TAILIEUCHUNG - Báo cáo hóa học: " Research Article On the Strengthened Jordan’s Inequality"

Tuyển tập báo cáo các nghiên cứu khoa học quốc tế ngành hóa học dành cho các bạn yêu hóa học tham khảo đề tài: Research Article On the Strengthened Jordan’s Inequality | Hindawi Publishing Corporation Journal ofInequalities and Applications Volume 2007 Article ID 74328 8 pages doi 2007 74328 Research Article On the Strengthened Jordan s Inequality Jian-Lin Li and Yan-Ling Li Received 21 July 2007 Revised 19 October 2007 Accepted 22 November 2007 Recommended by Lars-Erik Persson The main purpose of this paper is to present two methods of sharpening Jordan s inequality. The first method shows that one can obtain new strengthened Jordan s inequalities from old ones. The other method shows that one can sharpen Jordan s inequality by choosing proper functions in the monotone form of L Hopital s rule. Finally we improve a related inequality proposed early by Redheffer. Copyright 2007 . Li and . Li. 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. 1. Introduction The well-known Jordan s inequality states that sin x x 2 n x e 0 n 2 holds with equality if and only if x n 2 see 1 . It plays an important role in many areas of pure and applied mathematics. This inequality was first extended to the following V 2 n2 - 4x2 x e c n 1 x n 12n X 2 and then it was further extended to the following v 2 1 n2 - 4x2 x e 0 n 1 x n n3 2 which holds with equality if and only if x n 2 see 2-4 . Inequality is slightly stronger than inequality and is sharp in the sense that 1 n3 cannot be replaced by a larger constant. More recently the monotone form of L Hopital s rule see 5 Lemma has been successfully used by Zhu 6 7 and Wu and Debnath 8 9 to sharpen 2 Journal of Inequalities and Applications Jordan s inequality. For example it has been shown in 6 that if 0 x nl2 then n2 - 4x2 2 n-2 n2 - 4x2 n n3 x n n3 holds with equality if and only if x nl2. Furthermore the constants 1ln3 and n - 2 ln3 in are the best. Also in the process of sharpening Jordan s .

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