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.[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 certain Jacobi series.[6] Such functions occur in other areas of mathematics such as probability theory, numerical analysis, and elasticity.[7]

Definitions

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Functions

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A real valued function   defined over an interval   in the real line is called an absolutely monotonic function if it has derivatives   of all orders   and   for all   in  .[1] The function   is called a completely monotonic function if   for all   in  .[1]

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

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  .

Examples

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

  1.  , where   a non-negative constant, in the region  
  2.  , where   for all  , in the region  
  3.   in the region  
  4.   in the region  

Sequences

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A sequence   is called an absolutely monotonic sequence if its elements are non-negative and its successive differences are all non-negative, that is, if

 

where  .

A sequence   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

 

Examples

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The sequences   and   for   are completely monotonic sequences.

Some important properties

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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   can be extended to an analytic function on the interval defined by  .
  • A function that is absolutely monotonic on   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   that is absolutely monotonic on   can be represented there as a Laplace integral in the form
 
where   is non-decreasing and bounded on  .
  • A sequence   is completely monotonic if and only if there exists an increasing function   on   such that
 
The determination of this function from the sequence is referred to as the Hausdorff moment problem.

Further reading

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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.

See also

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References

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