Research Article | Open Access
New Sharp Bounds for the Bernoulli Numbers and Refinement of Becker-Stark Inequalities
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.
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.
In the following, we study on some trigonometric inequalities.
Mitrinovic  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.
Kuang  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).
On the other hand, Papenfuss  proposes an open problem described as the following statement.
Theorem E. Let , then Bach  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).
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 ). 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 ). 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
4. Proofs of Theorem 1.3 and G
4.1. Proof of Theorem 1.3
The following power series expansion can be found in : Then
4.2. Proof of Theorem G
In 2010, Zhu and Hua  proved for that
- W. Scharlau and H. Opolka, From Fermat to Minkowski, Undergraduate Texts in Mathematics, Springer, New York, NY, USA, 1985.
- M. Abramowitz and I. A. Stegun, Eds., Handbook of Mathematical Functions with Formulas, Graphs and Mathematical Tables, Dover Publications, New York, NY, USA, 1965.
- C. D'Aniello, “On some inequalities for the Bernoulli numbers,” Rendiconti del Circolo Matematico di Palermo, vol. 43, no. 3, pp. 329–332, 1994.
- H. Alzer, “Sharp bounds for the Bernoulli numbers,” Archiv der Mathematik, vol. 74, no. 3, pp. 207–211, 2000.
- D. S. Mitrinovic, Analytic Inequalities, Springer, New York, NY, USA, 1970.
- J.C. Kuang, Applied Inequalities, Shangdong Science and Technology Press, Jinan City, Shangdong Province, China, 3rd edition, 2004.
- M. C. Papenfuss, “E 2739,” The American Mathematical Monthly, vol. 85, p. 765, 1978.
- G. Bach, “Trigonometric inequality,” The American Mathematical Monthly, vol. 87, p. 62, 1980.
- L. Zhu and J. Hua, “Sharpening the Becker-Stark inequalities,” Journal of Inequalities and Applications, Article ID 931275, 4 pages, 2010.
- S. Ponnusamy and M. Vuorinen, “Asymptotic expansions and inequalities for hypergeometric functions,” Mathematika, vol. 44, no. 2, pp. 278–301, 1997.
- G. Pólya and G. Szegő, Problems and theorems in analysis. Vol. I, Springer, New York, NY, USA, 1972.
- A. Jeffrey, Handbook of Mathematical Formulas and Integrals, Elsevier Academic Press, San Diego, Calif, USA, 3rd edition, 2004.
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.