  The strong spectral order induces a natural partial ordering on the manifold
$H_{n}$ of monic hyperbolic polynomials of degree $n$. We prove that twisted
root maps associated with linear operators acting on $H_{n}$ are G\aa rding
convex on every polynomial pencil and we characterize the class of polynomial
pencils of logarithmic derivative type by means of the strong spectral order.
Let $A'$ be the monoid of linear operators that preserve hyperbolicity as well
as root sums. We show that any polynomial in $H_{n}$ is the global minimum of
its $A'$-orbit and we conjecture a similar result for complex polynomials.
