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.)