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

Let = (ₙ) ₍䃐 be a non-negative sequence increasing to +, () =₍ (n/ₙ), and D₀ () be the class of all Dirichlet series of the form F (s) =₍=₀^ aₙ (F) e^sₙ absolutely convergent in the half-plane Res0 we have (tx) (x) as x+, (t) t^- () as 0 for some fixed >0, 00, then for each p₀0, + and any positive function on c, 0) there exists a Dirichlet series F₀ () such that R^*, , (F) =p₀ and M (, F) () for all [₀, 0) ; (c) if () =0, then (R, , (F) ) ^1/ (R^*, , (F) ) ^1/+_ () for every Dirichlet series F₀ () ; (d) if () =0, then for each p₀[0, + there exists a Dirichlet series F₀ () such that R^*, , (F) =p₀ and (R, , (F) ) ^1/= (R ^*, , (F) ) ^1/+_ ().