For a convex function $f(\cdot)$ for $x_4 \ge x_3 \ge x_2 \ge x_1 \ge 0$ and $t,t^{'}\ge 0$ we are given that \begin{equation} \frac{f(x_4)-f(x_3)}{f(x_2)-f(x_1)} \ge \frac{f(x_4-t)-f(x_3-t^{'})}{f(x_2-t^{'})-f(x_1-t)}. \end{equation}
I need to prove that $t\ge t^{'}$.
For a linear function or a quadratic function this works well as we can prove that $t^{'}>t$ is impossible. However, I am not sure how to generalize this result for any convex function.
