞) →L(Cn, Cn) be a measurable mapping which satisfies the assumptions of Definition 1.5. Also let f ∈LSn. Then f is generalized spirallike with respect to A if and only if ℜ⟨[Df (z)]−1A(t) f (z), z⟩> 0, a.e. t ≥0, ∀z ∈Bn \ {0}. (3.1)

Corollary 3.7. Let δ ∈(0, π/2) and α : [0, ∞) →[−δ, δ] be a measurable func- tion. Also let f ∈LSn and A(t) = e−iα(t)In for t ≥0. Then f is generalized spirallike with respect to A if and only if ℜ[e−iα(t)⟨[Df (z)]−1 f (z), z⟩] > 0, a.e. t ≥0, ∀z ∈Bn \ {0}. (3.3)

Theorem 4.7. Let α ∈[0, 1], b, η > 0 and λ : [0, ∞) →C be a measurable function such that |λ(t)| ≤b and ℜλ(t) ≥η for t ≥0. Also let f : U →C be a generalized spirallike function with respect to λ and let Fα : Bn →Cn be given by Fα(z) =  f (z1), ˜z  f (z1) z1 α , z = (z1, ˜z) ∈Bn. Then Fα is generalized spirallike with respect to A, where A(t) = λ(t)In.

Theorem 3.5. This completes the proof. We next give an example which shows the difference between spirallikeness and generalized spirallikeness with respect to a measurable diagonal operator. Example 4.8. Let α ∈(−π/2, π/2) and f : U →C be given by f (z) = z (1 −z)2e−iα cos α , z ∈U. If f is generalized spirallike with respect to a uniformly bounded measurable func- tion a : [0, ∞) →C with ℜa(t) ≥δ, t ≥0, for some δ > 0, then there exists a measurable function r(t) such that r(t) > 0 and a(t) =