On the order of an entire function of several complex variables represented by multiple Dirichlet series
Let \( f_r(s_1, s_2) =\sum_{m,n=1}^\infty a_{mn}^{(r)}\exp(s_1\lambda_m^{(r)}+s_2\mu_n^{(r)})\), \(r =1,2,\dots,q,\) be \(q\) entire functions represented by multiple Dirichlet series of finite non zero orders \((\rho_r,\rho_r)\). By taking asymptotic behaviour of their coefficients, the author obtains the relations between the orders of such functions. If \(\lambda_m^{(r)}\sim\lambda_m\), \(\mu_m^{(r)}\sim\mu_m\), and \(\sum_{r=1}^q\alpha_r\{\log(1/|a_{mn}^{(r)}|)\}^{-1}\sim \{\log(1/|a_{mn}|)\}^{-1}\), where \(0 < \alpha_r < 1\), \(\sum_{r=1}^q\alpha_r= 1,\) then the function \(f(s_1, s_2) =\sum_{m,n=1}^\infty a_{mn}\exp(s_1\lambda_m+s_2\mu_n)\) is an entire function of the order \((\rho,\rho)\) satisfying the inequality \(\rho\leq\sum_{r=1}^q\alpha_r\rho_r\); if \(\sum_{r=1}^q\alpha_r\log(1/|a_{mn}^{(r)}|)\sim \log(1/|a_{mn}|)\), then \(1/\rho\geq\sum_{r=1}^q\alpha_r/\rho_r\); if \(\prod_{r=1}^\infty[\log(1/|a_{mn}^{(r)}|)]^{\alpha_r}\sim\log(1/|a_{mn}|)\), then \(\rho\leq\prod_{r=1}^q\rho_r^{\alpha_r}\). NEWLINENEWLINENEWLINEThe result can be extended to several complex variables.
scientific article; zbMATH DE number 1625348