I have a specific problem that Ive generalized here for simplicity.
Let $F(x)=\int^{g(x)}_0h(x,y)dy $
Suppose $F(0)=0$ (with $g(0)>0$)
Now suppose that $h$ is increasing in $y$. Then, it follows that:
$F(x) \leq g(x) h(x,g(x)) $
Does it therefore have to be the case that $h(0,g(0)) = 0$?