Abstract
The Schur (resp. Carath´eodory) class consists of all the analytic functions
f on the unit disk with |f| ≤1 (resp. Re f > 0 and f(0) = 1). The Schur parameters
γ0, γ1, . . . (|γj| ≤1) are known to parametrize the coefficients of functions in the Schur
class.
By employing a recursive formula for it, we describe the n-th coefficient of a
Carath´eodory function in terms of n independent variables γ1, . . . , γn with |γj| ≤1. The
mapping properties of those correspondences are also studied.
Results & Lemmas (10)
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Lemma 2.1.
Lemma 2.1. For each n ≥2, there exists a function Gn(γ1,..., γn−1) of n −1 complex variables γ1,..., γn−1 such that the following equality…
Lemma 2.1. For each n ≥2, there exists a function Gn(γ1, . . . , γn−1) of n −1 complex variables γ1, . . . , γn−1 such that the following equality holds: Fn(γ1, . . . , γn) = (1 −|γ1|2) · · · (1 −|γn−1|2)γn + Gn(γ1, . . . , γn−1). The following result will be the basis of our arguments below. It is not new (see, for instance, (1.3.47) in [7]) but a proof is given for convenience of the reader.
Lemma 2.2.
Lemma 2.2. Let ω(z) = c1z + c2z2 + · · · be a function in S0 with its Schur vector (0, γ1, γ2,... ). Then cn = Fn(γ1, γ2,..., γn) for n =…
Lemma 2.2. Let ω(z) = c1z + c2z2 + · · · be a function in S0 with its Schur vector (0, γ1, γ2, . . . ). Then cn = Fn(γ1, γ2, . . . , γn) for n = 1, 2, 3, . . ..
Lemma 3.1.
Lemma 3.1. bn = Qn(c1,..., cn) for n = 1, 2,....
Lemma 3.1. bn = Qn(c1, . . . , cn) for n = 1, 2, . . . .
Lemma 3.2.
Lemma 3.2. cn = Rn(b1,..., bn) for n = 1, 2,.... We note that the polynomial mapping ⃗Q: Cn →Cn It might be interesting to observe that the…
Lemma 3.2. cn = Rn(b1, . . . , bn) for n = 1, 2, . . . . We note that the polynomial mapping ⃗Q : Cn →Cn It might be interesting to observe that the polynomials Qn and Rn are related by a very simple relation.
Proposition 3.3.
Proposition 3.3. Rn(x1,..., xn) = −Qn(−x1,..., −xn) for n ≥1.
Proposition 3.3. Rn(x1, . . . , xn) = −Qn(−x1, . . . , −xn) for n ≥1.
Lemma 3.4.
Lemma 3.4. The mappings ⃗Qn and ⃗Rn are both polynomial automorphisms of Cn and they are inverses to each other; namely, ⃗Qn ◦⃗Rn = ⃗Rn…
Lemma 3.4. The mappings ⃗Qn and ⃗Rn are both polynomial automorphisms of Cn and they are inverses to each other; namely, ⃗Qn ◦⃗Rn = ⃗Rn ◦⃗Qn = idCn. 4. Main results We now define a sequence of functions Tn = Tn(γ1, . . . , γn) of n complex variables γ1, . . . , γn by (4.1) Tn(γ1, . . . , γn) = Qn(F1(γ1), F2(γ1, γ2), . . . , Fn(γ1, . . . , γn)), where Fk(γ1, . . . , γk) and Qn(x1, . . . , xn) are defined by (2.1) and (3.2) respectively. For instance, T1 = γ1, T2 = γ2 1 + γ2(1 −|γ1|2), T3 = γ3 1 + (
Theorem 4.1.
Theorem 4.1. Let n be a positive integer. The coefficient body Xn(P) of order n for the Carath´eodory class P is expressed as 1 × 2Vn, where…
Theorem 4.1. Let n be a positive integer. The coefficient body Xn(P) of order n for the Carath´eodory class P is expressed as {1} × 2Vn, where Vn is a convex body in Cn. Moreover, ⃗Tn(γ1, . . . , γn) = (T1(γ1), T2(γ1, γ2), . . . , Tn(γ1, . . . , γn)). is a continuous mapping of D n onto Vn and satisfies ⃗Tn(Dn) = Int Vn and ⃗Tn(∂D n) = ∂Vn. In addition, ⃗Tn : Dn →Int Vn is a real analytic diffeomorphism but ⃗Tn is not injective on the boundary ∂D n of D n for n = 2, 3, . . . .
Proposition 4.2.
Proposition 4.2. Let n be a positive integer. The coefficient body Xn(S0) of order n for S0 is described by Xn(S0) = 0 × Un, where Un is a…
Proposition 4.2. Let n be a positive integer. The coefficient body Xn(S0) of order n for S0 is described by Xn(S0) = {0} × Un, where Un is a convex body in Cn. Moreover, ⃗Fn maps D n continuously onto Un and satisfies ⃗Fn(Dn) = Int Un and ⃗Fn(∂D n) = ∂Un. Furthermore, ⃗Fn : Dn →Int Un is a real analytic diffeomorphism but ⃗Fn is not injective on ∂D n for n = 2, 3, . . . . Before the proof, we make a preliminary observation.
Lemma 4.3.
Lemma 4.3. Let (γ1,..., γn) ∈Dn−1 × C and (γ′ 1,..., γ′ n) ∈Cn. If ⃗Fn(γ1,..., γn) = ⃗Fn(γ′ 1,..., γ′ n), then (γ1,..., γn) = (γ′ 1,..., γ′…
Lemma 4.3. Let (γ1, . . . , γn) ∈Dn−1 × C and (γ′ 1, . . . , γ′ n) ∈Cn. If ⃗Fn(γ1, . . . , γn) = ⃗Fn(γ′ 1, . . . , γ′ n), then (γ1, . . . , γn) = (γ′ 1, . . . , γ′ n).
Theorem 5.1.
Theorem 5.1. The functions Tn(γ1,..., γn) defined in (4.1) are described by the following recursive formula with the initial condition T0 =…
Theorem 5.1. The functions Tn(γ1, . . . , γn) defined in (4.1) are described by the following recursive formula with the initial condition T0 = 1 : Tn(γ1, . . . , γn) = γ1Tn−1(γ1, . . . , γn−1) + n−1 X k=1 Tk(γ2, . . . , γk+1) h (1 + γ1)Tn−k−1(γ1, . . . , γn−k−1) −(1 + ¯γ1)Tn−k(γ1, . . . , γn−k) i . We remark that the transformation g1 from g above was already considered by Brown [2] in a more general context. References
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