Completely monotone function

A function $f : (0, \infty) \to [0, \infty)$ is said to be completely monotone (totally monotone, completely monotonic, totally monotonic) if $(-1)^k f^{(k)} \ge 0$ for $x > 0$ and $k = 0, 1, 2, ...$

Representation

By Bernstein's theorem, a function $f$ is completely monotone if and only if it is the Laplace transform of a nonnegative measure, \[ f(z) = \int_{[0, \infty)} e^{-s z} m(\mathrm d s) . \] Here $m$ is an arbitrary Radon measure such that the above integral is finite for all $z > 0$.

Examples

The following functions of $z$ are completely monotone:

• $z^s$ for $s \le 0$,
• $e^{-t z}$ for $t \ge 0$,
• $\log(1 + \frac{1}{z})$,
• $e^{1 / z}$.

Properties

If $f_1, f_2$ are completely monotone and $c > 0$, then also $c f_1$, $f_1 + f_2$, $f_1 f_2$ are completely monotone.

If $f$ is completely monotone and $g$ is a Bernstein function, then $f \circ g$ is completely monotone.

References

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