Sharp estimates in the classes of Schur, Carathéodory, and Borel functions (Q736031)

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.