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 New Inequalities of Shafer-Fink Type for Arc Hyperbolic Sine | Hindawi Publishing Corporation Journal of Inequalities and Applications Volume 2008 Article ID 368275 5 pages doi 2008 368275 Research Article New Inequalities of Shafer-Fink Type for Arc Hyperbolic Sine Ling Zhu Department of Mathematics Zhejiang Gongshang University Hangzhou 310018 China Correspondence should be addressed to Ling Zhu zhuling0571@ Received 2 July 2008 Revised 25 September 2008 Accepted 17 November 2008 Recommended by Martin J. Bohner In this paper we extend some Shafer-Fink-type inequalities for the inverse sine to arc hyperbolic sine and give two simple proofs of these inequalities by using the power series quotient monotone rule. Copyright 2008 Ling Zhu. 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 Mitrinovic in 1 page 247 gives us a result as follows. Theorem . Let x 0. Then 6 V1 x - V1 - x arcsin x --- 4 c1 x v1 - x 3x 2 V1 - x2 Fink in 2 obtains the following theorem. Theorem . Let 0 x 1. Then 3x nx 2 V1 - x2 2 V1 - x2 Furthermore 3 and n are the best constants in . The author of this paper improves the upper bound of inverse sine and obtains see 3 4 the following theorem. 2 Journal of Inequalities and Applications Theorem . Let 0 x 1. Then 3x 6 V1 x V1 x _ ----- ---- arcsin x 2 V 1 x2 4 V 1 x -f1 x J n a 2 1 2 V1 x VĨ ã nx 4 V1 x V1 x 2 V1 x2 Furthermore 3 and n 6 and n V2 1 2 are the best constants in . Malesevic in 5 6 obtains the following theorem using l-method and computer respectively. Theorem . For x e 0 1 the following inequality is true arcsin x n 2 V2 n 2V2 V1 x V1 x V2 n 4 n 2V2 V1 x V1 x In 7 Malesevic obtains inequality by further method on computer. Zhu in 8 shows new simple proof of inequality and obtains the following further result. Theorem . Let 0 x 1. Then a 2 V1 x V1 x . fi 2
