Suppose we have a non-negative random variable $X$ with density $p(x)$,and its characteristic function, evaluated at a complex number $z$, being $\phi(z)=E[e^{z X}]=\int_{0}^{\infty}e^{zx}p(x)dx$.

It seems that if $z$ has strictly negative real part, then $|\phi(z)|<1$. My question is, given any such $z$ with a strictly negative real parts, can we estimate how small the value $|\phi(z)|<1$ is as compared with one? Also how does the magnitude $|\phi(z)|<1$ affected by the properties of the density $p(x)$?

I am no expert in Fourier or Complex analysis, and would be very grateful if someone can point to the relevant part of the theory. Many many thanks.