Sharpening Jordan’s inequality and the Yang Le inequality

Abstract

In this work, the following inequality:

$$\frac{sinx}{x}\le \frac{2}{\pi }+\frac{\pi -2}{{\pi }^3}({\pi }^2-4x^2)\text{,}x\in (0,\pi /2]$$

is established. An application of this inequality gives an improvement of the Yang Le inequality [C.J. Zhao, Generalization and strengthening of the Yang Le inequality, Math. Practice Theory 30 (4) (2000) 493–497 (in Chinese)]:

$$(n-1)\sum _{k=1}^n{cos}^2\lambda A_k-2cos\lambda \pi \sum _{1\le i<j\le n}cos\lambda A_{i}cos\lambda A_j\le 4\left(\genfrac{}{}{0ex}{}{n}{2}\right){({\lambda }^3+\frac{\lambda (1-{\lambda }^2)}{2}\pi )}^2\text{,}$$

where $A_i>0(i=1,2,\dots ,n),{\sum }_{i=1}^n{A}_i\le \pi ,0\le \lambda \le 1$, and $n\ge 2$ is a natural number.