Suppose we have a density function $f(t)$ of a random variable and $f \in C^1(R)$. If character function of $f$ is $\phi_f(x) \asymp O(x^{-\beta})$ and $f$ satisfies some restrictive conditions like $|f'| \leq \epsilon |f|$, Is there any method to minimize the tail $\beta$, i.e, to make the character function decay as slow as possible?

Thank you and appreciate your reading.

Suppose we have a density function $f(t)$ of a random variable and $f \in C^1(R)$. If characteristic function of $f$ is $\phi_f(x) \asymp O(x^{-\beta})$ and $f$ satisfies some restrictive conditions like $|f'| \leq \epsilon |f|$, Is there any method to minimize the tail $\beta$, i.e, to make the character function decay as slow as possible?

Thank you and appreciate your reading.