On the relative growth of entire Dirichlet series

Анотація

Let Λ=(λ_n) be an increasing to +∞ sequence of non-negative numbers, S(Λ) be a class of entire Dirichlet series F(s)=∑_n=1^∞ f_n exp{sλ_n} and M_F(σ)=sup{|F(σ+it)|: t∈ ℝ}. By L we denote a class of continuous non-negative on (-∞,+∞) functions α such that α(x)= α(x₀)≥ 0 for x ≤ x₀ and α(x)↑+∞ as x₀≤ x → +∞. For α ∈ L, β ∈ L and F ∈ S(Λ) the quantities
 ρ_α,β[F]=lim_{σ→ +∞}α(ln M_F(σ))/β(σ) and
λ_α,β[F]=lim_{σ→ +∞}α(ln M_F(σ))/β(σ) are called the generalized (α,β)-order and the generalized lower (α,β)-order of F. Define the generalized (α,β)-order and the generalized lower (α,β)-order of the function F ∈ S(Λ) with respect to a function G(s)=∑_n=1^∞g_n exp{sλ_n}∈ S(Λ) as follows
ρ_α,β[F]_G=lim_{σ→ +∞}α(M^-1_G(M_F(σ)))/β(σ) and λ_α,β[F]_G=lim_{σ→ +∞}α(M^-1_G(M_F(σ)))/β(σ).
Under the condition 0<λ_α,α[G]=ρ_α,α[G]<+∞, formulas have been found for calculating ρ_α,β[F]_G and λ_α,β[F]_G through the coefficients f_n and g_n.