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

Hua-feng Ge

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: 2 ( 2 𝑛 ) ! / ( 𝜋 2 𝑛 ( 2 2 𝑛 − 1 ) ) < | 𝐵 2 𝑛 | ≤ ( 2 ( 2 2 𝑘 − 1 ) / 2 2 𝑘 ) 𝜁 ( 2 𝑘 ) ( 2 𝑛 ) ! / ( 𝜋 2 𝑛 ( 2 2 𝑛 − 1 ) ) , 𝑛 = 𝑘 , 𝑘 + 1 , … , 𝑘 ∈ 𝑁 + , 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.