Reversibly greater downside risk aversion

It is shown that, for the Arrow–Pratt measure ru=-u′′/u′\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$r_u=-u''/u'$$\end{document} and the third-order measure Du=u′′′/u′-3ru2,\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$D_u=u'''/u'-3r_u^2,$$\end{document} an increase in both risk-preference measures, when utility changes from u to v,  yields a strict partial ordering by greater downside risk aversion, in that v is then a risk-averse and downside risk-averse transformation of u. More decisively, the result is reversible and, so, a decrease in both measures yields an ordering of utility functions by less downside risk aversion.