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Estimates for the arctangent function related to Shafer's inequality

Cristinel Mortici, H. M. Srivastava (2014)

Colloquium Mathematicae

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.

Estimates of the remainder in Taylor’s theorem using the Henstock-Kurzweil integral

Erik Talvila (2005)

Czechoslovak Mathematical Journal

When a real-valued function of one variable is approximated by its n th degree Taylor polynomial, the remainder is estimated using the Alexiewicz and Lebesgue p -norms in cases where f ( n ) or f ( n + 1 ) are Henstock-Kurzweil integrable. When the only assumption is that f ( n ) is Henstock-Kurzweil integrable then a modified form of the n th degree Taylor polynomial is used. When the only assumption is that f ( n ) C 0 then the remainder is estimated by applying the Alexiewicz norm to Schwartz distributions of order 1.

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