Extremal properties of outer polynomial factors

If $p(s)$ is a positive polynomial of degree $2d$, then its outer factor $q(s)$has the property that the magnitude of each of its coefficients is larger than the magnitude of the corresponding coefficient of any other factor. In fact, this extremal property holds over vector-valued factorizations $r(s)^{*}r(s)=p(s)$. Corollaries include a result for symmetric functions and complex conjugate pairs.