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Zbl 1122.26014
Zhu, Ling
Sharpening Jordan's inequality and Yang Le inequality. II. (English)
[J] Appl. Math. Lett. 19, No. 9, 990-994 (2006). ISSN 0893-9659

Summary: Two refined forms of Jordan's inequality: $$\frac{2} {\pi}+\frac{1}{\pi^3} (\pi^2-4x^2)+\frac{12-\pi^2}{16\pi^5}(\pi^2-4x^2)^2\le\frac{\sin x}{x}\le\frac {2}{\pi}+\frac{1}{\pi^3}(\pi^2-4x^2)+\frac{\pi-3}{\pi^5}(\pi^2-4x^2)^2\tag a$$ and $$\frac{2}{\pi} +\frac{1}{\pi^3}(\pi^2-4x^2)+\frac{4(\pi-3)}{\pi^3}\left(x-\frac{\pi} {2}\right)^2\le\frac{\sin x}{x}\le\frac{2}{\pi}+\frac{1}{\pi^3}(\pi^2-4x^2)+\frac{12-\pi^2}{\pi^3}\left(x-\frac{\pi}{2}\right)^2\tag b$$ are established, where $x\in (0,\pi/2]$. The applications of the two results above give some new improvement of the Yang Le inequality. [For part I see ibid. 19, No. 3, 240--243 (2006; Zbl 1097.26012).]

MSC 2000:: *26D05 Inequalities for trigonometric functions and polynomials
42A05 Trigonometric polynomials

Keywords: lower and upper bounds

Citations: Zbl 1097.26012

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