Submultiplicative function

Remarks

For $S = (\N_{\ge 1}, +)$ the condition reads $a_{m+n} \le a_m a_n$ for a sequence $(a_n)_{n \ge 1}$, and one speaks of a submultiplicative sequence. Example: on the multiplicative semigroup $(0, \infty)$, let $f(x) = x + 1$. Then $f(xy) = xy + 1 \le xy + x + y + 1 = f(x) f(y)$, since $x, y > 0$. Example: for a square matrix $A$ and a matrix norm with $\norm{BC} \le \norm{B} \norm{C}$, the sequence $a_n = \norm{A^n}$ is submultiplicative. Non-example: the constant function $f = 1/2$ on any semigroup fails the condition, since $1/2 > 1/4 = f(x) f(y)$. If $f$ takes only positive values, then $f$ is submultiplicative exactly when $\log f$ satisfies $\log f(xy) \le \log f(x) + \log f(y)$.