On a property of the entire dirichlet series with decreasing coefficients

Abstract

The classS ^*_Ψ (A) of the entire Dirichlet series$F(s) = \sum\nolimits_{n = 0}^\infty {a_n exp(s\lambda _n )}$ is studied, which is defined for a fixed sequence$A = (a_n ),0< a_n \downarrow 0,\sum\nolimits_{n = 0}^\infty {a_n< + \infty } ,$ 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) ↑ +∞ asx →+ ∞. In this class, the necessary and sufficient conditions are given for the relation ϕ(InM(σ,F))∼ϕ(In μ(σ,F)) to hold as σ → +∞, where$M(\sigma ,F) = sup\{ |F(\sigma + it)|:t \in \mathbb{R}\} ,\mu (\sigma ,F) = max\{ a_n exp(\sigma \lambda _n ):n \in \mathbb{Z}_ + \}$, and ϕ is a positive continuous function increasing to +∞ on (0,+∞), forwhich ln ϕ(x) is a concave function andϕ(lnx) is a slowly increasing function.