I am not familiar with the definition of total positivity. I am not sure about the link between *log-concavity* and *total positivity*.


* In a paper [On Variation-Diminishing Integral Operators of the Convolution Type](https://www.jstor.org/stable/87970) of Schoenberg, he defines *Pólya frequency functions* as a *totally positive* one.
* In [Estimating a Polya Frequency Function](https://dept.stat.lsa.umich.edu/%7Emichaelw/PPRS/2007ims.pdf), Pal, Woodroofe, and Meyer define *Pólya frequency functions* as *log-concave* functions.


Then, my question is the following, what are the links between *total positivity* and *log-concavity* ? It seems that *total positivity* implies *log-concavity*, do we have the converse statement ?



Total positivity, log-concavity and Pólya frequency
$\newcommand{\R}{\mathbb R}$For a positive integer $r$, a measurable function $f\colon\R\to\R$ is called a Pólya frequency function of order $r$ (abbreviated as PF$\_r$) if the matrix $(f(x\_i-y\_j))\_{i,j,=1}^r$ is totally positive for all real $x\_i$ and $y\_j$ such that $x\_1\le\dots\le x\_r$ and $y\_1\le\dots\le y\_r$, that is, if $\det(f(x\_i-y\_j))\_{i,j,=1}^m\ge0$ for all integers $m=1,\dots,r$ and all real $x\_i$ and $y\_j$ such that $x\_1\le\dots\le x\_m$ and $y\_1\le\dots\le y\_m$. So, clearly PF$\_1\supseteq\,$PF$\_2\supseteq\cdots$.


Consider now the case $r=2$. Note that the set of all matrices of the form $(x\_i-y\_j)\_{i,j,=1}^2$ for real $x\_i$ and $y\_j$ such that $x\_1\le x\_2$ and $y\_1\le y\_2$ coincides with the set of all matrices of the form
$\begin{pmatrix}a&b\\c&d \end{pmatrix}$ for real $a,b,c,d$ such that $a+d=b+c$ and $\{a,d\}\subseteq[b,c]$.


So, if $f$ is log concave, then $f$ is PF$\_2$. Vice versa, if $f$ is PF$\_2$, then, taking $a=d=(b+c)/2$, we see that $f$ is midpoint log concave, and hence, being measurable, $f$ is log concave (see e.g. [Theorem II](https://www.ams.org/journals/tran/1919-020-01/S0002-9947-1919-1501114-0/)).


Thus, a function $f\colon\R\to\R$ is PF$\_2$ iff it is log concave.


However, it is easy to find examples of log-concave functions that are not in PF$\_3$, and hence not in PF$\_r$ for any natural $r\ge3$; for instance, take $f=1\_{(0,8)}$ and
\begin{equation}
 (x\_1,x\_2,x\_3,y\_1,y\_2,y\_3)= (0, 1, 3, -5, -4, 0);
\end{equation}
then $\det(f(x\_i-y\_j))\_{i,j,=1}^3=-1\not\ge0$.


So, the definitions of Pólya frequency functions in the two papers referred to in your post are equivalent to each other for $r=2$ -- but not for $r\ge3$.


