tailieunhanh - Báo cáo hóa học: " Sharp Cusa and Becker-Stark inequalities"

Tuyển tập các báo cáo nghiên cứu về hóa học được đăng trên tạp chí hóa hoc quốc tế đề tài : Sharp Cusa and Becker-Stark inequalities | Chen and Cheung Journal of Inequalities and Applications 2011 2011 136 http content 2011 1 136 RESEARCH 3 Journal of Inequalities and Applications a SpringerOpen Journal Open Access Sharp Cusa and Becker-Stark inequalities Chao-Ping Chen1 and Wing-Sum Cheung2 Correspondence chenchaoping@ 1School of Mathematics and Informatics Henan Polytechnic University Jiaozuo City 454003 Henan Province People s Republic of China Full list of author information is available at the end of the article Springer Abstract We determine the best possible constants ỠMa and b such that the inequalities 2 cosx e sin x ---- x 2 cos x ữ and n2 tan x n2 n 2 4x2 x n2 4x2 are valid for 0 X n 2. Our results sharpen inequalities presented by Cusa Becker and Stark. Mathematics Subject Classification 2000 26D05. Keywords Inequalities trigonometric functions 1. Introduction For 0 X n 2 it is known in the literature that sin x 2 cos x 1 x 3 Inequality 1 was first mentioned by the German philosopher and theologian Nicolaus de Cusa 1401-1464 by a geometrical method. A rigorous proof of inequality 1 was given by Huygens 1 who used 1 to estimate the number n. The inequality is now known as Cusa s inequality 2-5 . Further interesting historical facts about the inequality 1 can be found in 2 . It is the first aim of present paper to establish sharp Cusa s inequality. Theorem 1. For 0 X n 2 2 cos x e sin x 2 cos x ữ x 3 with the best possible constants 0 n n 2 . and Ử 1. ln 3 2 yy 2 2011 Chen and Cheung licensee Springer. This is an Open Access article distributed under the terms of the Creative Commons Attribution License http licenses by which permits unrestricted use distribution and reproduction in any medium provided the original work is properly cited. Chen and Cheung Journal of Inequalities and Applications 2011 2011 136 http content 2011 1 136 Page 2 of 6 Becker and .

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