On a property of the entire dirichlet series with decreasing coefficients - Sheremeta, M. M. (Ukrains’kyi Matematychnyi Zhurnal, 1993)
The class S_Ψ^ * (A) of the entire Dirichlet series F(s) = olimits_n = 0^∞ a_n exp(sλ _n ) is studied, which is defined for a fixed sequence A = (a_n ), 0 < a_n 0,olimits_n = 0^∞ a_n< + ∞ , by the conditions 0 ≤ λ_n ↗ +∞ and λ_n ≤ (1n^+(1/a_n )) imposed on the parameters λ_n, where ψ is a positive continuous function on (0, +∞) such that ψ(x) ↑ +∞ and x/ψ(x) ↑ +∞ as x →+ ∞. In this class, the necessary and sufficient conditions are given for the relation ϕ(M(σ, F)) ∼ ϕ(μ(σ, F)) to hold as σ → +∞, where M(σ ,F) = sup{ |F(σ + it)|:t R} ,μ (σ ,F) = max{ a_n exp(σ λ _n ):n Z_ + }, and ϕ is a positive continuous function increasing to +∞ on (0, +∞), forwhich ϕ(x) is a concave function and ϕ(x) is a slowly increasing function.