bibkey: maweigert2025logfunctions
authors: Rourou Ma; Julian Weigert
year: 2025
title: "Complete monotonicity of log-functions"
doi: 10.1007/s13324-025-01136-9
url: https://arxiv.org/abs/2505.04225v2
claim: "The paper studies the functions p(log x_1, ..., log x_s)/(x_1 ... x_s) with p a polynomial of total degree at most n, shows that their completely monotone members form a semialgebraic cone, proves that in one variable and degree two the regions cut out by the sign conditions on single derivatives form a descending chain, and conjectures that for every degree these regions form a descending chain converging to the completely monotone cone."
strata_touched:
  - D5/S3/Analytic/LogFunctionMonotoneChain
license: citation-only
triage: anchor

Complete monotonicity of log-functions

R. Ma and J. Weigert, arXiv:2505.04225v2; Analysis and Mathematical Physics 15(6) (2025), 141. Quotations are from the v2 TeX source, with the macros \RR, \NN, \F, \CM and \dd written out and \emph dropped.

Complete monotonicity:

Consider an infinitely differentiable function $f:{\mathbb{R}}_{>0}^s\to \mathbb{R}$. The function $f$ is said to be completely monotone (on the positive orthant) if $\forall \alpha \in {\mathbb{N}}_0^s:\forall x\in {\mathbb{R}}_{>0}^s:{\prod }_{i=1}^s{\left(\frac{-\partial }{\partial x_i}\right)}^{{\alpha }_i}f(x_1,\dots ,x_s)\ge 0$

The function class:

${\mathcal{F}}_{s,n}:=\left\{f\in {\mathcal{C}}^{\infty }({\mathbb{R}}_{>0}^s,\mathbb{R})|\exists p\in \mathbb{R}[\mathbf{y}{]}_n:\forall x\in {\mathbb{R}}_{>0}^s:f(x)=\frac{p(\log (x_1),\dots ,\log (x_s))}{x_1\cdots x_s}\right\}$

Here $\mathbb{R}[\mathbf{y}{]}_n$ is the space of real polynomials in $s$ variables of total degree at most $n$, and ${\mathrm{CM}}_{s,n}:={\mathrm{CM}}_s\cap {\mathcal{F}}_{s,n}$.

The conjecture (Conjecture conj:shrinking):

For any $n\in \mathbb{N}$ the semialgebraic sets $D_k=\left\{f\in {\mathcal{F}}_{1,n}|\forall x\in \mathbb{R}:{\left(\frac{-\mathrm{d}}{\mathrm{d}x}\right)}^{k}f(x)\ge 0\right\}$ form a descending chain under inclusion, i.e. $D_k\supseteq D_{k+1}$, which converges to the region ${\mathrm{CM}}_{1,n}={\bigcap }_{k=1}^{\infty }{D}_k$.

The members of ${\mathcal{F}}_{1,n}$ are functions on ${\mathbb{R}}_{>0}$, and the example preceding the conjecture states the sign conditions for $x\in {\mathbb{R}}_{>0}$.

Verified locator

Source: https://arxiv.org/abs/2505.04225v2

• URL: https://arxiv.org/abs/2505.04225v2 (the e-print is a tar archive):
  • member Intro.tex (75 lines, md5 273a80101f0cc7861d6932d5cf5568c8): complete monotonicity, l. 1–4; the class ${\mathcal{F}}_{s,n}$, l. 19–23; ${\mathrm{CM}}_{s,n}$, l. 33;
  • member infiniteOrderDeriv.tex (159 lines, md5 dee5d979f738ec3f47c4320538a59922): the example with the sign conditions on ${\mathbb{R}}_{>0}$, l. 148–152; the conjecture, l. 154–159.
• DOI: 10.1007/s13324-025-01136-9 (https://doi.org/10.1007/s13324-025-01136-9; Crossref lists Analysis and Mathematical Physics, volume 15, issue 6, article 141, published 6 November 2025).