Cross-posted on [maths.stackexchange][1]

I have the following integral:
\begin{equation}
f(\mu) = \int_0^\infty e^{-\mu t}\varphi_X(t)dt
\end{equation}
where $\varphi_X(t)$ is the characteristic function of some undetermined probability distribution and $\mu$ is some complex variable with strictly positive real part $\mu_r>0$.

It is easy enough to prove that this function is bounded above, $|f(\mu)| \le \frac{1}{\mu_r}$.

However, numerics suggest that for 'nice' distributions (symmetric, and non-increasing on $[0,\infty)$), the integral is bounded below by $|f(\mu_r)|\le|f(\mu)|$.

**So my questions are:**

>1) Are there established results on finding lower bounds of the Laplace transform of a characteristic function. (My trawling of google scholar and the like haven't produced anything. It seems like such an elementary problem that surely someone has considered it before.)

>2) If not what general techniques are used to look for lower bounds on integrals such as this?


Background:
The integral comes from investigating how non-identical frequency distribution of linear oscillators a particular collective behavior problem. This has occurred before in such problems and has been treated asymptotically for strictly real $\mu$ (ref. 1) and commented on for complex $\mu$ (appendix ref. 2). Due to the parameter regions we are interested in (around $\Re{\mu} =1$) the approximations made previously no longer hold.


Many Thanks, Pete.

References:

Ref. 1 - RE. Mirollo, SH Strogatz. *Amplitude Death in Limit Cycle Oscillators*. http://www.springerlink.com/index/ln051rp502550471.pdf

Ref. 2 - PC. Matthews, RE. Mirollo, SH Strogatz. *Dynamics of a large system of coupled nonlinear oscillators*. http://www.sciencedirect.com/science/article/pii/016727899190129W


  [1]: http://math.stackexchange.com/questions/310813/lower-bounds-of-laplace-transform-of-characteristic-functions

*edit: (typo) corrected to non-increasing*