#### Abstract

We obtain new sharp bounds for the Bernoulli numbers: , , and establish sharpening of Papenfuss's inequalities, the refinements of Becker-Stark, and Steckin's inequalities. Finally, we show a new simple proof of Ruehr-Shafer inequality.

#### 1. Introduction

The classical Bernoulli numbers can be defined by (see [1]) Reference [2] shows a upper bound for On the other hand, [3] presents a lower bound for as follows:

On the basis of (1.2) and (1.3), Alzer [4] obtains the further results.

Theorem A. *For all integers one has
**
with the best possible constants and .*

In this paper, we obtain new bounds for the Bernoulli numbers as follows.

Theorem 1.1. *Let , , then
**
The equality holds in (1.5) if and only if . Furthermore, 2 and are the best constants in (1.5).*

In the following, we study on some trigonometric inequalities.

Mitrinovic [5] gives us a result which belongs to Steckin.

Theorem B. *If , then
**
Now, we show a upper bound for and obtain the following sharp Steckin's inequalities.*

Theorem 1.2. *If , then
**
or
**
Furthermore, and are the best constants in (1.7) and (1.8).*

Kuang [6] gives us the further results described as Becker-Stark inequalities

Theorem C. *Let , then
**
Furthermore, and are the best constants in (1.9).*

Let in (1.9), then Theorem is equivalent to.

Theorem D. *Let , then
**
Furthermore, 8 and are the best constants in (1.10). *

Clearly, Becker-Stark inequalities (1.10) are the generalization of the strengthened Steckin's inequalities (1.8).

On the other hand, Papenfuss [7] proposes an open problem described as the following statement.

Theorem E. *Let , then
** Bach [8] prove Theorem and obtain a further result.*

Theorem F. *Let , then
*

In this section, we first obtain sharp Papenfuss-Bach inequalities described as Theorem 1.3.

Theorem 1.3. *Let , then
**
Furthermore, 64 and are the best constants in (1.13).*

The inequalities (1.13) are equivalent to That is,Then, integrating the three functions in (1.15) from 0 to , where , we obtain the following refinement of Becker-Stark inequalities.

Theorem 1.4 (Refinement of Becker-Stark Inequalities). * Let , then
**
An application of Theorem 1.4 leads to Theorem 1.5 ( the refinement of Steckin's inequalities).*

Theorem 1.5 (Refinement of Steckin's Inequalities). *If , then
**Finally, we will show a new proof of Ruehr-Shafer inequality.*

Theorem G (Ruehr-Shafer Inequality, see [8]). *Let , then
*

#### 2. Two Lemmas

Lemma 2.1 (see [9, Lemma 2.1]). *The function is decreasing, where is the Riemann's zeta function.*

Lemma 2.2 (see [10]). *Let and be real numbers, and let the power series and be convergent for . If for and if is strictly decreasing for , then the function is strictly decreasing on .*

#### 3. Proof of Theorem 1.1

Using the representation (cf. [11, page 266]), we have From Lemma 2.1, we know that is decreasing and . Then, the proof of Theorem 1.1 is complete.

#### 4. Proofs of Theorem 1.3 and G

##### 4.1. Proof of Theorem 1.3

The following power series expansion can be found in [12]: Then

Let where Then, , and . So, is decreasing by Lemma 2.1. Therefore, is decreasing on by Lemma 2.2. At the same time, and , so 64 and are the best constants in (1.13).

##### 4.2. Proof of Theorem G

By (4.1) and (4.2), we have so, (1.18) is equivalent to that is, or

From Lemma 2.1, we have that is decreasing or holds. By (3.1), we get so, (4.8) holds.

#### 5. Remark

In 2010, Zhu and Hua [9] proved for that

Now, we can compare the results of (5.1) with (1.16). In fact, we can easy check that So, (5.1) is equivalent to (1.16).