Sharpening Redheffer-type inequalities for circular functions (Q1021814)

The author proves the following inequalities: (1) Let \(0<x\leq \pi\). Then \[ \Big(\frac{\pi^2-x^2}{\sqrt{\pi^4+3x^4}}\Big)^{\alpha} \leq \frac{\sin x}{x}\leq \Big(\frac{\pi^2-x^2}{\sqrt{\pi^4+3x^4}}\Big)^{\beta} \] holds if and only if \(\alpha \geq \pi^2/6\) and \(\beta \leq 1\). (2) Let \(0<x\leq \pi/2\). Then \[ \Big(\frac{\pi^2-4x^2}{\sqrt{\pi^4+48x^4}}\Big)^{\pi^2/6} \leq \cos x \leq \Big(\frac{\pi^2-4x^2}{\sqrt{\pi^4+48x^4}}\Big)^{3/4} \] holds. (3) Let \(0<x<\pi/2\). Then \[ \Big(\frac{\sqrt{\pi^4+48x^4}}{\pi^2-4x^2}\Big)^{1/2} \leq \frac{\tan x}{x} \leq \Big(\frac{\sqrt{\pi^4+48x^4}}{\pi^2-4x^2}\Big)^{\pi^2/6} \] holds.