View source for Absolutely monotonic function

''absolutely monotone function''

An infinitely-differentiable [[Function|function]] on an interval <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a120/a120030/a1200301.png" /> such that it and all its derivatives are non-negative on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a120/a120030/a1200302.png" />. These functions were first investigated by S.N. Bernshtein in [[#References|[a1]]] and the study was continued in greater detail in [[#References|[a3]]]. The terminology also seems due to Bernshtein, [[#References|[a2]]], although the name was originally applied to differences rather than derivatives. A companion definition says that a function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a120/a120030/a1200303.png" />, infinitely differentiable on an interval <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a120/a120030/a1200304.png" />, is completely monotonic on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a120/a120030/a1200305.png" /> if for all non-negative integers <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a120/a120030/a1200306.png" />,

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a120/a120030/a1200307.png" /></td> </tr></table>

Of course, this is equivalent to saying that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a120/a120030/a1200308.png" /> is absolutely monotonic on the union of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a120/a120030/a1200309.png" /> and the interval obtained by reflecting <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a120/a120030/a12003010.png" /> with respect to the origin.

Both the extensions and applications of the theory of absolutely monotonic functions derive from two major theorems. The first, sometimes known as the little Bernshtein theorem, asserts that a function that is absolutely monotonic on a closed interval <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a120/a120030/a12003011.png" /> can be extended to an [[Analytic function|analytic function]] on the interval defined by <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a120/a120030/a12003012.png" />. In a similar manner, a function that is absolutely monotonic on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a120/a120030/a12003013.png" /> can be extended to a function that is not only analytic on the real line but is even the restriction of an [[Entire function|entire function]] to the real line. The big Bernshtein theorem states that a function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a120/a120030/a12003014.png" /> that is absolutely monotonic on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a120/a120030/a12003015.png" /> can be represented there as a [[Laplace integral|Laplace integral]] in the form

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a120/a120030/a12003016.png" /></td> </tr></table>

where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a120/a120030/a12003017.png" /> is non-decreasing and bounded on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a120/a120030/a12003018.png" />. For either or both theorems see [[#References|[a3]]], [[#References|[a6]]], and [[#References|[a7]]].

Questions of analyticity based on the signs of derivatives of functions have been extensively studied. See [[#References|[a5]]] for references to earlier work and [[#References|[a4]]] for more recent results and references.

====References====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  S. Bernstein,   "Sur la définition et les propriétés des fonctions analytique d'une variable réelle"  ''Math. Ann.'' , '''75'''  (1914)  pp. 449–468</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top">  S. Bernstein,   "Lecons sur les proprietes extremales et la meilleure approximation des fonctions analytiques d'une variable reelle" , Gauthier-Villars  (1926)</TD></TR><TR><TD valign="top">[a3]</TD> <TD valign="top">  S. Bernstein,   "Sur les fonctions absolument monotones"  ''Acta Math.'' , '''52'''  (1928)  pp. 1–66</TD></TR><TR><TD valign="top">[a4]</TD> <TD valign="top">  G.G. Bilodeau,   "Sufficient conditions for analyticity"  ''Real Analysis Exch.'' , '''19'''  (1993/4)  pp. 135–145</TD></TR><TR><TD valign="top">[a5]</TD> <TD valign="top">  R.P. Boas Jr.,   "Signs of derivatives and analytic behavior"  ''Amer. Math. Monthly'' , '''78'''  (1971)  pp. 1085–1093</TD></TR><TR><TD valign="top">[a6]</TD> <TD valign="top">  W.F. Donoghue Jr.,   "Monotone matrix functions and analytic continuation" , Springer  (1974)</TD></TR><TR><TD valign="top">[a7]</TD> <TD valign="top">  D.V. Widder,   "The Laplace transform" , Princeton Univ. Press  (1946)</TD></TR></table>