Superadditive function

Remarks

Example: $f(x) = x^2$ on $[0, \infty)$ is superadditive. Check: $(x + y)^2 = x^2 + y^2 + 2xy \ge x^2 + y^2$, since $xy \ge 0$.

Non-example: $f(x) = \sqrt{x}$ on $[0, \infty)$ is not superadditive, since $\sqrt{2} < 2 = \sqrt{1} + \sqrt{1}$.

A function $f$ is superadditive if and only if $-f$ is subadditive. Every convex function $f \colon [0, \infty) \to \R$ with $f(0) \le 0$ is superadditive. For a superadditive sequence, $a_{m+n} \ge a_m + a_n$, the limit of $a_n / n$ exists in $(-\infty, \infty]$ and equals $\sup_n a_n / n$.