Bernstein's theorem on monotone functions

<h2 id="bernstein-functions">Bernstein functions</h2>
<p>Nonnegative functions whose <a href="/facts/Derivative/7Ito70Dh">derivative</a> is completely monotone are called <i>Bernstein functions</i>.  Every Bernstein function has the <a href="/facts/L%25C3%25A9vy%25E2%2580%2593Khintchine_representation/m9A3YQW8">Lévy–Khintchine representation</a>:

f
        (
        t
        )
        =
        a
        +
        b
        t
        +
        
          ∫
          
            0
          
          
            ∞
          
        
        
          (
          
            1
            −
            
              e
              
                −
                t
                x
              
            
          
          )
        
        μ
        (
        d
        x
        )
        ,
      
    
    {\displaystyle f(t)=a+bt+\int _{0}^{\infty }\left(1-e^{-tx}\right)\mu (dx),}

where 
  
    
      
        a
        ,
        b
        ≥
        0
      
    
    {\displaystyle a,b\geq 0}
  
 and 
  
    
      
        μ
      
    
    {\displaystyle \mu }
  
 is a measure on the positive real half-line such that

∫
          
            0
          
          
            ∞
          
        
        
          (
          
            1
            ∧
            x
          
          )
        
        μ
        (
        d
        x
        )
        <
        ∞
        .
      
    
    {\displaystyle \int _{0}^{\infty }\left(1\wedge x\right)\mu (dx)<\infty .}

</p>
<h2 id="see-also">See also</h2>
<ul><li><a href="/facts/Absolutely_and_completely_monotonic_functions_and_sequences/MsM2rSK4">Absolutely and completely monotonic functions and sequences</a></li></ul>

<ul><li><a href="/facts/S._N._Bernstein/m7aCQ2Y6">S. N. Bernstein</a> (1928). <a href="https://doi.org/10.1007%2FBF02592679">"Sur les fonctions absolument monotones"</a>. <i>Acta Mathematica</i>. 52: 1–66. <a href="/facts/Doi_(identifier)/muM9Etpq">doi</a>:<a href="https://doi.org/10.1007%2FBF02592679">10.1007/BF02592679</a>.</li>
<li>D. Widder (1941). <i>The Laplace Transform</i>. Princeton University Press.</li>
<li>Rene Schilling, Renming Song and <a href="/facts/Zoran_Vondra%25C4%258Dek/klELH434">Zoran Vondraček</a> (2010). <i>Bernstein functions</i>. De Gruyter.</li></ul>

Bernstein's theorem on monotone functions open-in-new

Bernstein's theorem on monotone functions