Unification and refinements of Jordan, Adamovi\'c-Mitrinovi\'cand and Cusa's inequalities

TL;DR

This paper introduces new sharp bounds for the sine over x function, unifying and refining classical inequalities, and applies these results to derive improved inequalities for inverse sine, bivariate means, and sine integral estimates.

Contribution

It provides novel, sharper bounds for (x), unifying several classical inequalities and applies these to improve related inequalities and estimates.

Findings

New sharp bounds for (x) that unify and refine existing inequalities.

Derived new Shafer-Fink type inequalities for arc sine.

Provided more accurate estimates for sine integrals.

Abstract

In this paper, we find some new sharp bounds for $\left(\sin x\right) /x$, which unify and refine Jordan, Adamovi\'{c}-Mitrinovi\'{c}and and Cusa's inequalities. As applications of main results, some new Shafer-Fink type inequalities for arc sine function and ones for certain bivariate means are established, and a simpler but more accurate estimate for sine integral is derived.