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 $(2+\cos(x))/3 -(2/3-2/π)Υ(x)$, where $Υ(x) >0$ for $x\in (0, π/2)$, $Υ(0)=0$ and $Υ(π/2)=1$, such that $\sin x/x$ and the proposed bounds coincide at $x=0$ and $x=π/2$. The hierarchy of the obtained bounds is discussed, along with graphical study. Also, alternative proofs of the main result are given.

Figures (2)

• Figure 1: Plots of "lower bounds of Theorems 1 and 2 $-\sin x/x$"
• Figure 2: Plots of "upper bounds of Theorems 1 and 2 $-\sin x/x$"

Theorems & Definitions (11)

• Theorem 1
• Theorem 2
• Lemma 1
• Lemma 2
• Lemma 3
• Lemma 4
• Lemma 5
• proof
• Lemma 6
• proof