---
tag: QQSPWH4NG9
kind: concept
name: Absolutely monotonic function
msc: [26A48]
specializes: [0V33BNKZ9S]
sources:
  - url: https://en.wikipedia.org/wiki/Bernstein%27s_theorem_on_monotone_functions
---

## Summary

An absolutely monotonic function on an interval is a [[smooth function|0V33BNKZ9S]] 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|ZBS1JA62QP]] as mixtures of exponential functions.

## Explanations

### L1 human

The function is nonnegative, increasing, convex, and so on for every higher derivative.
The exponential function $e^x$ is the main example: all its derivatives equal $e^x > 0$.
A polynomial with nonnegative coefficients is absolutely monotonic on $[0, \infty)$.
A power series with nonnegative coefficients is absolutely monotonic on $[0, r)$, where $r$ is its radius of convergence.