On the relative growth of entire Dirichlet series with respect to Dirichlet series absolutely converging in half-plane

Let Λ = (λ _n) be an increasing to +∞ sequence of non-negative numbers, λ₀ = 0, and by S(Λ, A) we denote a class of Dirichlet series \(F\left(s\right)=\sum_{n=0}^{\infty }{f}_{n}\text{exp}\left\{s{\uplambda }_{n}\right\},s=\sigma +it,\) $F\left(s\right)={\sum }_{n=0}^{\infty }{f}_n\text{exp}\left\{s,{\mathrm{\lambda }}_n\right\},s=\sigma +it,$ with the abscissa of the absolute convergence σ_a = A ∈ (−∞,+∞]. For σ < A we put M_F (σ) = sup{|F(σ + it)| : t ∈ ℝ}. The growth of the function F ∈ S(Λ,+∞) with respect to the function \(G\left(s\right)=\sum_{n=0}^{\infty }{g}_{n}\text{exp}\left\{s{\uplambda }_{n}\right\} \in S\left(\Lambda ,0\right)\) $G\left(s\right)={\sum }_{n=0}^{\infty }{g}_n\text{exp}\left\{s,{\mathrm{\lambda }}_n\right\}\in S\left(\mathrm{\Lambda },,,0\right)$ is identified with the growth of the function \(\left|1/\right|{M}_{G}^{-1}\left({M}_{F}\left(\sigma \right)\right)|\) $\left|1,/\right|M_G^{-1}\left(M_F,\left(\sigma \right)\right)|$ as σ → +∞. In terms of generalized orders, the connection between the growth of this function and the behavior of the coefficients f_n and g_n has been studied.