PomeLemma 8.7.9.F28CED8C4FAF For continuous admissible $u,v$ and $a>0$, if
\[ e^{-av}H_ m(v)\geq e^{-au}H_ m(u), \]
then $u\geq v$.
Lemma 8.7.9F28CED8C4FAF
Proof. Fix $\delta >0$. On $\{ u<v-\delta \} $, the assumed inequality gives
\[ H_ m(u)\leq e^{a(u-v)}H_ m(v) \leq e^{-a\delta }H_ m(v-\delta ). \]
The constant $e^{-a\delta }$ is strictly smaller than one, so 8.7.8 applied to $u$ and $v-\delta $ gives $u\geq v-\delta $. Letting $\delta \downarrow 0$ proves the assertion. $\square$
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