We say that a real univariate polynomial $p=a_0+a_1t+\ldots+a_nt^n$ with all $a_i>0$ is unimodular or log-concave if the sequence $a_0,\ldots,a_n$ is. The following implications are well known:
$p$ has only real zeros $\Rightarrow$ $p$ is log-concave $\Rightarrow$ $p$ is unimodular.
Now I am interested in the following situation. Let $f,g,h$ be real univariate polynomials with positive coefficients such that $f=g\cdot h$ and assume that $f$ and $g$ both satisfy one of the three conditions above. Can we conclude that then also $h$ does?
This is certainly true if $f$ is real rooted, so my hope is that it might hold for the other two properties as well.