Sharp estimates in the classes of Schur, Carathéodory, and Borel functions. (English. Russian original) Zbl 1180.32001
Dokl. Math. 79, No. 3, 418-420 (2009); translation from Dokl. Akad. Nauk, Ross. Akad. Nauk 426, No. 6, 727-729 (2009).
Consider the hyperball
\[ S_R=\left\{z=(z_1,z_2,\dots,z_n): |z_1|^2+|z_2|^2+\dots+|z_n|^2<R^2\right\}. \]
Let \(f(z)= \sum_{m_1,\dots,m_n=0}^\infty a_{m_1,\dots,m_n} z_1^{m_1}\cdots z_n^{m_n}\) \((a_{0\dots 0}=1)\) be a holomorphic function in the hyperball \(S_R\).
In the paper, the author obtains sharp estimates for \(\sqrt{|a_{10\dots0}|^2+\dots+|a_{0\ldots01}|^2}\) that are a strengthening of estimates for the Taylor coefficients \(a_{m_1,m_2,\dots,m_n}\) \((m_1+\cdots+m_n=1)\), when \(f(z)\) is a Carathéodory, Schur or Borel function, respectively.
\[ S_R=\left\{z=(z_1,z_2,\dots,z_n): |z_1|^2+|z_2|^2+\dots+|z_n|^2<R^2\right\}. \]
Let \(f(z)= \sum_{m_1,\dots,m_n=0}^\infty a_{m_1,\dots,m_n} z_1^{m_1}\cdots z_n^{m_n}\) \((a_{0\dots 0}=1)\) be a holomorphic function in the hyperball \(S_R\).
In the paper, the author obtains sharp estimates for \(\sqrt{|a_{10\dots0}|^2+\dots+|a_{0\ldots01}|^2}\) that are a strengthening of estimates for the Taylor coefficients \(a_{m_1,m_2,\dots,m_n}\) \((m_1+\cdots+m_n=1)\), when \(f(z)\) is a Carathéodory, Schur or Borel function, respectively.
Reviewer: Dorina Raducanu (Brasov)
MSC:
| 32A10 | Holomorphic functions of several complex variables |
Keywords:
holomorphic function; hyperball; Taylor coefficients; Carathéodory, Schur and Borel functionsReferences:
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