New sharp bounds for the Bernoulli numbers and refinement of Becker-Stark inequalities

We obtain new sharp bounds for the bernoulli numbers: 2(2n) !/(π2n (22n -1))<| b2n |≤ (2(2 2k -1) / 22k) (2k)(2n)! /(π2n (2 2n -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. copyright © 2012 hua-feng ge.