Results & Lemmas (7)
Each result is auto-extracted from the paper via OCR - the math is shown verbatim and may carry occasional transcription artifacts. Click to expand the full statement.
Lemma 1.1
Lemma 1.1 Let Φ ∶U →U be a holomorphic function of the form Then for every m ∈ℕ1, p ∈ℕ0, satisfying the condition 0 ≤2p < m. The estimates…
Lemma 1.1 Let Φ ∶U →U be a holomorphic function of the form Then for every m ∈ℕ1, p ∈ℕ0, satisfying the condition 0 ≤2p < m. The estimates are optimal.
Theorem 2.1
Theorem 2.1 Let 휑 be a function from the family and has the form Then, for every k ∈ℕ1 and every 훾∈ℂ, there holds the inequality The…
Theorem 2.1 Let 휑 be a function from the family and has the form Then, for every k ∈ℕ1 and every 훾∈ℂ, there holds the inequality The estimate is sharp.
Theorem 2.2
Theorem 2.2 Let G ⊂ℂn be a bounded complete n-circular domain and let f ∈NG ∩F0,k(G,ℂ), k ∈ℕ2. If the expansion of the function f into a…
Theorem 2.2 Let G ⊂ℂn be a bounded complete n-circular domain and let f ∈NG ∩F0,k(G,ℂ), k ∈ℕ2. If the expansion of the function f into a series of m-homoge- nous polynomials Qf,m has the form (1.1), with Qf,0 = 1, then for the homogeneous polyno- mials Qf,2k, Qf,k and every 휆∈ℂ there holds the following sharp estimate:
Theorem 2.3
Theorem 2.3 Let G ⊂ ℂn be a bounded complete n-circular domain and let k ∈ℕ2. If the expansion of the function f ∈MG ∩F0,k(G,ℂ), into a…
Theorem 2.3 Let G ⊂ ℂn be a bounded complete n-circular domain and let k ∈ℕ2. If the expansion of the function f ∈MG ∩F0,k(G,ℂ) , into a series of m-homogenous polynomi- als Qf,m has the form (1.1), with Qf,0 = 1, then for the homogeneous polynomials Qf,2k, Qf,k and 휆∈ℂ there holds the following sharp estimate:
Theorem 2.4
Theorem 2.4 Let G ⊂ ℂn be a bounded complete n-circular domain and let f ∈Mk G ∩F0,k(G,ℂ), k ∈ℕ2. If the expansion of the function f into a…
Theorem 2.4 Let G ⊂ ℂn be a bounded complete n-circular domain and let f ∈Mk G ∩F0,k(G,ℂ), k ∈ℕ2. If the expansion of the function f into a series of m-homoge- nous polynomials Qf,m has the form (1.1), with Qf,0 = 1, then for the homogeneous polyno- mials Qf,2k, Qf,k and 휆∈ℂ there holds the following sharp estimate: The equality in the above inequality realize the same functions f = ̃f, f = ̂f as in the previ- ous Theorem 2.2. 3 Applications In this section we apply Theorems 2.3 and 2.4, to
Theorem 3.1
Theorem 3.1 For mappings F ∈̃Sk(픹n) ∩F1,k(𝔹n, ℂn), k ∈ℕ2, the parameter 휆∈ℂ and points z ∈픹n 0 there holds the following sharp estimate…
Theorem 3.1 For mappings F ∈̃Sk(픹n) ∩F1,k(𝔹n, ℂn), k ∈ℕ2, the parameter 휆∈ℂ and points z ∈픹n {0} there holds the following sharp estimate where Tz ∈(ℂn)∗, z ∈𝔹n{0}, is arbitrary functional satisfying the conditions ‖‖Tz‖‖ = 1, Tz(z) = ‖z‖.
Theorem 3.2
Theorem 3.2 For mappings F ∈̃S∗(픹n) ∩F1,k(𝔹n, ℂn), k ∈ℕ2, points z ∈픹n 0 and parameter 휆∈ℂ, there holds the following sharp estimate where…
Theorem 3.2 For mappings F ∈̃S∗(픹n) ∩F1,k(𝔹n, ℂn), k ∈ℕ2, points z ∈픹n {0} and parameter 휆∈ℂ, there holds the following sharp estimate where Tz ∈(ℂn)∗, z ∈𝔹n{0}, is arbitrary functional satisfying the conditions: ‖‖Tz‖‖ = 1, Tz(z) = ‖z‖. In the other way a similar theorem was proved by Xu [30]. 4 Final remarks It is possible to allow also k = 1 in definition of (j, k)-symmetrical, j ∈ℤ , functions, from F(G,ℂm). Then 휀= 1 and we should take the convention Fj,1(G, ℂm) = F(G, ℂm) for j ∈ℤ. Co
Related Papers