Dear Colleagues
I need an inequality for the ratio of two Bernoulli numbers, see attached picture. Could you please help me to find it? Thank you very much.
Best regards
Feng Qi (F. Qi)
Dear Peter and Feng,
Glad that Feng found the proof, I was actually surprised about what Feng was exactly looking for. It turned out to be that he wanted a proof for this inequality.
Definitely, the proof comes from Riemann Zeta function which asymptotically approaches one at x goes to infinity. The Riemann Zeta function at even arguments is expressed in terms of the Bernoulli numbers.
Best
Dear colleagues
What I wanted is a site, where the inequality existed, or a proof of the inequality. I did not find a site, but a proof.
Now we may pose a new question: can one discover an explicit lower bound for the ratio $|B2n+2|/|B2n|$ of two consecutive Bernoulli numbers with even indexes?
Best regards
Feng QI (F. Qi)
In the beginning, I did mean that I needed a site to cite. Because I did not find the site, I desired to find a proof, and I luckily made a real success.
Now we should pay our attention on discovering an explicit lower bound for the ratio $|B2n+2|/|B2n|$ of two consecutive Bernoulli numbers with even indexes.
Let us turn our attention mainly on mathematics.
I am glad to announce a news: I have obtained a double inequality for bounding the ratio $|B2n+2|/|B2n|$ of two consecutive Bernoulli numbers with even indexes.
Why do I need " these shallow games with 19th century numbers"? Here I do not know how to simply tell you in English the rationale. If you contact me, I may send you a completed manuscript by e-mail.
Anyway, I appreciate everyone who ever wrote and helped me here, because I learned very much from you, including mathematics, english, and personality. Thank all!
Thank Peter T Breuer for so much new information to me.
Moreover, I need a reference containing the integral analogue of Cauchy-Binet identity (a formula looks like the one in the attached picture), but I can not find such a reference published in English. Could you please help me? Anyway, thank you in advance.
Feng
[1] Feng Qi, A double inequality for ratios of Bernoulli numbers, ResearchGate Dataset, available online at http://dx.doi.org/10.13140/2.1.2367.2962.
[2] Feng Qi, A double inequality for ratios of Bernoulli numbers, RGMIA Research Report Collection 17 (2014), Article 103, 4 pages; Available online at http://rgmia.org/v17.php.
The following formally published papers are related to this question:
[1] Feng Qi, A double inequality for the ratio of two non-zero neighbouring Bernoulli numbers, Journal of Computational and Applied Mathematics 351 (2019), 1--5; available online at https://doi.org/10.1016/j.cam.2018.10.049
[2] Feng Qi, Notes on a double inequality for ratios of any two neighbouring non-zero Bernoulli numbers, Turkish Journal of Analysis and Number Theory 6 (2018), no. 5, 129--131; available online at https://doi.org/10.12691/tjant-6-5-1
This paper can help you. "A double inequality for the ratio of two non-zero neighbouring Bernoulli numbers" By Feng Qi
In the Journal of Computational and Applied Mathematics
Volume 351, 1 May 2019, Pages 1-5
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