Jordan and Kober Inequalities, PWW

Jordan's Inequality

$\displaystyle x\in\left(0,\frac{\pi}{2}\right)\,\Longrightarrow\,\sin x\gt\frac{2x}{\pi}.$

Indeed, following Feng Yuefeng,

Jordan's inequality, PWW

$\displaystyle \begin{align} OB=OM+MP\ge OA&\Longrightarrow\,\overset{\frown}{PBQ}\ge\overset{\frown}{PAQ}\ge PQ\\ &\Longrightarrow\,\pi\sin x\ge 2x\\ &\Longrightarrow\,\sin x\ge\frac{2x}{\pi}. \end{align}$

Kober's Inequality

$\displaystyle x\in\left(0,\frac{\pi}{2}\right)\,\Longrightarrow\,\cos x\gt 1-\frac{2x}{\pi}.$

Indeed:

Substitute $x:=\displaystyle \frac{\pi}{2}-x\,$ into Jordan's inequality.

References

  1. R. B. Nelsen, Proofs Without Words II, MAA, 2000, p 79

 

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