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On the Cusa–Huygens inequality
Revista de la Real Academia de Ciencias Exactas, Físicas y Naturales. Serie A. Matemáticas ( IF 2.9 ) Pub Date : 2021-01-01 , DOI: 10.1007/s13398-020-00978-1 Yogesh J. Bagul , Christophe Chesneau , Marko Kostić
Revista de la Real Academia de Ciencias Exactas, Físicas y Naturales. Serie A. Matemáticas ( IF 2.9 ) Pub Date : 2021-01-01 , DOI: 10.1007/s13398-020-00978-1 Yogesh J. Bagul , Christophe Chesneau , Marko Kostić
Sharp bounds of various kinds for the famous unnormalized sinc function defined by $$(\sin x)/x$$ ( sin x ) / x are useful in mathematics, physics and engineering. In this paper, we reconsider the Cusa–Huygens inequality by solving the following problem: given real numbers $$a,b, c\in {{\mathbb {R}}}$$ a , b , c ∈ R and $$T\in (0,\pi /2],$$ T ∈ ( 0 , π / 2 ] , we find the necessary and sufficient conditions such that the inequalities $$\begin{aligned} \frac{\sin x}{x}>a+b\cos ^{c}x,\quad x\in (0,T) \end{aligned}$$ sin x x > a + b cos c x , x ∈ ( 0 , T ) and $$\begin{aligned} \frac{\sin x}{x}
中文翻译：
关于库萨-惠更斯不等式
由 $$(\sin x)/x$$ ( sin x ) / x 定义的著名非归一化 sinc 函数的各种锐界在数学、物理和工程中很有用。在本文中，我们通过解决以下问题重新考虑 Cusa-Huygens 不等式：给定实数 $$a,b, c\in {{\mathbb {R}}}$$ a , b , c ∈ R and $$ T\in (0,\pi /2],$$ T ∈ ( 0 , π / 2 ] ，我们找到充分必要条件使得不等式 $$\begin{aligned} \frac{\sin x}{ x}>a+b\cos ^{c}x,\quad x\in (0,T) \end{aligned}$$ sin xx > a + b cos cx , x ∈ ( 0 , T ) and $$ \begin{对齐} \frac{\sin x}{x}
更新日期：2021-01-01
中文翻译：
关于库萨-惠更斯不等式
由 $$(\sin x)/x$$ ( sin x ) / x 定义的著名非归一化 sinc 函数的各种锐界在数学、物理和工程中很有用。在本文中，我们通过解决以下问题重新考虑 Cusa-Huygens 不等式：给定实数 $$a,b, c\in {{\mathbb {R}}}$$ a , b , c ∈ R and $$ T\in (0,\pi /2],$$ T ∈ ( 0 , π / 2 ] ，我们找到充分必要条件使得不等式 $$\begin{aligned} \frac{\sin x}{ x}>a+b\cos ^{c}x,\quad x\in (0,T) \end{aligned}$$ sin xx > a + b cos cx , x ∈ ( 0 , T ) and $$ \begin{对齐} \frac{\sin x}{x}
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