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The inverse tangent function can be bounded by different inequalities, forexample by Shafer's inequality. In this publication, we propose a new sharpdouble inequality, consisting of a lower and an upper bound, for the inversetangent function. In particular, we sharpen Shafer's inequality and calculatethe best corresponding constants. The maximum relative errors of the obtainedbounds are approximately smaller than 0.27% and 0.23% for the lower and upperbound, respectively. Furthermore, we determine an upper bound on the relativeerrors of the proposed bounds in order to describe their tightnessanalytically. Moreover, some important properties of the obtained bounds arediscussed in order to describe their behavior and achieved accuracy.

A Sharp Double Inequality for the Inverse Tangent Function