## Sharpening of Jordan’s inequalities and its applications.(English)Zbl 1089.26007

In this paper, we establish the following inequalities $\frac{\sin r}{r}+\frac{\sin r-r\cos r}{2r^3}\,(r^2-x^2)\leq \frac{\sin x}{x}\leq\frac{\sin r}{r}+\frac{r-\sin r}{r^3}\,(r^2-x^2)$ for $$x\in (0,r]$$, $$r\leq \pi/2$$. An application of the inequalities above leads to the following refinement of Yang Le inequalities:
$4C^2_n\left[\frac{\sin r}r\,\frac\lambda2\,\pi +\frac{\sin r-r\cos r}{2r^3}\left(r^2\frac \lambda2\,\pi-\frac{\lambda^3}8 \,\pi^3\right)\right]^2\cos^2\frac \lambda 2\,\pi$
$\leq(n-1)\sum^n_{k=1}\cos^2\lambda A_k-2\cos\lambda\pi\sum_{1\leq i<j\leq n}\cos\lambda A_i\cos\lambda A_j$
$\leq 4C^2_n\left[\frac{\sin r}r\,\frac\lambda2\,\pi +\frac{r-\sin r}{r^3}\left(r^2\frac \lambda2\,\pi-\frac{\lambda^3}8 \,\pi^3\right)\right]^2,$
where $$A_i>0$$, $$(i=1,2,\dots,n)$$, $$\sum^n_{i=1}A_i\leq\pi$$, $$0\leq \lambda\leq 1$$ and $$n\geq 2$$ is a natural number.

### MSC:

 26D05 Inequalities for trigonometric functions and polynomials
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