A new refined form of Jordan's inequality and its applications (Q2488689)

The main result is the following improvement of the classical inequality of Jordan: \[ \frac{\sin x}{x}>\frac{2}{\pi}+\frac{1}{\pi^{3}}\left( \pi^{2}-4x^{2}\right) +\frac{4(\pi-3)}{\pi^{3}}\left( x-\frac{\pi}{2}\right) ^{2}, \] for all \(x\in(0,\pi/2).\) Several applications are included.