Consider $a(t)\in\mathbf{L}^{2}(\mathbb{R})$ and $a(t)>0$, is a low pass smooth function with $\hat{a}(f)=0, f>f_{max}$. Can we have a upper bound on the following, $\Big\frac{a'(t)}{a(t)}\Big$? Using Bernstein's theorem we can upper bound $a'(t)$ alone based on $f_{max}$ but how can we upper bound the ratio mentioned here. Any suggestions for it.

$a(t)=\cos(t+iϵ)\cos(t−iϵ)$ is a low pass signal and $a^\prime/a$ can be as large as you wish at the point $t=\pi/2,$ if $\epsilon$ is sufficiently small. 

