I'm trying to find two nondecreasing functions $f$ and $g$ such that $$(\alpha x - \beta y)(f(x) - g(y)) \ge 0$$ for every $x \ge 0$, $y \ge 0$ and arbitrary constants $\alpha >0$ and $\beta > 0$. The functions $f$ and $g$ can equal ($f=g$), if it helps, but $f$ and $g$ cannot explicitly contain $\alpha$ and $\beta$.
It is obviously true that $(\alpha x - \beta y)(f(\alpha x) - f(\beta y)) \ge 0$ for any $f$ nondecreasing, but this is exactly what I don't want.
Maybe there is a trivial solution but I can't see it. Thank you.