Sharpening Jordan’s inequality and Yang Le inequality. II. (English) Zbl 1122.26014

Summary: Two refined forms of Jordan’s inequality: \[ \frac{2} {\pi}+\frac{1}{\pi^3} (\pi^2-4x^2)+\frac{12-\pi^2}{16\pi^5}(\pi^2-4x^2)^2\leq\frac{\sin x}{x}\leq\frac {2}{\pi}+\frac{1}{\pi^3}(\pi^2-4x^2)+\frac{\pi-3}{\pi^5}(\pi^2-4x^2)^2\tag{a} \] and \[ \frac{2}{\pi} +\frac{1}{\pi^3}(\pi^2-4x^2)+\frac{4(\pi-3)}{\pi^3}\left(x-\frac{\pi} {2}\right)^2\leq\frac{\sin x}{x}\leq\frac{2}{\pi}+\frac{1}{\pi^3}(\pi^2-4x^2)+\frac{12-\pi^2}{\pi^3}\left(x-\frac{\pi}{2}\right)^2\tag{b} \] 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).]


26D05 Inequalities for trigonometric functions and polynomials
42A05 Trigonometric polynomials, inequalities, extremal problems


Zbl 1097.26012
Full Text: DOI


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