Proposition 8.7.8103D0E7E46B0

Proof. Choose $b>1$ so that $(1-b^{-1})^ m>c$, and put $p=P(bu-(b-1)v)$ and $z=b^{-1}p+(1-b^{-1})v\leq u$. On $C=\{ p=bu-(b-1)v\} $ the contact comparison gives

Restricting to $\{ u<v\} $ and using the hypothesis shows that $H_ m(p)$ vanishes on $C\cap \{ u<v\} $. Its support lies in $C$, where $p<v$ is equivalent to $u<v$. Thus $H_ m(p)$ vanishes on $\{ p<v\} $. The preceding domination proposition gives $p\geq v$. Since $p\leq bu-(b-1)v$, it follows that $u\geq v$. $\square$