Let $F$ be a CDF with $f\equiv F'$ having a positive support (edit), i.e. $\text{supp}(f) \subseteq \mathbb{R}_+$. Then $Q\equiv F^{-1}$ is its quantile function and $q\equiv Q'$, where we know $q(p) = \frac{1}{f(Q(p))}$. I am trying to prove under which conditions (if any) $$ \frac{d}{d p } \frac{Q(p)}{pq(p)}<0, \quad \forall p \in(0,1).$$
I cannot find any CDF $F$ (under the positive support assumption) for which this result does not hold (I tried both log-concave and log-convex distributions, with different signs of skewness). However I am unable to prove it holds generally. Any thoughts?