Absolutely and completely monotonic functions and sequences

A real valued function $f(x)$ defined over an interval $I$ in the real line is called an absolutely monotonic function if it has derivatives $f^\{(n)\}(x)$ of all orders $n=0,1,2,\backslash ldots$ and $f^\{(n)\}(x)\; \backslash ge\; 0$ for all $x$ in $I$. The function $f(x)$ is called a completely monotonic function if $(-1)^nf^\{(n)\}(x)\; \backslash ge\; 0$ for all $x$ in $I$.

The two notions are mutually related. The function $f(x)$ is completely monotonic if and only if $f(-x)$ is absolutely monotonic on $-I$ where $-I$ the interval obtained by reflecting $I$ with respect to the origin. (Thus, if $I$ is the interval $(a,b)$ then $-I$ is the interval $(-b,-a)$.)

In applications, the interval on the real line that is usually considered is the closed-open right half of the real line, that is, the interval $[0,\; \backslash infty)$.

The following functions are absolutely monotonic in the specified regions.{{cite book |first=David Vernon|last= Widder |title=The Laplace Transform |date=1946|isbn=9780486477558|oclc=630478002 |publisher=Princeton University Press }}{{rp|142-143}}

1. $f(x)=c$, where $c$ a non-negative constant, in the region
2. $f(x)\; =\; \backslash sum\_\{k=0\}^\backslash infty\; a\_k\; x^k$, where $a\_k\backslash ge\; 0$ for all $k$, in the region
3. $f(x)\; =\; -\backslash log\; (-x)$ in the region
4. $f(x)=\backslash sin^\{-1\}x$ in the region $0\backslash le\; x\backslash le\; 1$

A sequence $\backslash \{\backslash mu\_n\backslash \}\_\{n=0\}^\backslash infty$ is called an absolutely monotonic sequence if its elements are non-negative and its successive differences are all non-negative, that is, if

::$\backslash Delta^k\backslash mu\_n\backslash ge\; 0,\; \backslash quad\; n,k\; =\; 0,1,2,\backslash ldots$

where $\backslash Delta^k\backslash mu\_n\; =\; \backslash sum\_\{m=0\}^k\; (-1)^m\; \{k\; \backslash choose\; m\}\backslash mu\_\{n+k-m\}$.

A sequence $\backslash \{\backslash mu\_n\backslash \}\_\{n=0\}^\backslash infty$ is called a completely monotonic sequence if its elements are non-negative and its successive differences are alternately non-positive and non-negative,{{rp|101}} that is, if

::$(-1)^k\backslash Delta^k\backslash mu\_n\backslash ge\; 0,\; \backslash quad\; n,k\; =\; 0,1,2,\backslash ldots$

The sequences $\backslash left\backslash \{\backslash frac\{1\}\{n+1\}\backslash right\backslash \}\_0^\backslash infty$ and $\backslash \{c^n\backslash \}\_0^\backslash infty$ for $0\backslash le\; c\; \backslash le\; 1$ are completely monotonic sequences.

Both the extensions and applications of the theory of absolutely monotonic functions derive from theorems.

• Bernstein's little theorem: A function that is absolutely monotonic on a closed interval $[a,b]$ can be extended to an analytic function on the interval defined by .
• A function that is absolutely monotonic on $[0,\backslash infty)$ can be extended to a function that is not only analytic on the real line but is even the restriction of an entire function to the real line.
• Bernstein's theorem on monotone functions: A function $f(x)$ that is absolutely monotonic on $(-\backslash infty,0]$ can be represented there as a Laplace integral in the form

::$f(x)\; =\; \backslash int\_0^\backslash infty\; e^\{xt\}\backslash ,\; d\backslash mu(t)$

:where $\backslash mu(t)$ is non-decreasing and bounded on $[0,\backslash infty)$.

• A sequence $\backslash \{\backslash mu\_n\backslash \}\_0^\backslash infty$ is completely monotonic if and only if there exists an increasing function $\backslash alpha(t)$ on $[0,1]$ such that

::$\backslash mu\_n\; =\; \backslash int\_0^1\; t^n\; \backslash ,\; d\backslash alpha(t),\; \backslash quad\; n=0,1,2,\backslash ldots$

:The determination of this function from the sequence is referred to as the Hausdorff moment problem.

The following is a selection from the large body of literature on absolutely/completely monotonic functions/sequences.

• {{cite book |last1=René L. Schilling, Renming Song and Zoran Vondraček |title=Bernstein Functions Theory and Applications |date=2010 |publisher=De Gruyter |isbn=978-3-11-021530-4 |pages=1–10}} (Chapter 1 Laplace transforms and completely monotone functions)
• {{cite book |last1=D. V. Widder |title=The Laplace Transform |date=1946 |publisher=Princeton University Press}} See Chapter III The Moment Problem (pp. 100 - 143) and Chapter IV Absolutely and Completely Monotonic Functions (pp. 144 - 179).
• {{cite book |last1=Milan Merkle |title=Analytic Number Theory, Approximation Theory, and Special Functions |date=2014 |publisher=Springer |pages=347–364 |arxiv=1211.0900 }} (Chapter: "Completely Monotone Functions: A Digest")
• {{cite journal |last1=Arvind Mahajan and Dieter K Ross |title=A note on completely and absolutely monotonic functions |journal=Canadian Mathematical Bulletin |date=1982 |volume=25 |issue=2 |pages=143–148 |doi=10.4153/CMB-1982-020-x |url=https://www.cambridge.org/core/services/aop-cambridge-core/content/view/414A95BC1647B805D081DF734DF02C8F/S0008439500064055a.pdf/a-note-on-completely-and-absolutely-monotone-functions.pdf |access-date=28 December 2023}}
• {{cite journal |last1=Senlin Guo, Hari M Srivastava and Necdet Batir |title=A certain class of completely monotonic sequences |journal=Advances in Difference Equations |date=2013 |volume=294 |pages=1–9 |doi=10.1186/1687-1847-2013-294 |doi-access=free |url=https://advancesincontinuousanddiscretemodels.springeropen.com/counter/pdf/10.1186/1687-1847-2013-294.pdf |access-date=29 December 2023}}
• {{cite journal |last1=Yajima |first1=S. |last2=Ibaraki |first2=T. |title=A Theory of Completely Monotonic Functions and its Applications to Threshold Logic |journal=IEEE Transactions on Computers |date=March 1968 |volume=C-17 |issue=3 |pages=214–229 |doi=10.1109/tc.1968.229094}}

• Bernstein's theorem on monotone functions
• Hausdorff moment problem
• Monotonic function
• Cyclical monotonicity

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