In this paper we introduce some new classes of functions which are a translational regular asymptotic behavior. In this sense we continue the study of the translational regularly varying functions. This results are closely connected with the Karamata’s theory of regularly varying functions. On the other hand, in this paper we give some theorems of Tauberian nature via the translational regularly varying functions. Applications of new Tauberian theorems and a method of the Monotone Density theorem for Stieltjes transform are considered. This results are connection with the Karamata’s Tauberian theorems, with the Karamata’s Hauptsatz, as and with the classical statements of Hardy and Littlewood.