Notes on a Double Inequality for Ratios of any Two Neighbouring Non-zero Bernoulli Numbers

Abstract

In the paper, the author notes on a double inequality published in "Feng Qi, A double inequality for the ratio of two non-zero neighbouring Bernoulli numbers,

Increasing property and logarithmic convexity of functions involving Riemann zeta function

“…is increasing from (1, ∞) onto 1 2 , 1 . By this result, Yang and Tian [33] extended and sharpened the double inequality established in [12,17] for bounding the ratio…”

“…for n ∈ N. In 2020, Zhu [34] used this result once to discuss those conclusions in [12,17]. (4) In 2015, Adell-Lekuona [2] and Alzer-Kwong [3] proved that the Dirichlet eta function η(x) is concave on (0, ∞).…”

“…(1) In 1998, Wang [27] proved that the Dirichlet eta function η(x) is logarithmically concave on (0, ∞). In 2018, Qi [12,17] used this result to establish a double inequality for bounding the ratio…”

“…(3) In 2010, Zhu and Hua [35] proved that the sequence λ(n) for n ∈ N is decreasing. In 2018, Qi [12,17] used also this result while he established a double inequality for bounding the ratio…”

Partial solutions to several conjectures on completely monotonic degrees for remainders in asymptotic expansions of the digamma function

“…for n ≥ 0 were proved to be completely monotonic on (0, ∞), where an empty sum is understood to be 0 and the Bernoulli numbers B n are generated [20,23,26] by…”