Let $X$ be a non-negative random variable that is drawn from a cumulative distribution function $F(\cdot)$, pdf $f(\cdot)$ and mean $E[x]$. $k$, $c$, $v_l$ and $v_h$ $(v_h>v_l)$ are non-negative parameters, where $c < v_h-v_l$. For convenience, define $$a=\frac{v_lk}{v_h\gamma}$$ and $b=k/\gamma$.
I am interested in showing that the function $R$ is monotonically increasing in $v_l$ (or finding sufficient conditions on the distribution function that will lead to monotonicity).
$$R = v_h(k-\gamma\int_{a}^{b}F(x)dx),$$ where $\gamma \in [0,1]$ is defined implicitly by $$\frac{v_h-v_l}{c}\int_{0}^{a}xf(x)dx=E[x].$$
Intermediate results: (1) $\gamma$ is unimodal in $v_l$. Proving monotonicity is straightforward when $\gamma$ is increasing in $v_l$, but is it so when $\gamma$ decreases in $v_l$? (2) $R$ monotonically increases in $v_l$ when $X$ is drawn from a uniform or Pareto distribution with $\alpha > 1$. I also numerically verified that it is the case when $X$ is drawn from a Gamma distribution.
