{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,10,7]],"date-time":"2025-10-07T12:11:55Z","timestamp":1759839115761,"version":"3.41.2"},"reference-count":12,"publisher":"Wiley","issue":"1","license":[{"start":{"date-parts":[[2011,9,8]],"date-time":"2011-09-08T00:00:00Z","timestamp":1315440000000},"content-version":"vor","delay-in-days":0,"URL":"http:\/\/creativecommons.org\/licenses\/by\/3.0\/"}],"content-domain":{"domain":["onlinelibrary.wiley.com"],"crossmark-restriction":true},"short-container-title":["Journal of Applied Mathematics"],"published-print":{"date-parts":[[2012,1]]},"abstract":"<jats:p>We obtain new sharp bounds for the Bernoulli numbers: 2(2<jats:italic>n<\/jats:italic>)!\/(<jats:italic>\u03c0<\/jats:italic><jats:sup>2<jats:italic>n<\/jats:italic><\/jats:sup>(2<jats:sup>2<jats:italic>n<\/jats:italic><\/jats:sup> \u2212 1)) &lt; |<jats:italic>B<\/jats:italic><jats:sub>2<jats:italic>n<\/jats:italic><\/jats:sub>| \u2264 (2(2<jats:sup>2<jats:italic>k<\/jats:italic><\/jats:sup> \u2212 1)\/2<jats:sup>2<jats:italic>k<\/jats:italic><\/jats:sup>)<jats:italic>\u03b6<\/jats:italic>(2<jats:italic>k<\/jats:italic>)(2<jats:italic>n<\/jats:italic>)!\/(<jats:italic>\u03c0<\/jats:italic><jats:sup>2<jats:italic>n<\/jats:italic><\/jats:sup>(2<jats:sup>2<jats:italic>n<\/jats:italic><\/jats:sup> \u2212 1)), <jats:italic>n<\/jats:italic> = <jats:italic>k<\/jats:italic>, <jats:italic>k<\/jats:italic> + 1, \u2026,\u2009<jats:italic>k<\/jats:italic> \u2208 <jats:italic>N<\/jats:italic><jats:sup>+<\/jats:sup>, and establish sharpening of Papenfuss\u2032s inequalities, the refinements of Becker\u2010Stark, and\nSteckin\u2032s inequalities. Finally, we show a new simple proof of Ruehr\u2010Shafer inequality.<\/jats:p>","DOI":"10.1155\/2012\/137507","type":"journal-article","created":{"date-parts":[[2011,9,8]],"date-time":"2011-09-08T20:36:26Z","timestamp":1315514186000},"update-policy":"https:\/\/doi.org\/10.1002\/crossmark_policy","source":"Crossref","is-referenced-by-count":14,"title":["New Sharp Bounds for the Bernoulli Numbers and Refinement of Becker\u2010Stark Inequalities"],"prefix":"10.1155","volume":"2012","author":[{"given":"Hua-feng","family":"Ge","sequence":"first","affiliation":[]}],"member":"311","published-online":{"date-parts":[[2011,9,8]]},"reference":[{"key":"e_1_2_6_1_2","doi-asserted-by":"publisher","DOI":"10.1007\/978-1-4757-1867-6"},{"volume-title":"Handbook of Mathematical Functions with Formulas, Graphs and Mathematical Tables","year":"1965","author":"Abramowitz M.","key":"e_1_2_6_2_2"},{"key":"e_1_2_6_3_2","doi-asserted-by":"publisher","DOI":"10.1007\/BF02844246"},{"key":"e_1_2_6_4_2","doi-asserted-by":"publisher","DOI":"10.1007\/s000130050432"},{"key":"e_1_2_6_5_2","doi-asserted-by":"publisher","DOI":"10.1007\/978-3-642-99970-3"},{"volume-title":"Applied Inequalities","year":"2004","author":"Kuang J.C.","key":"e_1_2_6_6_2"},{"key":"e_1_2_6_7_2","article-title":"E 2739","volume":"85","author":"Papenfuss M. 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