Cusa-Huygens不等式的新改进
New Refinements of Cusa-Huygens inequality
AI总结：
本文通过构造带函数$Υ(x)$的简单逼近式，给出了$\sin x/x$在端点重合的精确新界，改进并推广了Cusa-Huygens不等式，同时分析界的层级并提供替代证明。
摘要：
本文利用简单函数对Cusa-Huygens不等式进行改进和推广。特别地，我们确定了形式为$(2+\cos(x))/3-(2/3-2/π)Υ(x)$的$\sin(x)/x$的精确界，其中对于$x\in(0,π/2)$有$Υ(x)>0$，且$Υ(0)=0$、$Υ(π/2)=1$，使得$\sin x/x$与所提出的界在$x=0$和$x=π/2$处重合。文中讨论了所得界的层级关系并进行了图形研究，还给出了主要结果的替代证明。
英文摘要：
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.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
