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

Zhu, Ling

Math. Inequal. Appl. 9, No. 1, 103-106 (2006).

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

Keywords:

lower and upper bounds; Jordan inequality; Yang Le inequality