Absolutely monotonic function

absolutely monotone function

An infinitely-differentiable function on an interval $I$ such that it and all its derivatives are non-negative on $I$. These functions were first investigated by S.N. Bernshtein in [a1] and the study was continued in greater detail in [a3]. The terminology also seems due to Bernshtein, [a2], although the name was originally applied to differences rather than derivatives. A companion definition says that a function $f$, infinitely differentiable on an interval $I$, is completely monotonic on $I$ if for all non-negative integers $n$,

\begin{equation*} (-1) ^ {n} f ^ {(n)} (x) \geq 0 \text { on } I. \end{equation*}

Of course, this is equivalent to saying that $f ( - x )$ is absolutely monotonic on the union of $I$ and the interval obtained by reflecting $I$ 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 $[a , b]$ can be extended to an analytic function on the interval defined by $x - a | < b - a$. In a similar manner, a function that is absolutely monotonic on $[ 0 , \infty )$ 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 states that a function $f$ that is absolutely monotonic on $( - \infty , 0 ]$ can be represented there as a Laplace integral in the form

\begin{equation*} f ( x ) = \int _ { 0 } ^ { \infty } e ^ { x t } d \mu ( t ), \end{equation*}

where $\mu$ is non-decreasing and bounded on $[ 0 , \infty )$. For either or both theorems see [a3], [a6], and [a7].

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

References

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