A Note on an Analogue of the Sălăgean Class

Let F (s) = e s + k=1 f k e s k be a Dirichlet series that is absolutely convergent in the half-plane 0 = {s : Re s < 0}.This paper provides the conditions under which the inequality Re F (j+1) (s) F (j) (s) > holds true for [0, 1), s 0 and j 0, where ( k ) is a sequence of positive numbers with 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 + ( 0 e 2s + 1 e s + 2 )w = 0.