The mean values of an analytic function represented by Dirichlet series (Q1058004)

The author introduces the notions of logarithmic mean L(\(\sigma)\) and generalized logarithmic mean G(\(\sigma)\) of an analytic function f(s) represented by Dirichlet series in the half plane \(\sigma <A\). He also defines the growth constants \[ \lim_{\sigma \to A}\left\{ \begin{matrix} \sup \\ \inf \end{matrix} \right\}\frac{\log L(\sigma)}{\sigma^{\rho}}=\left\{ \begin{matrix} \alpha \\ \beta \end{matrix} \right. ;\quad \lim_{\sigma \to A}\left\{ \begin{matrix} \sup \\ \inf \end{matrix} \right\}\frac{\log L(\sigma)}{(1- e^{\sigma -A})w(\sigma)}=\left\{ \begin{matrix} c\\ d\end{matrix} \right.; \] \[ \lim_{\sigma \to A}\left\{ \begin{matrix} \sup \\ \inf \end{matrix} \right\}\frac{(1-e^{\sigma -A})w(\sigma)}{\sigma^{\rho}}=\left\{ \begin{matrix} \gamma \\ \delta \end{matrix}\right. \] where w(x) is a positive indefinitely increasing function connected with L(\(\sigma)\) and \(\rho\) is the order of f such that \(0<\rho <\infty\). They get growth estimates for L(\(\sigma)\) and G(\(\sigma)\) in terms of \(\rho\) and these growth constants and also establishes an exact asymptotic relation between log L(\(\sigma)\) and w(\(\sigma)\).