ON CERTAIN CLASSES OF DIRICHLET SERIES WITH REAL COEFFICIENTS ABSOLUTELY 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 = {s: Re s < 0} Dirichlet series F(s) = esh + P∞ k=1 fk exp{sλk} such that Re n (μ-1)F′(s)-μF′′(s)/h (μ-1)F(s)-μF′(s)/h o > α for all s ∈ Π0, and let ΣDh(μ, α) be a class of absolutely convergent in half-plane Π0 Dirichlet series F(s) = e-sh + P∞ k=1 fk exp{sλk} such that Re n (μ-1)F′(s)+μF′′(s)/h (μ-1)F(s)+μF′(s)/h o < -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 ΣDh(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 + P∞ k=1 fk exp{sλk} 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 ∞P 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+ P∞ k=1 fk exp{sλk} belongs to ΣDh(μ, α), it is sufficient, and in the case when fk(μλk/h + μ - 1) ≤ 0 for all k ≥ 1, it is necessary that ∞P 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. [ABSTRACT FROM AUTHOR]