Difference of convex functions

Question

Let $f_1(x)$, $f_2(x)$, and $f_3(x)$ all be decreasing and convex functions, $x\geq 0$. Furthermore, let's assume $1\geq f_1(x)> f_2(x)> f_3(x)\geq 0$ for all $x\geq 0$. Let $g(x)=\frac{f_1(x)-f_2(x)}{f_1(x)-f_3(x)}$. If we know the difference between $f_1(x)-f_2(x)$ and $f_1(x)-f_3(x)$ is decreasing in $x$, e.g. $\left(f_1(x+\epsilon)-f_3(x+\epsilon)\right)-\left(f_1(x+\epsilon)-f_2(x+\epsilon)\right)\leq \left(f_1(x)-f_3(x)\right)-\left(f_1(x)-f_2(x)\right)$ for $\epsilon>0$, how does $g(x)$ behave with regard to $x$? What are some sufficient conditions that would make $g(x)$ monotonically increasing or decreasing in $x$?