Absolutely and completely monotonic functions and sequences

<h2 id="definitions">Definitions</h2>
<h3>Functions</h3>
<p>A real valued function 
  
    
      
        f
        (
        x
        )
      
    
    {\displaystyle f(x)}
  
 defined over an interval 
  
    
      
        I
      
    
    {\displaystyle I}
  
 in the real line is called an absolutely monotonic function if it has derivatives 
  
    
      
        
          f
          
            (
            n
            )
          
        
        (
        x
        )
      
    
    {\displaystyle f^{(n)}(x)}
  
 of all orders 
  
    
      
        n
        =
        0
        ,
        1
        ,
        2
        ,
        …
      
    
    {\displaystyle n=0,1,2,\ldots }
  
 and 
  
    
      
        
          f
          
            (
            n
            )
          
        
        (
        x
        )
        ≥
        0
      
    
    {\displaystyle f^{(n)}(x)\geq 0}
  
 for all 
  
    
      
        x
      
    
    {\displaystyle x}
  
 in 
  
    
      
        I
      
    
    {\displaystyle I}
  
.<a class="footnote-ref" id="fnref:8" href="#fn:8"><sup>8</sup></a> The function 
  
    
      
        f
        (
        x
        )
      
    
    {\displaystyle f(x)}
  
 is called a completely monotonic function if 
  
    
      
        (
        −
        1
        
          )
          
            n
          
        
        
          f
          
            (
            n
            )
          
        
        (
        x
        )
        ≥
        0
      
    
    {\displaystyle (-1)^{n}f^{(n)}(x)\geq 0}
  
 for all 
  
    
      
        x
      
    
    {\displaystyle x}
  
 in 
  
    
      
        I
      
    
    {\displaystyle I}
  
.<a class="footnote-ref" id="fnref:9" href="#fn:9"><sup>9</sup></a>
</p><p>The two notions are mutually related. The function 
  
    
      
        f
        (
        x
        )
      
    
    {\displaystyle f(x)}
  
 is completely monotonic if and only if 
  
    
      
        f
        (
        −
        x
        )
      
    
    {\displaystyle f(-x)}
  
  is absolutely monotonic on  
  
    
      
        −
        I
      
    
    {\displaystyle -I}
  
 where 
  
    
      
        −
        I
      
    
    {\displaystyle -I}
  
 the interval obtained by reflecting 
  
    
      
        I
      
    
    {\displaystyle I}
  
 with respect to the origin. (Thus, if 
  
    
      
        I
      
    
    {\displaystyle I}
  
 is the interval 
  
    
      
        (
        a
        ,
        b
        )
      
    
    {\displaystyle (a,b)}
  
 then 
  
    
      
        −
        I
      
    
    {\displaystyle -I}
  
 is the interval 
  
    
      
        (
        −
        b
        ,
        −
        a
        )
      
    
    {\displaystyle (-b,-a)}
  
.)
</p><p>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
        ,
        ∞
        )
      
    
    {\displaystyle [0,\infty )}
  
.
</p>
<h4>Examples</h4>
<p>The following functions are absolutely monotonic in the specified regions.<a class="footnote-ref" id="fnref:10" href="#fn:10"><sup>10</sup></a>: 142–143 
</p>
<ol><li>
  
    
      
        f
        (
        x
        )
        =
        c
      
    
    {\displaystyle f(x)=c}
  
,  where 
  
    
      
        c
      
    
    {\displaystyle c}
  
 a non-negative constant, in the region 
  
    
      
        −
        ∞
        <
        x
        <
        ∞
      
    
    {\displaystyle -\infty <x<\infty }
  
</li>
<li>
  
    
      
        f
        (
        x
        )
        =
        
          ∑
          
            k
            =
            0
          
          
            ∞
          
        
        
          a
          
            k
          
        
        
          x
          
            k
          
        
      
    
    {\displaystyle f(x)=\sum _{k=0}^{\infty }a_{k}x^{k}}
  
, where 
  
    
      
        
          a
          
            k
          
        
        ≥
        0
      
    
    {\displaystyle a_{k}\geq 0}
  
 for all 
  
    
      
        k
      
    
    {\displaystyle k}
  
, in the region 
  
    
      
        0
        ≤
        x
        <
        ∞
      
    
    {\displaystyle 0\leq x<\infty }
  
</li>
<li>
  
    
      
        f
        (
        x
        )
        =
        −
        log
        ⁡
        (
        −
        x
        )
      
