We are given a positive, non-decreasing function $f$ defined on natural numbers with $f(0) = 0$. $f$ has a submodularity-like property:
$f(x+y) \leq f(x) + f(y) $
for all natural numbers $x$ and $y$.
Can we then show that for any natural numbers $x,y$, we have
$f(x) \geq \frac{x}{x+y}f(x+y)$?