Let $f$, defined on $\mathbb{R_+}$, be concave and increasing, and $g$, also defined on $\mathbb{R_+}$ be convex and increasing, with $f(0) = g(0) = 0$.
What kind of minimal conditions do we need on $f$ and $g$ to guarantee that $Argmax (f(x)-g(x)) \leq Argmax (f(\alpha x) - g(x))$ for any $\alpha \geq 1$.
Any help welcome.