Suppose for some value of $z$ I have an inequality
$$f \left( \phi(z) + \gamma_1(z)\right) - f\left(\phi(z) + \gamma_2(z)\right) < c,$$
where $\phi$ is decreasing in $z$ and $\gamma_i$ is increasing in $z$ for $i=1,2$. All values are positive.
I am interested in how this inequality is preserved as we vary $z$. Specifically, suppose this inequality holds for some $z_0$. When will it also hold for any $z<z_0$?
For instance, when $f$ and $\gamma_i$ are linear, decreasing $z$ should preserve the inequality. However, what if $f$ is concave? My conjecture is that the inequality is still preserved in such a case. What methods can I use to prove this and understand how the inequality is preserved for other assumptions on $f,\phi,\gamma_i$?
Even if we assume that $f(x)=x$ for each $x$, the required inequality transforms to the inequality
$$\gamma_1(z) - \gamma_2(z) < c$$
for increasing functions $\gamma_i$. If it holds for $z=z_0$ then it may fails for some $z<0$. For instance, let $\gamma_1(z)=z$, $\gamma_2(z)=2z$, $z_0=0$, $z=-1$, and $c=1$.