Consider a polynomial $\sum\limits\_{k=0}^n a\_kx^k$ with $a\_k\geq 0$ and $x\geq 0$.
In this [comment](https://mathoverflow.net/questions/229028/log-concave-polynomial-is-a-log-concave-function), [Richard Stanley](https://mathoverflow.net/users/2807/richard-stanley) mentions that polynomials with only real roots are log concave functions. Can somebody provide a reference for this result? I can't find it anywhere. I am in particular interested to know if a similar result holds for multivariate polynomials.


Also, does anybody know other results about the log-concavity of polynomials as functions?


(To be clear I'm **not** talking about log-concave polynomials in the sense that their coefficients form a log-concave sequence.)


Edit: I've added restrictions such that the polynomial is never negative.



Are polynomials with only real zeros log concave functions?
This fact is trivial. Your assumptions imply that $$f(x)=ax^m\prod(1+x/x\_k),$$
where $a>0$ and $x\_k>0$. All zeros $-x\_k$ are negative because you assume that
$a\_k\geq 0$, so there are no positive zeros. Now every factor in this product is log-concave, therefore the product is log-concave.


