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Complete monotonicity of log-functions

Rourou Ma, Julian Weigert

TL;DR

This work studies complete monotonicity on the positive orthant for the log-based function family $\mathcal{F}_s$, proving that for each fixed degree $n$ the finite-dimensional cone $\mathrm{CM}_{s,n}$ is linearly isomorphic to the nonnegative polynomial cone $\mathrm{Psd}_{s,n}$, hence $\mathrm{CM}_{s,n}$ is semialgebraic and testable in finite time. The key tool is a linear map $\mathcal{L}$ built from a matrix $A$ whose entries involve derivatives of the gamma function, yielding an explicit representation of $\mathcal{F}_{s,n}$ in terms of multivariate polynomials and enabling a PSD-based certificate via $A^{-1}$. Two complementary viewpoints are developed: a Laplace-transform criterion (via the Bernstein–Hausdorff–Widder–Choquet theorem) and an infinite-order-derivative framework that transfers CM to the nonnegativity of a limiting polynomial $\widetilde{g}_\infty$, connected to $\mathcal{L}^{-1}(f)$. The results provide a practical, finite-time algorithm for CM testing in this function family and open avenues for nested semialgebraic approximations, with potential applications to positivity constraints arising in scattering amplitudes and related physics.

Abstract

In this article we investigate the property of complete monotonicity within a special family $\mathcal{F}_s$ of functions in $s$ variables involving logarithms. The main result of this work provides a linear isomorphism between $\mathcal{F}_s$ and the space of real multivariate polynomials. This isomorphism identifies the cone of completely monotone functions with the cone of non-negative polynomials. We conclude that the cone of completely monotone functions in $\mathcal{F}_s$ is semi-algebraic. This gives a finite time algorithm to decide whether a function in $\mathcal{F}_s$ is completely monotone

Complete monotonicity of log-functions

TL;DR

This work studies complete monotonicity on the positive orthant for the log-based function family , proving that for each fixed degree the finite-dimensional cone is linearly isomorphic to the nonnegative polynomial cone , hence is semialgebraic and testable in finite time. The key tool is a linear map built from a matrix whose entries involve derivatives of the gamma function, yielding an explicit representation of in terms of multivariate polynomials and enabling a PSD-based certificate via . Two complementary viewpoints are developed: a Laplace-transform criterion (via the Bernstein–Hausdorff–Widder–Choquet theorem) and an infinite-order-derivative framework that transfers CM to the nonnegativity of a limiting polynomial , connected to . The results provide a practical, finite-time algorithm for CM testing in this function family and open avenues for nested semialgebraic approximations, with potential applications to positivity constraints arising in scattering amplitudes and related physics.

Abstract

In this article we investigate the property of complete monotonicity within a special family of functions in variables involving logarithms. The main result of this work provides a linear isomorphism between and the space of real multivariate polynomials. This isomorphism identifies the cone of completely monotone functions with the cone of non-negative polynomials. We conclude that the cone of completely monotone functions in is semi-algebraic. This gives a finite time algorithm to decide whether a function in is completely monotone
Paper Structure (5 sections, 18 theorems, 89 equations, 2 figures)

This paper contains 5 sections, 18 theorems, 89 equations, 2 figures.

Table of Contents

  1. Introduction
  2. The univariate quadratic case
  3. Positivity of the inverse Laplace transform
  4. Infinite order derivatives
  5. Acknowledgments

Key Result

Theorem 1.3

Let $p=\sum_{i=0}^nc_iy^i\in \mathbb{R}[y]_n$ and let $f:x \mapsto \frac{p(\log(x))}{x}$ be the corresponding function in $\mathcal{F}_n$. Write $c=(c_0,\ldots,c_n)^T$ and $Y=(1, y,\ldots,y^n)^T$, then $f$ is completely monotone if and only if

Figures (2)

  • Figure 1: Outer approximations of the complete monotonicity cone in $c_0,c_1,c_2$ for the family of functions $f(x)=\frac{c_2 \log(x)^2+ c_1 \log(x)+c_0}{x}$. It shows the nested structure for the area of nonnegativity of signed derivatives of $f(x)$. The CM region in parameter space is the intersection of all of these regions.
  • Figure 2: Outer approximations of the complete monotonicity cone in $c_0,c_1,c_2$ for the family of functions $f(x)=\frac{\log(x)^4+c_2 \log(x)^2+ c_1 \log(x)+c_0}{x}$. There seems be the similar nested structure as in Fig.\ref{['fig:ShrinkCone2']} which seems to converge to the CM region of $f(x)$ visually, see Conjecture \ref{['conj:shrinking']}.

Theorems & Definitions (42)

  • Definition 1.1
  • Theorem 1.3
  • Theorem 1.4
  • Definition 2.1
  • Lemma 2.2
  • proof
  • Definition 2.3
  • Lemma 2.4
  • proof
  • Proposition 2.5
  • ...and 32 more