    
    {\displaystyle f(x)=-\log(-x)}
  
 in the region 
  
    
      
        −
        1
        ≤
        x
        <
        0
      
    
    {\displaystyle -1\leq x<0}
  
</li>
<li>
  
    
      
        f
        (
        x
        )
        =
        
          sin
          
            −
            1
          
        
        ⁡
        x
      
    
    {\displaystyle f(x)=\sin ^{-1}x}
  
 in the region 
  
    
      
        0
        ≤
        x
        ≤
        1
      
    
    {\displaystyle 0\leq x\leq 1}
  
</li></ol>
<h3>Sequences</h3>
<p>A sequence 
  
    
      
        {
        
          μ
          
            n
          
        
        
          }
          
            n
            =
            0
          
          
            ∞
          
        
      
    
    {\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 
</p>

Δ
          
            k
          
        
        
          μ
          
            n
          
        
        ≥
        0
        ,
        
        n
        ,
        k
        =
        0
        ,
        1
        ,
        2
        ,
        …
      
    
    {\displaystyle \Delta ^{k}\mu _{n}\geq 0,\quad n,k=0,1,2,\ldots }

<p>where 
  
    
      
        
          Δ
          
            k
          
        
        
          μ
          
            n
          
        
        =
        
          ∑
          
            m
            =
            0
          
          
            k
          
        
        (
        −
        1
        
          )
          
            m
          
        
        
          
            
              (
            
            
              k
              m
            
            
              )
            
          
        
        
          μ
          
            n
            +
            k
            −
            m
          
        
      
    
    {\displaystyle \Delta ^{k}\mu _{n}=\sum _{m=0}^{k}(-1)^{m}{k \choose m}\mu _{n+k-m}}
  
.  
</p><p>A sequence 
  
    
      
        {
        
          μ
          
            n
          
        
        
          }
          
            n
            =
            0
          
          
            ∞
          
        
      
    
    {\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,<a class="footnote-ref" id="fnref:11" href="#fn:11"><sup>11</sup></a>: 101  that is, if
</p>

(
        −
        1
        
          )
          
            k
          
        
        
          Δ
          
            k
          
        
        
          μ
          
            n
          
        
        ≥
        0
        ,
        
        n
        ,
        k
        =
        0
        ,
        1
        ,
        2
        ,
        …
      
    
    {\displaystyle (-1)^{k}\Delta ^{k}\mu _{n}\geq 0,\quad n,k=0,1,2,\ldots }

<h4>Examples</h4>
<p>The sequences 
  
    
      
        
          
            {
            
              
                1
                
                  n
                  +
                  1
                
              
            
            }
          
          
            0
          
          
            ∞
          
        
      
    
    {\displaystyle \left\{{\frac {1}{n+1}}\right\}_{0}^{\infty }}
  
 and 
  
    
      
        {
        
          c
          
            n
          
        
        
          }
          
            0
          
          
            ∞
          
        
      
    
    {\displaystyle \{c^{n}\}_{0}^{\infty }}
  
 for 
  
    
      
        0
        ≤
        c
        ≤
        1
      
    
    {\displaystyle 0\leq c\leq 1}
  
 are completely monotonic sequences.
</p>
<h2 id="some-important-properties">Some important properties</h2>
<p>Both the extensions and applications of the theory of absolutely monotonic functions derive from theorems. 
</p>
<ul><li>The little Bernshtein theorem: A function that is absolutely monotonic on a closed interval 
  
    
      
        [
        a
        ,
        b
        ]
      
    
    {\displaystyle [a,b]}
  
 can be extended to an analytic function on the interval defined by 
  
    
      
        
          |
        
        x
        −
        a
        
          |
        
        <
        b
        −
        a
      
    
    {\displaystyle |x-a|<b-a}
  
.</li>
<li>A function that is absolutely monotonic on 
  
    
      
        [
        0
        ,
        ∞
        )
      
    
    {\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.</li>
<li>The <a href="/facts/Bernstein%2527s_theorem_on_monotone_functions/RW16PpPy">big Bernshtein theorem</a>: A function 
  
    
      
        f
        (
        x
        )
      
    
    {\displaystyle f(x)}
  
 that is absolutely monotonic on 
  
    
      
        (
        −
        ∞
        ,
        0
        ]
      
    
    {\displaystyle (-\infty ,0]}
  
 can be represented there as a Laplace integral in the form</li></ul>

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

where 
  
    
      
