Let $g$ be a symmetric unimodal probability distribution and $H$ be the right half plane.
We call $$f(z) = \int_{-\infty}^\infty \frac{1}{z-i t}g(t)dt$$ the dispersion function of $g$.

Now, one can relate $f$ to a variety of things; the Hilbert transform $g$, the Laplace transform of the associated characteristic function, etc. It has also appeared in coupled oscillator problems [here][1], [here][2] and in my current work on coupled oscillators.

Let $L$ be a vertical line in the complex plane (without loss of generality, take the real part to be $1$) and consider the mapping $L\rightarrow f(L) \rightarrow 1/f(L)$ (see figure).

![The mappings $f(L)$ and $\frac{1}{f(L)}][3]

As you can see in the right hand figure (in this case, the Gaussian, but the picture is similar for the uniform and triangle) $1/f(L)$ seems to be bounded in a strip.

The heavy black line is the inequality $$ 1 \le \Re\frac{1}{f(1+i y)}$$ which is easy to prove. The dashed line, not so much.

So the question:

Is it true for all dispersion functions of symmetric distributions that: $$\Re\left[\frac{1}{f(1+i y)}\right]\le \frac{1}{f(1)}$$ which is equivalent to the following $$ \left|f(1+iy) - \frac{f(1)}{2}\right|\ge \frac{f(1)}{2}$$ and $$\frac{d}{dy}\frac{1}{f(1+iy)} \le 0\quad \forall y>0$$.

We do however know that for symmetric, unimodal $g$:

 

• $f$ is confromal on $H$

• $f(\bar{z}) =\overline{f(z)}$ thus $f(x+i0)$ is real,

• $f(z)$ constrained the the positive real axis is a decreasing function,

• $$\Im f(z)\le 0 \quad\text{and}\quad \Im \frac{df}{dz}\ge0\qquad \forall y\ge0$$

• $$|\frac{df}{dz}| < \Re \frac{f(z)}{x}$$

I have tried a variety of approaches, but i seem to need some stronger relationship between the real and imaginary parts of $\frac{df}{dy}$.

From a distributional standpoint, picking $g$ to be a Cauchy distribution with the same modal height actually produces the dashed line, and picking $g$ to be the Dirac delta function produces the heavy black line.Maybe in some sense we can think of the space of unimodal distributions and identify the Cauchy and Dirac distributions as extremum.

Any references/thoughts/suggestions would be muchly appreaciated.

Cheers, Pete [1]: http://www.springerlink.com/index/ln051rp502550471.pdf [2]: http://www.sciencedirect.com/science/article/pii/016727899190129W [3]: https://i.sstatic.net/Th7Y8.png

Let $g$ be a symmetric unimodal probability distribution and $H$ be the right half plane.
We call $$f(z) = \int_{-\infty}^\infty \frac{1}{z-i t}g(t)dt$$ the dispersion function of $g$.

Now, one can relate $f$ to a variety of things; the Hilbert transform $g$, the Laplace transform of the associated characteristic function, etc. It has also appeared in coupled oscillator problems here, here and in my current work on coupled oscillators.

Let $L$ be a vertical line in the complex plane (without loss of generality, take the real part to be $1$) and consider the mapping $L\rightarrow f(L) \rightarrow 1/f(L)$ (see figure).

As you can see in the right hand figure (in this case, the Gaussian, but the picture is similar for the uniform and triangle) $1/f(L)$ seems to be bounded in a strip.

The heavy black line is the inequality $$ 1 \le \Re\frac{1}{f(1+i y)}$$ which is easy to prove. The dashed line, not so much.

