Absolutely and completely monotonic functions and sequences

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

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

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 ${\displaystyle [0,\infty )}$ .

The following functions are absolutely monotonic in the specified regions.^{[8]: 142–143}

1. ${\displaystyle f(x)=c}$ , where ${\displaystyle c}$  a non-negative constant, in the region ${\displaystyle -\infty <x<\infty }$ 
2. ${\displaystyle f(x)=\sum _{k=0}^{\infty }a_{k}x^{k}}$ , where ${\displaystyle a_{k}\geq 0}$  for all ${\displaystyle k}$ , in the region ${\displaystyle 0\leq x<\infty }$ 
3. ${\displaystyle f(x)=-\log(-x)}$  in the region ${\displaystyle -1\leq x<0}$ 
4. ${\displaystyle f(x)=\sin ^{-1}x}$  in the region ${\displaystyle 0\leq x\leq 1}$ 

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

${\displaystyle \Delta ^{k}\mu _{n}\geq 0,\quad n,k=0,1,2,\ldots }$ 

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

A sequence ${\displaystyle \{\mu _{n}\}_{n=0}^{\infty }}$  is called a completely monotonic sequence if its elements are non-negative and its successive differences are alternately non-positive and non-negative,^[8]: 101 that is, if

${\displaystyle (-1)^{k}\Delta ^{k}\mu _{n}\geq 0,\quad n,k=0,1,2,\ldots }$ 

The sequences ${\displaystyle \left\{{\frac {1}{n+1}}\right\}_{0}^{\infty }}$  and ${\displaystyle \{c^{n}\}_{0}^{\infty }}$  for ${\displaystyle 0\leq c\leq 1}$  are completely monotonic sequences.

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 ${\displaystyle [a,b]}$  can be extended to an analytic function on the interval defined by ${\displaystyle |x-a|<b-a}$ .
• A function that is absolutely monotonic on ${\displaystyle [0,\infty )}$  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.
• The big Bernshtein theorem: A function ${\displaystyle f(x)}$  that is absolutely monotonic on ${\displaystyle (-\infty ,0]}$  can be represented there as a Laplace integral in the form

${\displaystyle f(x)=\int _{0}^{\infty }e^{xt}\,d\mu (t)}$ 

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

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

${\displaystyle \mu _{n}=\int _{0}^{1}t^{n}\,d\alpha (t),\quad n=0,1,2,\ldots }$ 

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.

• René L. Schilling, Renming Song and Zoran Vondraček (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. arXiv:1211.0900. (Chapter: "Completely Monotone Functions: A Digest")
• Arvind Mahajan and Dieter K Ross (1982). "A note on completely and absolutely monotonic functions" (PDF). Canadian Mathematical Bulletin. 25 (2): 143–148. doi:10.4153/CMB-1982-020-x. Retrieved 28 December 2023.
• Senlin Guo, Hari M Srivastava and Necdet Batir (2013). "A certain class of completely monotonic sequences" (PDF). Advances in Difference Equations. 294: 1–9. doi:10.1186/1687-1847-2013-294. Retrieved 29 December 2023.
• Yajima, S.; Ibaraki, T. (March 1968). "A Theory of Completely Monotonic Functions and its Applications to Threshold Logic". IEEE Transactions on Computers. C-17 (3): 214–229. doi:10.1109/tc.1968.229094.

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