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, Journal of Computational and Applied Mathematics 351 (2019), 1-5; Available online at https://doi.org/10.1016/j.cam.2018.10.049.”

1. Introduction

We recall from [ ¹, p. 804, 23.1.1] and [ ², p. 3, (1.1)] that the Bernoulli numbers can be generated by

for . It is easy to verify that the function

is even in . Consequently, all the Bernoulli numbers for equal 0.

To discover explicit formulas, recurrent formulas, closed expressions, and integral representations of the Bernoulli numbers for is a classical topic. For recently published results, please refer to the papers ^{3, 4, 5, 6, 7, 8, 9, 10, 11, 12} and closely related references therein.

To bound the Bernoulli numbers for by inequalities is an alternative topic. In [ ¹, p. 805, 23.1.15], [ ¹³, Theorem 1.1], [ ², p. 14, (1.23) and p. 23, Exercise 1.2], and the papers ^{14, 15, 16}, some inequalities for bounding the Bernoulli numbers were established and collected. Most of these inequalities have been refined or sharpened in ¹⁷ by the double inequality

(1)

for , where and

are the best possible in the sense that they can not be replaced respectively by any bigger and smaller constants in the double inequality (1).

To study the differences and the ratios for is also an interesting topic. In the newly published paper ¹⁸, the ratios for which is equivalent to the differences were bounded by the double inequality

(2)

Motivated by the double inequality (2) and by the fact that the function is strictly increasing in for all we naturally pose a problem: what are the best constants and such that the double inequality

(3)

is valid for all ?

In [ ², p. 5, (1.14)], it was listed that

where the Riemann zeta function can be defined ^{19, 20, 21} by the series under the condition and by analytic continuation elsewhere.

(4)

for By virtue of (4), the double inequality (3) can be rewritten as

(5)

which can be further reformulated as

and

Let

Then

In order that the function is strictly increasing (or strictly decreasing, respectively) on , it is necessary and sufficient that

on , which can be rearranged as

Consequently, in order that the function for and the sequence with are strictly increasing (or strictly decreasing, respectively), it is necessary that (or

respectively). The double inequality (5) can also be reformulated as

and

Since

It follows that the necessary conditions are and

This implies that the right-hand side inequality in (2) is sharp, but the left-hand side inequality in (2) perhaps can be improved. In conclusion, we guess that the double inequality (3) is valid if and only if and

Since

and

we guess that the function

is strictly increasing (or strictly decreasing, respectively) if and only if (or , respectively).

The double inequality (2) has been cited and applied in the papers ^{22, 23, 24, 25, 26, 27, 28, 29}.

Can one generalize the inequality (2) to the case for the Bernoulli polynomials?

This paper and ¹⁸ are respectively extracted from the preprints ^{30, 31, 32, 33}.

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