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  "body": "\n## Definition\n\nLet $(S, \\cdot)$ be a [[semigroup|0XMEM406JM]].\nA function $f \\colon S \\to [0, \\infty)$ is *submultiplicative* if\n$$f(x \\cdot y) \\le f(x) \\, f(y) \\quad \\text{for all } x, y \\in S.$$\n\n## Remarks\n\nFor $S = (\\N_{\\ge 1}, +)$ 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.\nExample: on the multiplicative semigroup $(0, \\infty)$, let $f(x) = x + 1$.\nThen $f(xy) = xy + 1 \\le xy + x + y + 1 = f(x) f(y)$, since $x, y \u003e 0$.\nExample: for a square matrix $A$ and a [[matrix norm|54T6Y6DSY4]] with $\\norm{BC} \\le \\norm{B} \\norm{C}$, the sequence $a_n = \\norm{A^n}$ is submultiplicative.\nNon-example: the constant function $f = 1/2$ on any semigroup fails the condition, since $1/2 \u003e 1/4 = f(x) f(y)$.\nIf $f$ takes only positive values, then $f$ is submultiplicative exactly when $\\log f$ satisfies $\\log f(xy) \\le \\log f(x) + \\log f(y)$.\n",
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