Generalized and modified orders of growth for Dirichlet series absolutely convergent in a half-plane

Let λ=(λₙ)n₀ be a non-negative sequence increasing to +∞, τ(λ)=n→∞(ln n/λₙ), and D₀(λ) be the class of all Dirichlet series of the form F(s)=∑ₙ₌₀^∞ aₙ(F)esλₙ absolutely convergent in the half-plane Res<0 with aₙ(F)=0 for at least one integer n≥0. Also, let α be a continuous function on [x₀,+∞) increasing to +∞, β be a continuous function on $[a,0)$ such that β(σ)→+∞ as σ↑0, and γ be a continuous positive function on $[b,0)$. In the article, we investigate the growth of a Dirichlet series F₀(λ) depending on the behavior of the sequence (|aₙ(F)|) in terms of its α,β,γ-orders determined by the equalitiesR^*α,β,γ(F)=σ↑0α(max₀,γ(σ)lnμ(σ)\)β(σ),Rα,β,γ(F)=σ↑0α(max₀,γ(σ)ln M(σ)\)β(σ),where μ(σ)=max\|aₙ(F)|eσλₙ n≥0\ and M(σ)=\|F(s )| Res=σ\ are the maximal term and the supremum modulus of the series F, respectively. In particular, if for every fixed $t>0$ we have α(tx)~ α(x) as x→+∞, β(tσ)~ t-ρβ(σ) as σ↑0 for some fixed ρ>0, 0<σ↑0γ(tσ)/γ(σ) ≤σ↑0γ(tσ)/γ(σ)<+∞,Φ(σ)=α⁻¹(β(σ))/γ(σ) for all σ∈[σ₀,0),Φ(x)=maxσ-Φ(σ) σ∈[σ₀,0)\ for all x, and Δ_Φ(λ)=n→∞( -ln n/Φ(λₙ)), then: (a) for each Dirichlet series F₀(λ) we haveR^*α,β,γ(F)=n→ +∞(ln⁺|aₙ(F)|-Φ (λₙ))^ρ; (b) if τ(λ)>0, then for each p₀∈[0,+∞] and any positive function Ψ on $[c,0)$ there exists a Dirichlet series F₀(λ) such that R^*α,β,γ(F)=p₀ and M(σ,F)≥ Ψ(σ) for all σ∈[σ₀,0); (c) if τ(λ)=0, then (Rα,β,γ (F))1/ρ≤ (R^*α,β,γ(F))1/ρ+Δ_Φ(λ) for every Dirichlet series F₀(λ); (d) if τ(λ)=0, then for each p₀∈[0,+∞] there exists a Dirichlet series F₀(λ) such that R^*α,β,γ(F)=p₀ and (Rα,β,γ(F))1/ρ=(R ^*α,β,γ(F))1/ρ+Δ_Φ(λ).