2026-07-072026-07-07http://salesiana.dossiersoluciones.com/handle/123456789/165798The 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 Gårding 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.final version, to appear in Rocky Mountain J. Math.; 14 pages, no figures, LaTeX2eClassical Analysis and ODEsComplex Variables39B62 (Primary) 26C10, 30C15, 60E15 (Secondary)text