Reversibly greater downside risk aversion

It is shown that, for the Arrow-Pratt measure r(u) = -u"/u' and the third-order measure D-u = u"'/u' - 3r(u)(2), 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.