Absolutely monotonic function

Summary

An absolutely monotonic function on an interval is a smooth function whose derivatives of every order, including order $0$, are nonnegative there. The mirror notion is a completely monotone function, whose derivatives alternate in sign. On the half-lines $(-\infty, 0)$ and $(0, \infty)$ respectively, both classes are described by Bernstein's theorem on monotone functions as mixtures of exponential functions.