New Refinements of Cusa-Huygens inequality

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 $(2+\cos (x))/3-(2/3-2/\pi ){\rm Y}(x)$, where ${\rm Y}(x)>0$ for $x\in (0,\pi /2)$, ${\rm Y}(0)=0$ and ${\rm Y}(\pi /2)=1$, 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.