Subadditive function

Remarks

Example: $f(x) = \sqrt{x}$ on $[0, \infty)$ is subadditive. Check: both sides of $\sqrt{x + y} \le \sqrt{x} + \sqrt{y}$ are nonnegative, and squaring gives $x + y \le x + y + 2\sqrt{xy}$, which is true.

Non-example: $f(x) = x^2$ on $[0, \infty)$ is not subadditive, since $f(2) = 4 > 2 = f(1) + f(1)$.

The defining condition is a functional inequality. A subadditive sequence is a subadditive function on the positive integers, $a_{m+n} \le a_m + a_n$. Every concave function $f \colon [0, \infty) \to \R$ with $f(0) \ge 0$ is subadditive.