On the relative growth of Dirichlet series absolutely

Let \(\Lambda = (\lambda_n)\) be an increasing sequence of non-negative numbers tending to \(+\infty\), with \(\lambda_0 = 0\). We denote by \(S(\Lambda, 0)\) a class of Dirichlet series \(F(s) = \sum_{n=0}^{\infty} f_n \exp\{s \lambda_n\}, \quad s = \sigma + it,\) which have an abscissa of absolute convergence \(\sigma_a = 0\). For \(\sigma < 0\), we define \( M_F(\sigma) = \sup \{|F(\sigma + it)| : t \in \mathbb{R}\}. \) The growth of the function \(F \in S(\Lambda, 0)\) is analyzed in relation to the function \( G(s) = \sum_{n=0}^{\infty} g_n \exp\{s \lambda_n\} \in S(\Lambda, 0), \) via the growth of the function \(1/|M^{-1}_G(M_F(\sigma))|\) as \(\sigma \uparrow 0\). We investigate the connection between this growth and the behavior of the coefficients \(f_n\) and \(g_n\) in terms of generalized orders.