Tuyển tập các báo cáo nghiên cứu về hóa học được đăng trên tạp chí hóa hoc quốc tế đề tài : Refinements of Jordan’s inequality | Kuo Journal of Inequalities and Applications 2011 2011 130 http content 2011 1 130 RESEARCH 3 Journal of Inequalities and Applications a SpringerOpen Journal Open Access Refinements of Jordan s inequality Meng-Kuang Kuo Correspondence kuo53@. tw Center of General Education Jen-Teh Junior College of Medicine Nursing and Management No. 799 Sha-Luen Hu Xi-Zhou Li Hou-Loung Town Miaoli County Republic of China Abstract A method of sharpening Jordan s inequality proposed by Li-Li would be improved. Increasing lower bounds and decreasing upper bounds for strengthened Jordan s inequality can be constructed and the errors of lower-upper bounds for strengthened Jordan s inequality can be estimated. 2010 Mathematics Subject Classification 26D05 26D15. Keywords Jordan s inequality lower bound upper bound 1 Introduction The well-known Jordan s inequality 1 p. 33 reads that 2 tín 1 0 x n 1 1 n x 2 with equality holds if and only if x n 2. This inequality plays an important role in many areas of pure and applied mathematics. Jordan s inequality has been refined generalized and applied by many mathematicians. It was first extended to the following sínx 2 7 n2 - 4x2 0 x n 1 2 x n 12n 2 and then it was further refined to inequality sin 2 2 - 4 2 0 x n. 1 3 x n n3 2 For detailed information please refer to the expository and survey articles 2 and related references therein. In 3 Theorem or 2 a new method of sharpening Jordan s inequality was established by Li-Li which shows that one can obtain strengthened Jordan s inequalities from old ones. This result may be stated as follows. Theorem Li-Li Let g 0 n 2 0 1 be a continuous function. If sin x n 1 4 g x 0 x x then the double inequality 2 n sin x n- V 2 Kx f 1 h x 0 x n 2 1 5 2011 Kuo licensee Springer. This is an Open Access article distributed under the terms of the Creative Commons Attribution License http licenses by which permits .
