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Approximations to inverse tangent function.

Quan-Xi Qiao¹, Chao-Ping Chen¹

¹School of Mathematics and Informatics, Henan Polytechnic University, Jiaozuo, China.

Journal of Inequalities and Applications

|August 24, 2018

Summary

This summary is machine-generated.

This study introduces a precise Shafer-type inequality for the inverse tangent function. New approximations and bounds for arctan(x) were established using the Padé approximation method.

Keywords:

Approximation Inequality Inverse trigonometric function

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Area of Science:

Mathematical Analysis
Approximation Theory

Background:

The inverse tangent function (arctan x) is a fundamental function in mathematics.
Existing inequalities and approximations for arctan x are crucial for various applications.
Shafer-type inequalities provide specific bounds for certain functions.

Purpose of the Study:

To establish a sharp Shafer-type inequality for the inverse tangent function.
To develop new approximations for arctan x using the Padé approximation method.
To derive novel bounds for arctan x based on the established inequality and approximations.

Main Methods:

Application of the Padé approximation method to the inverse tangent function.
Development of a sharp Shafer-type inequality.
Derivation of new analytical bounds for arctan x.

Main Results:

A new sharp Shafer-type inequality for arctan x is presented.
Novel Padé approximations for arctan x are obtained.
New, tighter bounds for arctan x have been established.

Conclusions:

The presented inequality and approximations offer improved analytical tools for studying the inverse tangent function.
The new bounds for arctan x enhance the precision in related mathematical and numerical analyses.
This work contributes to the ongoing research in inequalities and function approximations.