        μ
        (
        t
        )
      
    
    {\displaystyle \mu (t)}
  
 is non-decreasing and bounded on 
  
    
      
        [
        0
        ,
        ∞
        )
      
    
    {\displaystyle [0,\infty )}
  
.
<ul><li>A sequence 
  
    
      
        {
        
          μ
          
            n
          
        
        
          }
          
            0
          
          
            ∞
          
        
      
    
    {\displaystyle \{\mu _{n}\}_{0}^{\infty }}
  
 is completely monotonic if and only if there exists an increasing function 
  
    
      
        α
        (
        t
        )
      
    
    {\displaystyle \alpha (t)}
  
 on 
  
    
      
        [
        0
        ,
        1
        ]
      
    
    {\displaystyle [0,1]}
  
 such that</li></ul>

μ
          
            n
          
        
        =
        
          ∫
          
            0
          
          
            1
          
        
        
          t
          
            n
          
        
        
        d
        α
        (
        t
        )
        ,
        
        n
        =
        0
        ,
        1
        ,
        2
        ,
        …
      
    
    {\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 <a href="/facts/Hausdorff_moment_problem/1LdQ8rur">Hausdorff moment problem</a>.
<h2 id="further-reading">Further reading</h2>
<p>The following is a selection from the large body of literature on absolutely/completely monotonic functions/sequences.
</p>
<ul><li>René L. Schilling, Renming Song and <a href="/facts/Zoran_Vondra%25C4%258Dek/klELH434">Zoran Vondraček</a> (2010). <i>Bernstein Functions Theory and Applications</i>. De Gruyter. pp. 1–10. <a href="/facts/ISBN_(identifier)/15AdSPa9">ISBN</a> 978-3-11-021530-4. (Chapter 1 Laplace transforms and completely monotone functions)</li>
<li>D. V. Widder (1946). <i>The Laplace Transform</i>. Princeton University Press. See Chapter III The Moment Problem (pp. 100 - 143) and Chapter IV Absolutely and Completely Monotonic Functions (pp. 144 - 179).</li>
<li>Milan Merkle (2014). <i>Analytic Number Theory, Approximation Theory, and Special Functions</i>. Springer. pp. 347–364. <a href="/facts/ArXiv_(identifier)/H6EtgnBe">arXiv</a>:<a href="https://arxiv.org/abs/1211.0900">1211.0900</a>. (Chapter: "Completely Monotone Functions: A Digest")</li>
<li>Arvind Mahajan and Dieter K Ross (1982). <a href="https://www.cambridge.org/core/services/aop-cambridge-core/content/view/414A95BC1647B805D081DF734DF02C8F/S0008439500064055a.pdf/a-note-on-completely-and-absolutely-monotone-functions.pdf">"A note on completely and absolutely monotonic functions"</a> (PDF). <i>Canadian Mathematical Bulletin</i>. 25 (2): 143–148. <a href="/facts/Doi_(identifier)/muM9Etpq">doi</a>:<a href="https://doi.org/10.4153%2FCMB-1982-020-x">10.4153/CMB-1982-020-x</a>. Retrieved 28 December 2023.</li>
<li>Senlin Guo, Hari M Srivastava and Necdet Batir (2013). <a href="https://advancesincontinuousanddiscretemodels.springeropen.com/counter/pdf/10.1186/1687-1847-2013-294.pdf">"A certain class of completely monotonic sequences"</a> (PDF). <i>Advances in Difference Equations</i>. 294: 1–9. <a href="/facts/Doi_(identifier)/muM9Etpq">doi</a>:<a href="https://doi.org/10.1186%2F1687-1847-2013-294">10.1186/1687-1847-2013-294</a>. Retrieved 29 December 2023.</li>
<li>Yajima, S.; Ibaraki, T. (March 1968). "A Theory of Completely Monotonic Functions and its Applications to Threshold Logic". <i>IEEE Transactions on Computers</i>. C-17 (3): 214–229. <a href="/facts/Doi_(identifier)/muM9Etpq">doi</a>:<a href="https://doi.org/10.1109%2Ftc.1968.229094">10.1109/tc.1968.229094</a>.</li></ul>
<h2 id="see-also">See also</h2>
<ul><li><a href="/facts/Bernstein%2527s_theorem_on_monotone_functions/RW16PpPy">Bernstein's theorem on monotone functions</a></li>
<li><a href="/facts/Hausdorff_moment_problem/1LdQ8rur">Hausdorff moment problem</a></li>
<li><a href="/facts/Monotonic_function/MjQK0YL2">Monotonic function</a></li>
<li><a href="/facts/Cyclical_monotonicity/vTxfXSbz">Cyclical monotonicity</a></li></ul>

