Let $X,Y$ be random variables with cdf's $F,G$ respectively (assume $X,Y$ nonnegative if necessary, the book I am reading is not clear about this hypothesis but from definitions it seems implicitly assumed). We say that $X<_* Y$ if $G^{-1}(F(x))/x$ is increasing for all $x\geq 0$. This is a partial order on distributions.
I would like to show a known result: that $X<_*Y$ if and only if for any $b>0$, $F(x)-G(bx)$ crosses zero at most once, with the sign change being from $-$ to $+$ if there is a crossing point. The book I am reading mentions this fact as 'easy to show'. It is indeed not too hard for $b=1$, which I can prove, but I cannot show the claim for $b\neq 1$. Thanks for any help or hint about how to get started.