Absolutely and completely monotonic functions and sequences explained

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.^{[1] [2] [3]} 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.^{[4] [5]} 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 certainJacobi series.^[6] Such functions occur in other areas of mathematics such as probability theory, numerical analysis, and elasticity.^[7]

Definitions

Functions

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 ,\ldots and f ^{(n)} (x) \ge 0 for all x in I .^[1] The function f(x) is called a completely monotonic function if ( -1 ) ^{nf} ^{(n)} (x) \ge 0 for all x in I .^[1]

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 , inf ty) .

Examples

The following functions are absolutely monotonic in the specified regions.^[8]

f(x) = c , where c a non-negative constant, in the region -inf ty <x < inf ty f(x) = \sum ^{inf ty} _{k =0} a _{k} x ^{k} , where a _{k\ge} 0 for all k , in the region 0 \le x < inf ty f(x) = -log ( - x) in the region -1 \le x < 0 f(x) = \sin ^{-1} x in the region 0 \le x\le 1

Sequences

A sequence

\{\mu _{n\}} ^{inf ty} ^{n =0} is called an absolutely monotonic sequence if its elements are non-negative and its successive differences are all non-negative, that is, if \Delta ^{k\mu} _{n\ge} 0 , n,k = 0 , 1 , 2 ,\ldots

where

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

A sequence

\{\mu _{n\}} ^{inf ty} ^{n =0} is called a completely monotonic sequence if its elements are non-negative and its successive differences are alternately non-positive and non-negative,^[8] that is, if( -1 ) ^{k\Delta} ^{k\mu} ^{n\ge} 0 , n,k = 0 , 1 , 2 ,\ldots

Examples

The sequences

\left\{ (1)/(n +1) \right\} ^{inf ty} _{0} and \{c ^{inf ty} _{0} for 0 \le c \le 1 are completely monotonic sequences.

Some important properties

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

• The little Bernshtein 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 |x - a| < b - a .

• A function that is absolutely monotonic on

[ 0 , inf ty) 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 big Bernshtein theorem: A function

f(x) that is absolutely monotonic on ( -inf ty, 0 ] can be represented there as a Laplace integral in the form f(x) = \int ^{inf ty} _{0} e ^{xt} d\mu(t)

where

\mu(t) is non-decreasing and bounded on [ 0 , inf ty) .

• A sequence

\{\mu _{n\}} ^{inf ty} ^{0} is completely monotonic if and only if there exists an increasing function \alpha(t) on [ 0 , 1 ] such that \mu _{n} = \int ^{1} _{0} t ^{n} d\alpha(t), n =0 , 1 , 2 ,\ldots

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

Further reading

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

• Book: . Bernstein Functions Theory and Applications . 2010 . De Gruyter . 978-3-11-021530-4 . 1–10. (Chapter 1 Laplace transforms and completely monotone functions)
• Book: D. V. Widder . The Laplace Transform . 1946 . Princeton University Press. See Chapter III The Moment Problem (pp. 100 - 143) and Chapter IV Absolutely and Completely Monotonic Functions (pp. 144 - 179).
• Book: Milan Merkle . Analytic Number Theory, Approximation Theory, and Special Functions . 2014 . Springer . 347–364 . 1211.0900 . (Chpter: "Completely Monotone Functions: A Digest")
• Arvind Mahajan and Dieter K Ross . A note on completely and absolutely monotonic functions . Canadian Mathematical Bulletin . 1982 . 25 . 2 . 143–148 . 10.4153/CMB-1982-020-x . 28 December 2023.
• Senlin Guo, Hari M Srivastava and Necdet Batir . A certain class of completely monotonic sequences . Advances in Difference Equations . 2013 . 294 . 1–9 . 10.1186/1687-1847-2013-294 . free . 29 December 2023.
• Yajima . S. . Ibaraki . T. . A Theory of Completely Monotonic Functions and its Applications to Threshold Logic . IEEE Transactions on Computers . March 1968 . C-17 . 3 . 214–229 . 10.1109/tc.1968.229094.

See also

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

Notes and References

1. Web site: Absolutely monotonic function . encyclopediaofmath.org . Encyclopedia of Mathematics . 28 December 2023.
2. S. Bernstein . Sur la définition et les propriétés des fonctions analytique d'une variable réelle . Mathematische Annalen . 1914 . 75 . 4 . 449–468. 10.1007/BF01563654 .
3. S. Bernstein . Sur les fonctions absolument monotones . Acta Mathematica . 1928 . 52 . 1–66. 10.1007/BF02592679 .
4. Senlin. Guo . Some Properties of Functions Related to Completely Monotonic Functions . Filomat . 2017 . 31 . 2 . 247–254 . 10.2298/FIL1702247G . 29 December 2023.
5. Senlin. Guo. Andrea. Laforgia. Necdet. Batir. Qiu-Ming . Luo . Completely Monotonic and Related Functions: Their Applications . Journal of Applied Mathematics . 2014 . 2014 . 1–3 . 10.1155/2014/768516 . free . 28 December 2023.
6. R. Askey . Summability of Jacobi series . Transactions of the American Mathematical Society . 1973 . 179 . 71–84. 10.1090/S0002-9947-1973-0315351-7 .
7. Book: William Feller . An Introduction to Probability Theory and Its Applications, Vol. 2 . 1971 . Wiley . New York . 3. 279852. 9780471257097.
8. Book: Widder, David Vernon. The Laplace Transform . 1946. 9780486477558. 630478002 . Princeton University Press .