<h2 id="references">References</h2>

<ol>
<li id="fn:1"><p>"Absolutely monotonic function". encyclopediaofmath.org. Encyclopedia of Mathematics. Retrieved 28 December 2023. <a href="https://encyclopediaofmath.org/wiki/Absolutely_monotonic_function" target="_blank">https://encyclopediaofmath.org/wiki/Absolutely_monotonic_function</a> <a href="#fnref:1" class="footnote-back-ref">↩</a></p></li>
<li id="fn:2"><p>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. <a href="/wiki/Doi_(identifier)" target="_blank">/wiki/Doi_(identifier)</a> <a href="#fnref:2" class="footnote-back-ref">↩</a></p></li>
<li id="fn:3"><p>S. Bernstein (1928). "Sur les fonctions absolument monotones". Acta Mathematica. 52: 1–66. doi:10.1007/BF02592679. <a href="https://doi.org/10.1007%2FBF02592679" target="_blank">https://doi.org/10.1007%2FBF02592679</a> <a href="#fnref:3" class="footnote-back-ref">↩</a></p></li>
<li id="fn:4"><p>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. <a href="https://www.pmf.ni.ac.rs/filomat-content/2017/31-2/31-2-7-1944.pdf" target="_blank">https://www.pmf.ni.ac.rs/filomat-content/2017/31-2/31-2-7-1944.pdf</a> <a href="#fnref:4" class="footnote-back-ref">↩</a></p></li>
<li id="fn:5"><p>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. <a href="https://downloads.hindawi.com/journals/jam/2014/768516.pdf" target="_blank">https://downloads.hindawi.com/journals/jam/2014/768516.pdf</a> <a href="#fnref:5" class="footnote-back-ref">↩</a></p></li>
<li id="fn:6"><p>R. Askey (1973). "Summability of Jacobi series". Transactions of the American Mathematical Society. 179: 71–84. doi:10.1090/S0002-9947-1973-0315351-7. <a href="/wiki/Doi_(identifier)" target="_blank">/wiki/Doi_(identifier)</a> <a href="#fnref:6" class="footnote-back-ref">↩</a></p></li>
<li id="fn:7"><p>William Feller (1971). An Introduction to Probability Theory and Its Applications, Vol. 2 (3 ed.). New York: Wiley. ISBN 9780471257097. OCLC 279852. <a href="9780471257097" target="_blank">9780471257097</a> <a href="#fnref:7" class="footnote-back-ref">↩</a></p></li>
<li id="fn:8"><p>"Absolutely monotonic function". encyclopediaofmath.org. Encyclopedia of Mathematics. Retrieved 28 December 2023. <a href="https://encyclopediaofmath.org/wiki/Absolutely_monotonic_function" target="_blank">https://encyclopediaofmath.org/wiki/Absolutely_monotonic_function</a> <a href="#fnref:8" class="footnote-back-ref">↩</a></p></li>
<li id="fn:9"><p>"Absolutely monotonic function". encyclopediaofmath.org. Encyclopedia of Mathematics. Retrieved 28 December 2023. <a href="https://encyclopediaofmath.org/wiki/Absolutely_monotonic_function" target="_blank">https://encyclopediaofmath.org/wiki/Absolutely_monotonic_function</a> <a href="#fnref:9" class="footnote-back-ref">↩</a></p></li>
<li id="fn:10"><p>Widder, David Vernon (1946). The Laplace Transform. Princeton University Press. ISBN 9780486477558. OCLC 630478002. {{cite book}}: ISBN / Date incompatibility (help) <a href="9780486477558" target="_blank">9780486477558</a> <a href="#fnref:10" class="footnote-back-ref">↩</a></p></li>
<li id="fn:11"><p>Widder, David Vernon (1946). The Laplace Transform. Princeton University Press. ISBN 9780486477558. OCLC 630478002. {{cite book}}: ISBN / Date incompatibility (help) <a href="9780486477558" target="_blank">9780486477558</a> <a href="#fnref:11" class="footnote-back-ref">↩</a></p></li>
</ol>

Absolutely and completely monotonic functions and sequences open-in-new

Absolutely and completely monotonic functions and sequences