A generalization of the Shafer-Fink inequality

TL;DR

This paper introduces a new technique using Weierstrass products and bisection formulas to generalize the Shafer-Fink inequality for the arctangent function, also exploring algebraic approximations via Chebyshev expansions.

Contribution

It presents a novel method for generalizing the Shafer-Fink inequality and offers new algebraic approximations for the arctangent function.

Findings

Generalized Shafer-Fink inequality derived

New algebraic approximations for arctangent proposed

Connections to Chebyshev expansions established

Abstract

In this article we show a tecnique based on the Weierstrass product for the sine and cosine function and the bisection formula for the cotangent function that leads to a generalization of the classical Shafer-Fink inequality $ \frac{3 x}{1+2\sqrt{1+x^2}} < \arctan x < \frac{\pi x}{1+2\sqrt{1+x^2}}$. Other algebraic approximations are also shown, including one that follows from the Chebyshev expansion for the arctangent function in the interval $[-1,1]$.