Reversibly greater downside risk aversion

It is shown that, for the Arrow–Pratt measure $$r_u=-u''/u'$$ r u = - u ′ ′ / u ′ and the third-order measure $$D_u=u'''/u'-3r_u^2,$$ D u = u ′ ′ ′ / u ′ - 3 r 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.