Difference of convex functions over a convex function

Question

If $f(x)$ is a decreasing and convex in $x$, $f(x)>0$, $x\geq 0$, and $$g(x)=\frac{f(x+y)-f(x+z)}{f(x)}$$ for any $z\geq y \geq 0$. Can we make some assumptions or add some constraints to $f(x)$ so that $g(x)$ is decreasing in $x$?

Answer 1

It's not true. For example, suppose $f(x) = 1+\epsilon - x$ on the interval $[0, 1]$ with $y = \epsilon$, $z = 2\epsilon$, and $0 < \epsilon < 1/2$.
Then $g(0) = \frac{\epsilon}{1+\epsilon}$ while $g(1-2\epsilon) = 1/3$, and $g(0) < g(1-2\epsilon)$.