New Refinements of Cusa-Huygens inequality
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arXiv
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| Autori principali: | , , , , |
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| Natura: | Preprint |
| Pubblicazione: |
2020
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| _version_ | 1866911830852501504 |
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| author | Chesneau, Christophe Kostic, Marko Malesevic, Branko Banjac, Bojan Bagul, Yogesh J. |
| author_facet | Chesneau, Christophe Kostic, Marko Malesevic, Branko Banjac, Bojan Bagul, Yogesh J. |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2009_01688 |
| institution | arXiv |
| publishDate | 2020 |
| record_format | arxiv |
| spellingShingle | New Refinements of Cusa-Huygens inequality Chesneau, Christophe Kostic, Marko Malesevic, Branko Banjac, Bojan Bagul, Yogesh J. Classical Analysis and ODEs 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. |
| title | New Refinements of Cusa-Huygens inequality |
| topic | Classical Analysis and ODEs |
| url | https://arxiv.org/abs/2009.01688 |