A Note on an Analogue of the Sălăgean Class

Let F (s) = e s + ∞ k=1 f k e sλ k be a Dirichlet series that is absolutely convergent in the half-plane Π 0 = {s : Re s < 0}.This paper provides the conditions under which the inequality Re F (j+1) (s) F (j) (s) > α holds true for α ∈ [0, 1), s ∈ Π 0 and j ≥ 0, where (λ k ) is a sequence of positive numbers with λ 1 > 1 increasing to +∞.The obtained result is applied to the study of the properties of a solution of the differential equation d 2 w ds 2 + (γ 0 e 2s + γ 1 e s + γ 2 )w = 0.