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The aim of this article is to give new refinements and sharpenings of Shafer's inequality involving the arctangent function. These are obtained by means of a change of variables, which makes the computations much easier than the classical approach.
When a real-valued function of one variable is approximated by its th degree Taylor polynomial, the remainder is estimated using the Alexiewicz and Lebesgue -norms in cases where or are Henstock-Kurzweil integrable. When the only assumption is that is Henstock-Kurzweil integrable then a modified form of the th degree Taylor polynomial is used. When the only assumption is that then the remainder is estimated by applying the Alexiewicz norm to Schwartz distributions of order 1.