So the question:

Is it true for all dispersion functions of symmetric distributions that: $$\Re\left[\frac{1}{f(1+i y)}\right]\le \frac{1}{f(1)}$$ which is equivalent to the following $$ \left|f(1+iy) - \frac{f(1)}{2}\right|\ge \frac{f(1)}{2}$$ and $$\frac{d}{dy}\frac{1}{f(1+iy)} \le 0\quad \forall y>0.$$

We do however know that for symmetric, unimodal $g$:

 

• $f$ is confromal on $H$

• $f(\bar{z}) =\overline{f(z)}$ thus $f(x+i0)$ is real,

• $f(z)$ constrained the the positive real axis is a decreasing function,

• $$\Im f(z)\le 0 \quad\text{and}\quad \Im \frac{df}{dz}\ge0\qquad \forall y\ge0$$

• $$|\frac{df}{dz}| < \Re \frac{f(z)}{x}$$

I have tried a variety of approaches, but I seem to need some stronger relationship between the real and imaginary parts of $\frac{df}{dy}$.

From a distributional standpoint, picking $g$ to be a Cauchy distribution with the same modal height actually produces the dashed line, and picking $g$ to be the Dirac delta function produces the heavy black line. Maybe in some sense we can think of the space of unimodal distributions and identify the Cauchy and Dirac distributions as extremum.

Any references/thoughts/suggestions would be muchly appreciated.

Cheers, 
Pete

Cross-posted on maths.stackexchange

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

edit: (typo) corrected to non-increasing

Let $g$ be a symmetric unimodal probability distribution and $H$ be the right half plane.
We call $$f(z) = \int_{-\infty}^\infty \frac{1}{z-i t}g(t)dt$$ the dispersion function of $g$.

Now, one can relate $f$ to a variety of things; the Hilbert transform $g$, the Laplace transform of the associated characteristic function, etc. It has also appeared in coupled oscillator problems [here][1], [here][2] and in my current work on coupled oscillators.

Let $L$ be a vertical line in the complex plane (without loss of generality, take the real part to be $1$) and consider the mapping $L\rightarrow f(L) \rightarrow 1/f(L)$ (see figure).

![The mappings $f(L)$ and $\frac{1}{f(L)}][3]

As you can see in the right hand figure (in this case, the Gaussian, but the picture is similar for the uniform and triangle) $1/f(L)$ seems to be bounded in a strip.

The heavy black line is the inequality $$ 1 \le \Re\frac{1}{f(1+i y)}$$ which is easy to prove. The dashed line, not so much.

So the question:

Is it true for all dispersion functions of symmetric distributions that: $$\Re\left[\frac{1}{f(1+i y)}\right]\le \frac{1}{f(1)}$$ which is equivalent to the following $$ \left|f(1+iy) - \frac{f(1)}{2}\right|\ge \frac{f(1)}{2}$$ and $$\frac{d}{dy}\frac{1}{f(1+iy)} \le 0\quad \forall y>0$$.

We do however know that for symmetric, unimodal $g$:

• $f$ is confromal on $H$

• $f(\bar{z}) =\overline{f(z)}$ thus $f(x+i0)$ is real,

• $f(z)$ constrained the the positive real axis is a decreasing function,

• $$\Im f(z)\le 0 \quad\text{and}\quad \Im \frac{df}{dz}\ge0\qquad \forall y\ge0$$

• $$|\frac{df}{dz}| < \Re \frac{f(z)}{x}$$

I have tried a variety of approaches, but i seem to need some stronger relationship between the real and imaginary parts of $\frac{df}{dy}$.

From a distributional standpoint, picking $g$ to be a Cauchy distribution with the same modal height actually produces the dashed line, and picking $g$ to be the Dirac delta function produces the heavy black line.Maybe in some sense we can think of the space of unimodal distributions and identify the Cauchy and Dirac distributions as extremum.

Any references/thoughts/suggestions would be muchly appreaciated.

Cheers, Pete [1]: http://www.springerlink.com/index/ln051rp502550471.pdf [2]: http://www.sciencedirect.com/science/article/pii/016727899190129W [3]: https://i.sstatic.net/Th7Y8.png

Cross-posted on maths.stackexchange

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-decreasing 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

Cross-posted on maths.stackexchange

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

edit: (typo) corrected to non-increasing