Absolutely and completely monotonic functions and sequences: Difference between revisions

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In mathematics, the notions of an '''absolutely monotonic function''' and a '''completely monotonic function''' are two very closely related concepts. Both imply very strong monotonicity properties. Both types of functions have derivatives of all orders. In the case of an absolutely monotonic function, the function as well as its derivatives of all orders must be non-negative in its ___domain of definition which would imply that the function as well as its derivatives of all orders are monotonically increasing functions in the ___domain of definition. In the case of a completely monotonic function, the function and its derivatives must be alternately non-negative and non-positive in its ___domain of definition which would imply that function and its derivatives are alternately monotonically increasing and monotonically decreasing functions.

Such functions were first studied by S. Bernshtein in 1914 and the terminology is also due to him.<ref name="Enc">{{cite web |title=Absolutely monotonic function |url=https://encyclopediaofmath.org/wiki/Absolutely_monotonic_function |website=encyclopediaofmath.org |publisher=Encyclopedia of Mathematics |access-date=28 December 2023}}</ref><ref>{{cite journal |last1=S. Bernstein |title=Sur la définition et les propriétés des fonctions analytique d'une variable réelle |journal=Mathematische Annalen |date=1914 |volume=75 |issue=4 |pages=449-468449–468|doi=10.1007/BF01563654 }}</ref><ref>{{cite journal |last1=S. Bernstein |title=Sur les fonctions absolument monotones |journal=Acta Mathematica |date=1928 |volume=52 |pages=11–66|doi=10.1007/BF02592679 |doi-66access=free }}</ref> There are several other related notions like the concepts of almost completely monotonic function, logarithmically completely monotonic function, strongly logarithmically completely monotonic function, strongly completely monotonic function and almost strongly completely monotonic function.<ref>{{cite journal |last1first=Senlin|last= Guo |title=Some Properties of Functions Related to Completely Monotonic Functions |journal=Filomat |date=2017 |volume=31 |issue=2 |pages=247247–254 - 254|doi=10.2298/FIL1702247G |url=https://www.pmf.ni.ac.rs/filomat-content/2017/31-2/31-2-7-1944.pdf |access-date=29 December 2023}}</ref><ref>{{cite journal |last1first1=Senlin|last1= Guo,|first2= Andrea|last2= Laforgia,|first3= Necdet|last3= Batir and|first4= Qiu-Ming |last4=Luo |title=Completely Monotonic and Related Functions: Their Applications |journal=Journal of Applied Mathematics |date=2014 |volume=2014 |pages=11–3 |doi=10.1155/2014/768516 |doi-3access=free |url=https://downloads.hindawi.com/journals/jam/2014/768516.pdf |access-date=28 December 2023}}</ref> Another related concept is that of a '''completely/absolutely monotonic sequence'''. This notion was introduced by Hausdorff in 1921.

The notions of completely and absolutely monotone function/sequence play an important role in several areas of mathematics. For example, in classical analysis they occur in the proof of the positivity of integrals involving Bessel functions or the positivity of Cesàro means of certain

Jacobi series.<ref>{{cite journal |last1=R. Askey |title=Summability of Jacobi series |journal=Transactions of the American Mathematical Society |date=1973 |volume=179 |pages=71 71–84|doi=10.1090/S0002-9947-1973-0315351-7 84}}</ref> Such functions occur in other areas of mathematics such as probability theory, numerical analysis, and elasticity.<ref>{{cite book |last1=William Feller |title=An Introduction to Probability Theory and Its Applications, Vol. 2 |date=1971 |publisher=Wiley |___location=New York |edition=3|oclc=279852|isbn=9780471257097}}</ref>

==Definitions==

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====Examples====

The following functions are absolutely monotonic in the specified regions.<ref name=dvw>{{cite book |last1first=David Vernon|last= Widder |title=The Laplace Transform |date=1946|isbn=9780486477558|oclc=630478002 |publisher=Princeton University Press |pages=142 - 143}}</ref>{{rp|142-143}}

# <math>f(x)=c</math>, where <math> c</math> a non-negative constant, in the region <math> -\infty <x < \infty </math>

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where <math>\Delta^k\mu_n = \sum_{m=0}^k (-1)^m {k \choose m}\mu_{n+k-m}</math>.

A sequence <math>\{\mu_n\}_{n=0}^\infty</math> is called a completely monotonic sequence if its elements are non-negative and its successive differences are alternately non-positive and non-negative,<ref name=dvw/>{{cite book rp|last1=David Vernon Widder |title=The Laplace Transform |date=1946 |publisher=Princeton University Press |page=101}}</ref> that is, if

::<math>(-1)^k\Delta^k\mu_n\ge 0, \quad n,k = 0,1,2,\ldots </math>

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

* TheBernstein's little Bernshtein theorem: A function that is absolutely monotonic on a closed interval <math>[a,b]</math> can be extended to an analytic function on the interval defined by <math>|x-a| < b-a</math>.

* A function that is absolutely monotonic on <math>[0,\infty)</math> 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.

*The [[Bernstein's theorem on monotone functions|big Bernshtein theorem]]: A function <math>f(x)</math> that is absolutely monotonic on <math>(-\infty,0]</math> can be represented there as a Laplace integral in the form

::<math> f(x) = \int_0^\infty e^{xt}\, d\mu(t)</math>

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==Further reading==

The following is a random 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=1-101–10}} (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=347-364347–364 |urlarxiv=https://arxiv.org/pdf/1211.0900.pdf |access-date=28 December 2023}} (ChpterChapter: "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=143143–148 |doi=10.4153/CMB-1481982-020-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 DifferentialDifference Equations |date=2013 |volume=294 |pages=11–9 |doi=10.1186/1687-91847-2013-294 |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=Shzo Yajima and Toshihide|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 19691968 |volume=C-17 |issue=3 |pages=214–229 |doi=10.1109/tc.1968.229094}}

==See also==