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Proximate order and type of entire functions of several complex variables. (English) Zbl 0844.32003

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 P. Lelong and 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.
Reviewer: Z.Ye (DeKalb)

MSC:

32A15 Entire functions of several complex variables

Citations:

Zbl 0583.32001
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References:

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