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Research Article Sharpening and Generalizations of Shafer’s Inequality for the Arc Tangent Function | Hindawi Publishing Corporation Journal of Inequalities and Applications Volume 2009 Article ID 930294 10 pages doi 10.1155 2009 930294 Research Article Sharpening and Generalizations of Shafer s Inequality for the Arc Tangent Function Feng Qi 1 Shi-Qin Zhang 2 and Bai-Ni Guo3 1 Department of Mathematics College of Science Tianjin Polytechnic University Tianjin City 300160 China 2 Department of Mathematics Nanyang Normal University Nanyang City Henan Province 473061 China 3 School of Mathematics and Informatics Henan Polytechnic University Jiaozuo City Henan Province 454010 China Correspondence should be addressed to Feng Qi qifeng618@gmail.com Received 18 March 2009 Revised 29 June 2009 Accepted 7 July 2009 Recommended by Martin J. Bohner We sharpen and generalize Shafer s inequality for the arc tangent function. From this some known results are refined. Copyright 2009 Feng Qi et al. 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 and Main Results In 1 the following elementary problem was posed showing that for x 0 3x arctan x ----- - 1 2V1 x2 1.1 In 2 the following three proofs for the inequality 1.1 were provided. Solution by Grinstein Direct computation gives dF x _ VTTx2 - 1 2 dx 1 x2 1 2-717x2 2 1.2 2 Journal of Inequalities and Applications where F x arctan x - 3x 1 2V1 x2 1.3 Now dF x dx is positive for all x 0 whence F x is an increasing function. Since F 0 0 it follows that F x 0 for x 0. Solution by Marsh It follows from cos ộ - 1 2 0 that 3 6 cos ộ 1 - J2 cos ộ 2 2 1.4 The desired result is obtained directly upon integration of the latter inequality with respect to ộ from 0 to arctan x for x 0. Solution by Konhauser The substitution x tan y transforms the given inequality into y 3 sin y 2 cos y which is a special case of an inequality discussed on 3 pages 105-106 . It may be .