I know that $\frac{\sin(ax)}{ax}$ can be bounded as follows
$$-\alpha \leq \frac{\sin(ax)}{ax} \leq 1$$
where $\alpha \approx \frac{2}{3\pi}$.
I am facing trouble trying to extend this to a product of sinc functions
$$\prod_{i=1}^d\frac{\sin(ax_i)}{ax_i}$$
How to derive a bound tighter than
$$-\alpha \leq \prod_{i=1}^d\frac{\sin(ax_i)}{ax_i} \leq 1$$