Growth estimates of analytic functions defined by Dirichlet series (Q1184899)

Let \(f\) be defined by a Dirichlet series with abscissa of convergence \(c\) and abscissa of absolute convergence \(A\). The function \(f\) is analytic in the half plane \(\sigma<A\). The authors study the relative growth of \(f(s)\) and its derivative \(f'(s)\) in terms of the maximum modulus, the maximum term and the mean value. For example they prove: Let \(f(s)\) be an analytic function for \(\sigma<A\) of order \(\rho\), \(0<\rho<\infty\) and of type \(\tau\), \(0<\tau <\infty\). Let \(G(\sigma,f)\) be any given growth function associated with \(f\), satisfying (a) \(\log G(\sigma,f)\) is a non-negative strictly increasing convex function of \(\sigma\), (b) \(\limsup_{\sigma\to A}{\log G(\sigma,f)\over(1-\exp(\sigma-A))^{- \rho}}=\tau\). (c) There exists a positive constant \(h\) depending on \(G(\sigma,f)\) such that \[ G(\sigma,f')/h\leq\inf[G(x,f)/(x-\sigma)]. \] (d) There exists a constant \(c\geq 0\) depending on \(G(\sigma,f)\) such that \(G(\sigma,f')\geq{dG\over d\sigma}(\sigma,f)-{c\over\sigma} G(\sigma,f)\) then \[ \rho\tau\leq\limsup_{\sigma\to A}{G(\sigma,f')/G(\sigma,f)\over 1-\exp(\sigma-A)^{\lambda-1}}\leq e\rho\tau h\left({\rho+1\over\rho}\right)^{\rho+1}. \] Some applications are also given.