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 $F(s) = \sum olimits_{n = 0}^∞ {a_n exp(sλ _n )}$ is studied, which is defined for a fixed sequence $A = (a_n ),\; 0 < a_n \downarrow 0,\sum olimits_{n = 0}^∞ {a_n< + ∞ } ,$ 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 $ϕ(\ln M(σ, F)) ∼ ϕ(\ln μ(σ, F))$ to hold as $σ → +∞$, where $M(σ ,F) = sup\{ |F(σ + it)|:t \in \mathbb{R}\} ,μ (σ ,F) = max\{ a_n exp(σ λ _n ):n \in \mathbb{Z}_ + \}$, and $ϕ$ is a positive continuous function increasing to $+∞$ on $(0, +∞)$, forwhich $\ln ϕ(x)$ is a concave function and $ϕ(\ln x)$ is a slowly increasing function.