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Journal of Applied Mathematics
Volume 2012 (2012), Article ID 137507, 7 pages
http://dx.doi.org/10.1155/2012/137507
Research Article

New Sharp Bounds for the Bernoulli Numbers and Refinement of Becker-Stark Inequalities

Department of Mathematics, Taizhou University, Linhai, Zhejiang 317000, China

Received 25 May 2011; Accepted 25 June 2011

Academic Editor: Kai Diethelm

Copyright © 2012 Hua-feng Ge. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

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).

References

  1. W. Scharlau and H. Opolka, From Fermat to Minkowski, Undergraduate Texts in Mathematics, Springer, New York, NY, USA, 1985. View at Zentralblatt MATH
  2. M. Abramowitz and I. A. Stegun, Eds., Handbook of Mathematical Functions with Formulas, Graphs and Mathematical Tables, Dover Publications, New York, NY, USA, 1965.
  3. C. D'Aniello, “On some inequalities for the Bernoulli numbers,” Rendiconti del Circolo Matematico di Palermo, vol. 43, no. 3, pp. 329–332, 1994. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  4. H. Alzer, “Sharp bounds for the Bernoulli numbers,” Archiv der Mathematik, vol. 74, no. 3, pp. 207–211, 2000. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  5. D. S. Mitrinovic, Analytic Inequalities, Springer, New York, NY, USA, 1970.
  6. J.C. Kuang, Applied Inequalities, Shangdong Science and Technology Press, Jinan City, Shangdong Province, China, 3rd edition, 2004.
  7. M. C. Papenfuss, “E 2739,” The American Mathematical Monthly, vol. 85, p. 765, 1978. View at Google Scholar
  8. G. Bach, “Trigonometric inequality,” The American Mathematical Monthly, vol. 87, p. 62, 1980. View at Google Scholar
  9. L. Zhu and J. Hua, “Sharpening the Becker-Stark inequalities,” Journal of Inequalities and Applications, Article ID 931275, 4 pages, 2010. View at Google Scholar · View at Zentralblatt MATH
  10. S. Ponnusamy and M. Vuorinen, “Asymptotic expansions and inequalities for hypergeometric functions,” Mathematika, vol. 44, no. 2, pp. 278–301, 1997. View at Publisher · View at Google Scholar
  11. G. Pólya and G. Szegő, Problems and theorems in analysis. Vol. I, Springer, New York, NY, USA, 1972.
  12. A. Jeffrey, Handbook of Mathematical Formulas and Integrals, Elsevier Academic Press, San Diego, Calif, USA, 3rd edition, 2004.