Absolutely and completely monotonic functions and sequences

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. 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. 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. Such functions occur in other areas of mathematics such as probability theory, numerical analysis, and elasticity.

Definitions

Functions

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

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

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 [math]\displaystyle{ [0, \infty) }[/math].

Examples

The following functions are absolutely monotonic in the specified regions.

1. [math]\displaystyle{ f(x)=c }[/math], where [math]\displaystyle{ c }[/math] a non-negative constant, in the region [math]\displaystyle{ -\infty \lt x \lt \infty }[/math]
2. [math]\displaystyle{ f(x) = \sum_{k=0}^\infty a_k x^k }[/math], where [math]\displaystyle{ a_k\ge 0 }[/math] for all [math]\displaystyle{ k }[/math], in the region [math]\displaystyle{ 0\le x \lt \infty }[/math]
3. [math]\displaystyle{ f(x) = -\log (-x) }[/math] in the region [math]\displaystyle{ -1 \le x \lt 0 }[/math]
4. [math]\displaystyle{ f(x)=\sin^{-1}x }[/math] in the region [math]\displaystyle{ 0\le x\le 1 }[/math]

Sequences

A sequence [math]\displaystyle{ \{\mu_n\}_{n=0}^\infty }[/math] is called an absolutely monotonic sequence if its elements are non-negative and its successive differences are all non-negative, that is, if

[math]\displaystyle{ \Delta^k\mu_n\ge 0, \quad n,k = 0,1,2,\ldots }[/math]

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

A sequence [math]\displaystyle{ \{\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, that is, if

[math]\displaystyle{ (-1)^k\Delta^k\mu_n\ge 0, \quad n,k = 0,1,2,\ldots }[/math]

Examples

The sequences [math]\displaystyle{ \left\{\frac{1}{n+1}\right\}_0^\infty }[/math] and [math]\displaystyle{ \{c^n\}_0^\infty }[/math] for [math]\displaystyle{ 0\le c \le 1 }[/math] 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 [math]\displaystyle{ [a,b] }[/math] can be extended to an analytic function on the interval defined by [math]\displaystyle{ |x-a| \lt b-a }[/math].
• A function that is absolutely monotonic on [math]\displaystyle{ [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 big Bernshtein theorem: A function [math]\displaystyle{ f(x) }[/math] that is absolutely monotonic on [math]\displaystyle{ (-\infty,0] }[/math] can be represented there as a Laplace integral in the form

[math]\displaystyle{ f(x) = \int_0^\infty e^{xt}\, d\mu(t) }[/math]

where [math]\displaystyle{ \mu(t) }[/math] is non-decreasing and bounded on [math]\displaystyle{ [0,\infty) }[/math].

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

[math]\displaystyle{ \mu_n = \int_0^1 t^n \, d\alpha(t), \quad n=0,1,2,\ldots }[/math]

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

Further reading

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

• Rene´ L. Schilling, Renming Song and Zoran Vondracˇek (2010). Bernstein Functions Theory and Applications. De Gruyter. pp. 1-10. ISBN 978-3-11-021530-4.  (Chapter 1 Laplace transforms and completely monotone functions)
• D. V. Widder (1946). The Laplace Transform. Princeton University Press.  See Chapter III The Moment Problem (pp. 100 - 143) and Chapter IV Absolutely and Completely Monotonic Functions (pp. 144 - 179).
• Milan Merkle (2014). Analytic Number Theory, Approximation Theory, and Special Functions. Springer. pp. 347-364. https://arxiv.org/pdf/1211.0900.pdf. Retrieved 28 December 2023.  (Chpter: "Completely Monotone Functions: A Digest")
• Arvind Mahajan and Dieter K Ross (1982). "A note on completely and absolutely monotonic functions". Canadian Mathematical Bulletin 25 (2): 143-148. https://www.cambridge.org/core/services/aop-cambridge-core/content/view/414A95BC1647B805D081DF734DF02C8F/S0008439500064055a.pdf/a-note-on-completely-and-absolutely-monotone-functions.pdf. Retrieved 28 December 2023. 
• Senlin Guo, Hari M Srivastava and Necdet Batir (2013). "A certain class of completely monotonic sequences". Advances in Differential Equations 294: 1-9. https://advancesincontinuousanddiscretemodels.springeropen.com/counter/pdf/10.1186/1687-1847-2013-294.pdf. Retrieved 29 December 2023. 
• Shzo Yajima and Toshihide Ibaraki (March 1969). "A Theory of Completely Monotonic Functions and its Applications to Threshold Logic". IEEE Transactions on Computers 17 (3). 

See also

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