Copyright © 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.