Proximate order and type of entire functions of several complex variables
The \((p,q)\)-order, \((p,q)\)-type, and index-pair for entire functions of several complex variables are defined in this paper. Using these definitions, the authors generalize two growth theorems given by \textit{P. Lelong} and \textit{L. Gruman} in their book `Entire functions of several complex variables' (Chapter one), Springer-Verlag, Berlin (1986; Zbl 0583.32001). Let \(p\) and \(q\) be two integers with \(p \geq q \geq 1\) and \(\psi (r) = \log^{[q - 1]}r\), and assume that \(f(z) = \sum^\infty_{k = 0} P_{n_k} (z)\) is an entire function of finite \((p,q)\)-order \(\rho > 0\) and \((p,q)\)-type \(\sigma\) with respect to any proximate order \(\rho (r)\), and let \(C_{n_k} = \sup_{\Gamma (z) \leq 1} |P_{n_k} (z) |\). The main result of this paper is \[ \limsup_{k \to \infty} \left[ {\psi (\log^{[p - 2]} n_k) \over \log^{[q - 1]} C_{n_k}^{- 1/n_k}} \right]^{\rho - A} = {\rho \over M}, \] where \(A\) is 1 if \(q = 2\), and zero otherwise; \(M = (\rho - 1)^{\rho - 1}/ \rho^\rho\), if \((p,q) = (2,2)\); \(M = 1/(\rho e)\), if \((p,q) = (2,1)\); and \(M = 1\), otherwise. For a more detailed explanation, we refer the reader to the original paper.
scientific article; zbMATH DE number 836167
scientific article