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

For $h>0$, α∈ [0,h) and μ∈ R denote by SDₕ(μ, α) a class of absolutely convergent in the half-plane Π₀=:\, Re\,s<0\ Dirichlet series F(s)=eˢʰ+∑ₖ₌₁∞fₖexpλₖ\ such that {Re\(μ-1)F'(s)-μ F''(s)/h/(μ-1)F(s)-μ F'(s)/h\>α for all s∈ Π₀,} and let Σ Dₕ(μ, α) be a class of absolutely convergent in half-plane Π₀ Dirichlet series F(s)=e⁻ˢʰ+∑ₖ₌₁∞fₖexpλₖ\ such that {Re\(μ-1)F'(s)+μ F''(s)/h/(μ-1)F(s)+μ F'(s)/h\<-α for all s∈ Π₀.} Then SDₕ(0, α) consists of pseudostarlike functions of order α and SDₕ(1, α) consists of pseudoconvex functions of order α. For functions from the classes SDₕ(μ, α) and Σ Dₕ(μ, α), estimates for the coefficients and growth estimates are obtained. {In particular, it is proved the following statements: 1) In order that function F(s)=eˢʰ+∑ₖ₌₁∞fₖexpλₖ\ belongs to SDₕ(μ, α), it is sufficient, and in the case when fₖ(μλₖ/h-μ+1)≤ 0 for all k≥ 1, it is necessary that} {∑ₖ₌₁∞|fₖ(μλₖ/h-μ+1)|(λₖ-α)≤ h-α,} {where h>0, α∈ [0, h) (Theorem 1).} 2) {In order that function F(s)=e⁻ˢʰ+∑ₖ₌₁∞fₖexpλₖ\ belongs to Σ Dₕ(μ, α), it is sufficient, and in the case when fₖ(μλₖ/h+μ-1)≤ 0 for all k≥ 1, it is necessary that {∑ₖ₌₁∞|fₖ(μλₖ/h+μ-1)|(λₖ+α)≤ h-α,} where h>0, α∈ [0, h) (Theorem~2).} Neighborhoods of such functions are investigated. Ordinary Hadamard compositions and Hadamard compositions of the genus m were also studied.