# Lower bounds of laplace transform of characteristic functions

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. It turns out that the solutions of these problems can be described in terms of the solutions \begin{equation} f(\mu) = \mathbb{E}\left(\frac{1}{\mu - \mathrm{i} X}\right) \end{equation} where is $X$ set of frequencies. If we consider the real part only, we can relate this to the convolution of a Cauchy distribution (with median $\mu_i$ and spread $\mu_r$) with the distribution of frequencies. Treating it as a convolution allows us to use Fourier transforms to reformulate the integral as above, which seemed to me a easier (or at least seemingly more natural) problem.

This has occurred before in such problems and has been treated asymptotically for strictly real $\mu$  and commented on for complex $\mu$ (appendix ). Due to the parameter regions we are interested in (around $\Re{\mu} =1$) the approximations made previously no longer hold.

Many Thanks, Pete.

References:

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

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

• Watson's lemma may give you some insight. Feb 26, 2013 at 1:40
• The link to Mirollo & Strogatz's paper at springerlink.com is broken, but it can be found at doi:10.1007/BF01013676. May 21, 2022 at 19:44

One could start from the formula $$f(\mu)=\mathbb E\left(\frac1{\mu-\mathrm iX}\right).$$ If $\mu=\mu_r+\mathrm i\nu$, this is also $$f(\mu)=\mathbb E\left(\frac{\mu_r+\mathrm i(X-\nu)}{\mu_r^2+(\nu-X)^2}\right).$$