On certain classes of Dirichlet series with real coefficients absolute convergent in a half-plane

For h>0, [0, h) and R denote by SDₕ (, ) a class of absolutely convergent in the half-plane ₀=\s: \, Re\, s for all s ₀, and let Dₕ (, ) be a class of absolutely convergent in half-plane ₀ Dirichlet series F (s) =e^-sh+₊=₁^fₖ\sₖ\ such that Re\{ (-1) F' (s) + F'' (s) /h{ (-1) F (s) + F' (s) /h\}0, [0, h) (Theorem 1). } 2) In order that function F (s) =e^{-sh+₊=₁^fₖ\sₖ\ belongs to Dₕ (, ), it is sufficient, and in the case when fₖ (ₖ/h+-1) 0 for all k 1, it is necessary that ₊=₁^|fₖ (ₖ{h+-1) | (ₖ+) h-, } where h>0, [0, h) (Theorem~2). } Neighborhoods of such functions are investigated. Ordinary Hadamard compositions and Hadamard compositions of the genus m were also studied.