The uncertainty principle in Fourier analysis is given by the inequality $$ \sigma_t \sigma_\omega \geq \frac{1}{2} $$
It is a well known result that this lower bound is saturated exactly only in the case of a Gaussian spectrum.
However, I read somewhere (I can't really find the reference) that it can be proved that it will be nearly saturated, i.e. there exists a constant $c_1$ of order $1$ such that $$ \sigma_t \sigma_\omega = c_1 $$ when the spectrum is unimodal and without long tails.
Can anyone provide some evidence of this? Is there any known proof of this fact, and any estimate of this $c_1$?