# Monotonic $C_\infty$ bounds on the sinc function

The sinc function ($$\sin(x)/x$$ with a value of $$1$$ at $$x=0$$) is bound between $$-1/x$$ and $$1/x$$, but these are not the tightest bounds at any point, in fact, it is quite clear that there are even other monotonic functions that would be more tight. The tightest monotonic bounds are the function formed by drawing a horizontal line towards zero from each local extremum until it hits the previous one - but this will be a piecewise function with a discontinuous derivative and without a nice closed form.

Any there tighter bounds than $$\pm 1/x$$ that are monotonic, continuous and infinitely differentiable, preferably with a simple closed form?

For $$x \in (0,\pi)$$, tight bounds are $$\text{left}=\pi ^{-\frac{\pi ^2}{3}} \left(\pi^2-x^2\right)^{\frac{\pi ^2}{6}}\leq \text{sinc}(x) \leq \pi ^{\frac{\pi ^2}{3}}\left(\pi ^2+x^2\right)^{-\frac{\pi ^2}{6}}=\text{right}$$
Expanded around $$x=0$$ $$\text{sinc}(x)-\text{left}=\frac{\left(15-\pi ^2\right) }{180\pi ^2} x^4 +O\left(x^6\right)$$ $$\text{right}-\text{sinc}(x)=\frac{\left(15+\pi ^2\right) }{180 \pi ^2}x^4+O\left(x^6\right)$$