PreprintCC BY-SA 4.0arXiv
Bernstein and Stieltjes properties of some functions involving logarithmic functions
Abstract
We show that 1/g(x), with g(x)=(x−1)/(xln x), is a complete Bernstein function and give its Lévy measure explicitly. For h(x)=((x2−1)ln(x+1))/(x2ln x), we prove that h is not a Stieltjes function, so 1/h is not a complete Bernstein function, but 1/h is a Bernstein function. The latter rests on the positivity of a function built from the convolution of the exponential integrals E1 and E2. We also give a related completely monotonic function that is not a Stieltjes function.
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BibTeX
@misc{qi2026bernstein,
title = {{Bernstein and Stieltjes properties of some functions involving logarithmic functions}},
author = {Feng Qi},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.40718},
eprint = {2610.40718},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.40718},
note = {CC BY-SA 4.0},
}
RIS
TY - UNPB
TI - Bernstein and Stieltjes properties of some functions involving logarithmic functions
AU - Feng Qi
PY - 2026
DA - 2026/10/29
PB - arXiv
DO - 10.48550/arxiv.2610.40718
UR - https://arxiv.org/abs/2610.40718
AB - We show that 1/g(x), with g(x)=(x−1)/(xln x), is a complete Bernstein function and give its Lévy measure explicitly. For h(x)=((x^2−1)ln(x+1))/(x^2ln x), we prove that h is not a Stieltjes function, so 1/h is not a complete Bernstein function, but 1/h is a Bernstein function. The latter rests on the positivity of a function built from the convolution of the exponential integrals E_1 and E_2. We also give a related completely monotonic function that is not a Stieltjes function.
ER -