New Refinements of Cusa-Huygens inequality

TL;DR

This paper refines the Cusa-Huygens inequality by establishing sharp bounds for sin(x)/x using simple functions, providing a hierarchy of bounds, graphical analysis, and alternative proofs.

Contribution

It introduces new sharp bounds for sin(x)/x that extend the classical inequality with a clear hierarchy and multiple proof techniques.

Findings

Derived sharp bounds for sin(x)/x involving os(x) and unction ppa(x)

Established the hierarchy of the bounds and their graphical representations

Provided alternative proofs for the main inequalities

Abstract

In the paper, we refine and extend Cusa-Huygens inequality by simple functions. In particular, we determine sharp bounds for $\sin(x) /x$ of the form $(2+\cos(x))/3 -(2/3-2/\pi)\Upsilon(x)$, where $\Upsilon(x) >0$ for $x\in (0, \pi/2)$, $\Upsilon(0)=0$ and $\Upsilon(\pi/2)=1$, such that $\sin x/x$ and the proposed bounds coincide at $x=0$ and $x=\pi/2$. The hierarchy of the obtained bounds is discussed, along with graphical study. Also, alternative proofs of the main result are given.