Hindawi Publishing Corporation Journal of Inequalities and Applications Volume 2011, Article ID 840206, 10 pages doi:10.1155/2011/840206 Research Article On Shafer and Carlson Inequalities Chao-Ping Chen, 1 Wing-Sum Cheung, 2 and Wusheng Wang 3 1 School of Mathematics and Informatics, Henan Polytechnic University, Jiaozuo City, Henan Province 454003, China 2 Department of Mathematics, The University of Hong Kong, Pokfulam Road, Hong Kong 3 Department of Mathematics, Hechi University, Yizhou, Guangxi 546300, China Correspondence should be addressed to Wing-Sum Cheung, wscheung@hku.hk Received 23 November 2010; Accepted 5 February 2011 Academic Editor: Martin Bohner Copyright q 2011 Chao-Ping Chen 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. We present a generalized and sharp version of Shafer’s inequality for the inverse tangent function and a new lower bound of Carlson’s inequality by means of a third order estimate of the inverse cosine function. 1. Introduction For x> 0, it is known in the literature that 3x 1  2 √ 1  x 2 < arctan x. 1.1 This inequality was first presented without proof by Shafer 1. Three proofs of it were later given in 2. Shafer’s inequality 1.1 was recently sharpened and generalized by Qi et al. in 3. In view of inequality 1.1, we now ask: for each a> 0, what is the largest number b and what is the smallest number c such that the inequalities bx 1  a √ 1  x 2 ≤ arctan x ≤ cx 1  a √ 1  x 2 1.2 are valid for all x ≥ 0? Theorem 2.1 below answers this question.