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Consider a polynomial $\sum\limits_{k=0}^n a_kx^k$ with $a_k\geq 0$ and $x\geq 0$. 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 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.

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    $\begingroup$ Such polynomials aren't even positive functions, so what is meant with log-convexity? $\endgroup$
    – Wojowu
    Commented May 4, 2017 at 16:26
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    $\begingroup$ What does the question mean? If a polynomial has a real zero, it is generally negative on parts of the real line... $\endgroup$
    – Igor Rivin
    Commented May 4, 2017 at 16:26
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    $\begingroup$ $\log( (x-a)(x-b))=\log(x-a)+\log(x-b)$ and sum of concave functions is concave, but you have to be careful about the domain. $\endgroup$
    – Mikhail Katz
    Commented May 4, 2017 at 16:28

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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.

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  • $\begingroup$ Thanks. Does a result like this exist for multivariate polynomials? $\endgroup$
    – user_lambda
    Commented May 4, 2017 at 17:03
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    $\begingroup$ @MTD: I don't know. At least this is not straightforward. But there is a literature on zeros of multivariate polynomials with positive coefficients. $\endgroup$
    – Alexandre Eremenko
    Commented May 8, 2017 at 14:03

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