The Paley problem for functions that are meromorphic in \(\mathbb{C}^ n\) (Q1360758)

Let \(S^n\), \(n\geq 1\), be the unit sphere in \(\mathbb{C}^n\), \(s_n\) be the area of \(S^n\) and \(ds_n\) be an element of the area of \(S^n\). The author proves the following variant of the Paley problem. Theorem. Let \(f= {g\over h}\) be a meromorphic function in \(\mathbb{C}^n\), \(g(0)= h(0)=1\), such that \[ \varliminf_{r\to\infty} {1\over\log r} \log \left({1 \over s_n} \int_{S^n} \log\biggl( \bigl|g(r\rho) \bigr|^2 +\bigl|h(r\rho) \bigr|^2 \biggr)^{1/2} ds_n(\rho) \right)= \lambda. \] Then \[ \varliminf_{r\to\infty} {\Bigl(\log \sup\biggl\{ \bigl|f(\rho w) \bigr|: |w|= r\biggr\} \Bigr)\over {1\over s_n} \int_{S^n} \log\biggl( \bigl|g(r\rho) \bigr|^2 +\bigl|h(r\rho) \bigr|^2 \biggr)^{1/2} ds_n(\rho)} \leq\prod^{n-1}_{k=1} \left(1+ {\lambda\over 2k} \right) \begin{cases} \pi\lambda/ \sin \pi\lambda,\;0\leq \lambda \leq {1\over 2},\\ \pi\lambda,\;\lambda> {1\over 2} \end{cases}. \] This estimate is the best one.