Hazard function and characterizations on distribution tails of nonnegative random variables

Some equivalent conditions on the classes of lighted-tailed and heavily heavy-tailed and lightly heavy-tailed d. f. s are introduced. The limit behavior of \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document} $$x^a \bar F(x)$$ \end{document} and \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document} $$e^{\lambda x} \bar F(x)$$ \end{document} are discussed. Some properties of the subclass DKc and subclass DKc are obtained.