bibkey: weiguo2014completemonotonicity
authors: C.-F. Wei and B.-N. Guo
year: 2014
title: Complete monotonicity of functions connected with the exponential function and derivatives
doi: 10.1155/2014/851213
url: https://doi.org/10.1155/2014/851213
claim: "Section 5, Conjecture 12 asserts complete monotonicity of both consecutive signed derivative ratios for every nonnegative integer index."
strata_touched:
  - D5/S3/Analytic/ExpRatioCM/Defs
  - D5/S3/Analytic/ExpRatioCM/Result
license: citation-only
triage: anchor

Complete monotonicity of exponential derivative ratios

Verified locator

DOI: 10.1155/2014/851213

Source: https://doi.org/10.1155/2014/851213

C.-F. Wei and B.-N. Guo, Complete monotonicity of functions connected with the exponential function and derivatives, Abstract and Applied Analysis 2014, Article ID 851213, 5 pp. Section 5, Conjecture 12; definitions (6) and (10), with the alternative expressions (27). Equation (30) verifies the indices zero and one.

Mathematical correspondence

For every nonnegative integer $i$, equation (6) defines

$$F_i(t)=(-1{)}^i{\left(\frac{1}{e^t-1}\right)}^{(i)},G_i(t)=(-1{)}^i{\left(\frac{1}{1-e^{-t}}\right)}^{(i)}.$$

Equation (10) defines ${\mathcal{F}}_i=F_{i+1}/F_i$ and ${\mathcal{G}}_i=G_{i+1}/G_i$. Conjecture 12 asserts that both ratios are completely monotonic on $(0,\infty )$ for every such index.

The six definitions in D5/S3/Analytic/ExpRatioCM/Defs implement these functions, their ratios, complete monotonicity and the universal assertion. Complete monotonicity includes smoothness on the open positive half-line and nonnegativity of every alternating derivative, including derivative order zero. Globally defined iterated derivatives are compared by neighborhood equality at positive arguments.

D5/S3/Analytic/ExpRatioCM/Result.result refutes the universal assertion. Positive.cm_of_le_two gives the positive range through index two, while Result.failure_on_every_tail excludes every index at least three by supplying simultaneous negative alternating derivatives of both ratios arbitrarily far to the right.

Literature scope

Guo–Qi, arXiv:2201.06970, Section 5, Proposition 5.1, strengthens the decreasing property to the range $(1,\infty )$ and does not address complete monotonicity of the ratios. No settlement was found in the bounded scope consisting of the title, DOI and conjecture queries, Proskurnikov–Runvik–Medvedev, arXiv:2302.01364, Appendix B, Formal Conjectures, conjectures.io, and the corresponding local mathematical and bibliographic entries. This scope does not establish exhaustive citation-index coverage or priority. The scope and its limitations are also stated in Problems/wei-guo-2014-exponential-derivative-ratio-complete-monotonicity.md.