Are polynomials with only real zeros log concave functions? In this comment, 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 the result holds for polynomials in more than one dimension.
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.)
 A: I'd like to outline possible two ways.
Method 1: Note, this only works when $p$ has negative roots. Then the coefficients of $p$ are positive. We know by Newton's Inequalities that these coefficients form an ultra log-concave sequence. Hence, by Shephard's Theorem (1960), we can realize $p$ as the relative Steiner polynomial $\text{Vol}(K + xL) = \sum_{k=0}^{n} \binom{n}{k} W_{k}(K;L) x^{k}$ of two convex bodies $K,L$. From there, the Brunn-Minkowski Inequality gives you log-concavity of $p$ as a univariate function of $x$.
Method 2: Use the fact that the roots of $p'$ interlacing the roots of $p$ (this is a consequence of the Mean-Value Theorem applied to the intervals $(\lambda_{i},\lambda_{i+1})$ where $\lambda_{1},\dots,\lambda_{n}$ are the roots of $p$). Then use Lemma 10.21 from a lecture note by Shayan O. Gharan on $p$ and $p'$. Note the Wronskian $W[p,p'] = p \cdot p'' - (p')^{2}$ is just the numerator of the second derivative of $\log p(x)$ (its denominator is $p^{2}$). Then use the fact that log-concavity of a function is equivalent to nonpositivity of the second derivative of $\log p(x)$. (Sorry I'm a bit lazy to type out the proof of Lemma 10.21 here).
Best wishes.
EDIT: I embarrassingly realized there is an extremely elementary method, which is just to write $p(x) = \prod_{i=1}^{n} (x - \lambda_{i})$ and then observe that $\log p(x) = \sum_{i=1}^{n} \log (x - \lambda_{i})$. The summation of concave functions is concave and it isn't difficult to verify that each $\log (x - \lambda_{i})$ is concave on their appropriate domain of definition.
