In my research there are two CDFs, $G(x)$, $H(x)$ which support is [0,1]. The CDFs are twice differentiable and $G(x) > H(x) > x$ on (0,1). -> stochatical dominance.
My crucial condition is
$\frac{g(x)}{h(x)}>\frac{1-G(x)}{1-H(x)}$
My questions is
is $\frac{g(x)}{h(x)}>\frac{1-G(x)}{1-H(x)}$ true if $G(x) > H(x) > x$ on (0,1)?
Inequalities about functions hardly imply inequalities about their derivatives. If we consider $$ G(x)=x+\frac{\sin^2(5\pi x)}{5\pi},\qquad H(x)=x+\frac{\sin^2(10\pi x)}{20\pi} $$ then both $G(x)$ and $H(x)$ are non-negative and weakly increasing on $(0,1)$, with $G(x)\geq H(x)\geq x$, but $\frac{G'(x)}{H'(x)}-\frac{1-G(x)}{1-H(x)}$ has a lot of changes of sign over $(0,1)$.
On the other hand the inequality $\frac{g(x)}{h(x)}\geq\frac{1-G(x)}{1-H(x)}$ stands some chance to hold if $G(x)> H(x) > x$ on $(0,1)$ and both $G(x)$ and $H(x)$ are concave on $[0,1]$. It appears to be true for the test functions $F(x)=x^{\alpha}(2-x^{\alpha})$ with $\alpha\in(0,1)$, for instance.